## Trig. Equations Examples using CAST Diagrams

These lessons, with videos, examples and step-by-step solutions help A Level Maths students learn to solve trigonometric problems.

Related Pages Trigonometric Functions Trigonometric Graphs Trigonometric Identities Lessons On Trigonometry More Lessons for A Level Maths

What is the CAST diagram? The Cast diagram helps us to remember the signs of the trigonometric functions in each of the quadrants. The CAST diagram is also called the Quadrant Rule or the ASTC diagram.

In the first quadrant, the values are all positive. In the second quadrant, only the values for sin are positive. In the third quadrant, only the values for tan are positive. In the fourth quadrant, only the values for cos are positive.

Note that the mnemonic CAST goes anticlockwise starting from the 4th quadrant.

The following diagram shows how the CAST diagram or Quadrant rule can be used. Scroll down the page for more examples and solutions. Using the Quadrant Rule to solve trig. equations

Quadrant Rule for solving trig equations with different ranges

Core (2) Graphs of Trigonometric Functions (4) - CAST Diagram or Unit Circle  ## Find Study Materials for

Business studies, combined science, computer science, english literature, environmental science, human geography, macroeconomics, microeconomics.

• Social Studies
• Browse all subjects
• Exam Revision
• Student Life
• Study Guide

## Create Study Materials  A trigonometric equation is an equation that consists of a trigonometric function.  These functions include sine, cosine, tangent, cotangent, secant and cosecant.  Depending on the type of trigonometric equation, they can be solved using a CAST diagram, the quadratic formula, one of the various trigonometric identities available, or the unit circle. Explore our app and discover over 50 million learning materials for free.

• Solving Trigonometric Equations
• Explanations
• StudySmarter AI
• Textbook Solutions
• Absolute Maxima and Minima
• Absolute and Conditional Convergence
• Accumulation Function
• Accumulation Problems
• Algebraic Functions
• Alternating Series
• Antiderivatives
• Application of Derivatives
• Approximating Areas
• Arc Length of a Curve
• Area Between Two Curves
• Arithmetic Series
• Average Value of a Function
• Calculus of Parametric Curves
• Candidate Test
• Combining Differentiation Rules
• Combining Functions
• Continuity Over an Interval
• Convergence Tests
• Cost and Revenue
• Density and Center of Mass
• Derivative Functions
• Derivative of Exponential Function
• Derivative of Inverse Function
• Derivative of Logarithmic Functions
• Derivative of Trigonometric Functions
• Derivatives
• Derivatives and Continuity
• Derivatives and the Shape of a Graph
• Derivatives of Inverse Trigonometric Functions
• Derivatives of Polar Functions
• Derivatives of Sec, Csc and Cot
• Derivatives of Sin, Cos and Tan
• Determining Volumes by Slicing
• Direction Fields
• Disk Method
• Divergence Test
• Eliminating the Parameter
• Euler's Method
• Evaluating a Definite Integral
• Evaluation Theorem
• Exponential Functions
• Finding Limits
• Finding Limits of Specific Functions
• First Derivative Test
• Function Transformations
• General Solution of Differential Equation
• Geometric Series
• Growth Rate of Functions
• Higher-Order Derivatives
• Hydrostatic Pressure
• Hyperbolic Functions
• Implicit Differentiation Tangent Line
• Implicit Relations
• Improper Integrals
• Indefinite Integral
• Indeterminate Forms
• Initial Value Problem Differential Equations
• Integral Test
• Integrals of Exponential Functions
• Integrals of Motion
• Integrating Even and Odd Functions
• Integration Formula
• Integration Tables
• Integration Using Long Division
• Integration of Logarithmic Functions
• Integration using Inverse Trigonometric Functions
• Intermediate Value Theorem
• Inverse Trigonometric Functions
• Jump Discontinuity
• Lagrange Error Bound
• Limit of Vector Valued Function
• Limit of a Sequence
• Limits at Infinity
• Limits at Infinity and Asymptotes
• Limits of a Function
• Linear Approximations and Differentials
• Linear Differential Equation
• Linear Functions
• Logarithmic Differentiation
• Logarithmic Functions
• Logistic Differential Equation
• Maclaurin Series
• Manipulating Functions
• Maxima and Minima
• Maxima and Minima Problems
• Mean Value Theorem for Integrals
• Models for Population Growth
• Motion Along a Line
• Motion in Space
• Natural Logarithmic Function
• Net Change Theorem
• Newton's Method
• Nonhomogeneous Differential Equation
• One-Sided Limits
• Optimization Problems
• Particle Model Motion
• Particular Solutions to Differential Equations
• Polar Coordinates
• Polar Coordinates Functions
• Polar Curves
• Population Change
• Power Series
• Related Rates
• Removable Discontinuity
• Riemann Sum
• Rolle's Theorem
• Second Derivative Test
• Separable Equations
• Separation of Variables
• Simpson's Rule
• Solid of Revolution
• Solutions to Differential Equations
• Surface Area of Revolution
• Symmetry of Functions
• Tangent Lines
• Taylor Polynomials
• Taylor Series
• Techniques of Integration
• The Fundamental Theorem of Calculus
• The Mean Value Theorem
• The Power Rule
• The Squeeze Theorem
• The Trapezoidal Rule
• Theorems of Continuity
• Trigonometric Substitution
• Vector Valued Function
• Vectors in Calculus
• Vectors in Space
• Washer Method
• Algorithmic Recurrence Relations
• Algorithms on Graphs
• Allocation Problems
• Bin-packing Algorithms
• Constructing Cayley Tables
• Critical Path Analysis
• Decision Analysis
• Dijkstra's Algorithm
• Dynamic Programming
• Eulerian graphs
• Flow Charts
• Floyd's Algorithm
• Formulating Linear Programming Problems
• Gantt Charts
• Graph Theory
• Group Generators
• Group Theory Terminology
• Kruskal's Algorithm
• Linear Programming
• Prim's Algorithm
• Simplex Algorithm
• The Minimum Spanning Tree Method
• The North-West Corner Method
• The Simplex Method
• The Travelling Salesman Problem
• Using a Dummy
• 2 Dimensional Figures
• 3 Dimensional Vectors
• 3-Dimensional Figures
• Angles in Circles
• Arc Measures
• Area and Volume
• Area of Circles
• Area of Circular Sector
• Area of Parallelograms
• Area of Plane Figures
• Area of Rectangles
• Area of Regular Polygons
• Area of Rhombus
• Area of Trapezoid
• Area of a Kite
• Composition
• Congruence Transformations
• Congruent Triangles
• Convexity in Polygons
• Coordinate Systems
• Distance and Midpoints
• Equation of Circles
• Equilateral Triangles
• Fundamentals of Geometry
• Geometric Inequalities
• Geometric Mean
• Geometric Probability
• Glide Reflections
• HL ASA and AAS
• Identity Map
• Inscribed Angles
• Isosceles Triangles
• Law of Cosines
• Law of Sines
• Linear Measure and Precision
• Parallel Lines Theorem
• Parallelograms
• Perpendicular Bisector
• Plane Geometry
• Projections
• Properties of Chords
• Proportionality Theorems
• Pythagoras Theorem
• Reflection in Geometry
• Regular Polygon
• Right Triangles
• SSS and SAS
• Segment Length
• Similarity Transformations
• Surface Area of Cone
• Surface Area of Cylinder
• Surface Area of Prism
• Surface Area of Sphere
• Surface Area of a Solid
• Surface of Pyramids
• Translations
• Triangle Inequalities
• Using Similar Polygons
• Vector Product
• Volume of Cone
• Volume of Cylinder
• Volume of Pyramid
• Volume of Solid
• Volume of Sphere
• Volume of prisms
• What is Point Slope Form
• Acceleration and Time
• Acceleration and Velocity
• Angular Speed
• Assumptions
• Calculus Kinematics
• Coefficient of Friction
• Connected Particles
• Conservation of Mechanical Energy
• Constant Acceleration
• Constant Acceleration Equations
• Converting Units Mechanics
• Damped harmonic oscillator
• Direct Impact and Newton's Law of Restitution
• Elastic Energy
• Elastic Strings and Springs
• Force as a Vector
• Newton's First Law
• Newton's Law of Gravitation
• Newton's Second Law
• Newton's Third Law
• Problems involving Relative Velocity
• Projectiles
• Relative Motion
• Resolving Forces
• Rigid Bodies in Equilibrium
• Statics and Dynamics
• Tension in Strings
• The Trajectory of a Projectile
• Variable Acceleration
• Vertical Oscillation
• Work Done by a Constant Force
• Basic Probability
• Charts and Diagrams
• Continuous and Discrete Data
• Frequency, Frequency Tables and Levels of Measurement
• Independent Events Probability
• Line Graphs
• Mean Median and Mode
• Mutually Exclusive Probabilities
• Probability Rules
• Probability of Combined Events
• Quartiles and Interquartile Range
• Systematic Listing
• ASA Theorem
• Absolute Value Equations and Inequalities
• Addition and Subtraction of Rational Expressions
• Addition, Subtraction, Multiplication and Division
• Algebraic Fractions
• Algebraic Notation
• Algebraic Representation
• Analyzing Graphs of Polynomials
• Angle Measure
• Angles in Polygons
• Approximation and Estimation
• Area and Perimeter of Quadrilaterals
• Area of Triangles
• Argand Diagram
• Arithmetic Sequences
• Average Rate of Change
• Bijective Functions
• Binomial Expansion
• Binomial Theorem
• Cayley Hamilton Theorem
• Circle Theorems
• Circles Maths
• Combinatorics
• Common Factors
• Common Multiples
• Completing the Square
• Complex Numbers
• Composite Functions
• Composition of Functions
• Compound Interest
• Compound Units
• Congruence Equations
• Conic Sections
• Construction and Loci
• Converting Metrics
• Convexity and Concavity
• Coordinate Geometry
• Coupled First-order Differential Equations
• Cubic Function Graph
• Data transformations
• De Moivre's Theorem
• Deductive Reasoning
• Definite Integrals
• Deriving Equations
• Determinant of Inverse Matrix
• Determinants
• Diagonalising Matrix
• Differential Equations
• Differentiation
• Differentiation Rules
• Differentiation from First Principles
• Differentiation of Hyperbolic Functions
• Direct and Inverse proportions
• Disjoint and Overlapping Events
• Disproof by Counterexample
• Distance from a Point to a Line
• Divisibility Tests
• Double Angle and Half Angle Formulas
• Drawing Conclusions from Examples
• Eigenvalues and Eigenvectors
• Equation of Line in 3D
• Equation of a Perpendicular Bisector
• Equation of a circle
• Equations and Identities
• Equations and Inequalities
• Estimation in Real Life
• Euclidean Algorithm
• Evaluating and Graphing Polynomials
• Even Functions
• Exponential Form of Complex Numbers
• Exponential Rules
• Exponentials and Logarithms
• Expression Math
• Expressions and Formulas
• Faces Edges and Vertices
• Factoring Polynomials
• Factorising expressions
• Fermat's Little Theorem
• Finding Maxima and Minima Using Derivatives
• Finding Rational Zeros
• Finding the Area
• First-order Differential Equations
• Fractional Powers
• Fractional Ratio
• Fractions and Decimals
• Fractions and Factors
• Fractions in Expressions and Equations
• Fractions, Decimals and Percentages
• Function Basics
• Functional Analysis
• Fundamental Counting Principle
• Fundamental Theorem of Algebra
• Generating Terms of a Sequence
• Geometric Sequence
• Graphical Representation
• Graphing Rational Functions
• Graphing Trigonometric Functions
• Graphs and Differentiation
• Graphs of Common Functions
• Graphs of Exponents and Logarithms
• Graphs of Trigonometric Functions
• Greatest Common Divisor
• Group Mathematics
• Growth and Decay
• Growth of Functions
• Harmonic Motion
• Highest Common Factor
• Imaginary Unit and Polar Bijection
• Implicit differentiation
• Inductive Reasoning
• Inequalities Maths
• Infinite geometric series
• Injective functions
• Instantaneous Rate of Change
• Integrating Polynomials
• Integrating Trigonometric Functions
• Integrating e^x and 1/x
• Integration
• Integration Using Partial Fractions
• Integration by Parts
• Integration by Substitution
• Integration of Hyperbolic Functions
• Invariant Points
• Inverse Hyperbolic Functions
• Inverse Matrices
• Inverse and Joint Variation
• Inverse functions
• Iterative Methods
• L'Hopital's Rule
• Law of Cosines in Algebra
• Law of Sines in Algebra
• Laws of Logs
• Leibnitz's Theorem
• Limits of Accuracy
• Linear Expressions
• Linear Systems
• Linear Transformations of Matrices
• Location of Roots
• Logarithm Base
• Lower and Upper Bounds
• Lowest Common Denominator
• Lowest Common Multiple
• Math formula
• Matrix Calculations
• Matrix Determinant
• Matrix Multiplication
• Metric and Imperial Units
• Mixed Expressions
• Modelling with First-order Differential Equations
• Modular Arithmetic
• Modulus Functions
• Modulus and Phase
• Multiples of Pi
• Multiplication and Division of Fractions
• Multiplicative Relationship
• Multiplying and Dividing Rational Expressions
• Natural Logarithm
• Natural Numbers
• Number Line
• Number Systems
• Number Theory
• Numerical Methods
• Odd functions
• Open Sentences and Identities
• Operation with Complex Numbers
• Operations with Decimals
• Operations with Matrices
• Operations with Polynomials
• Order of Operations
• Parallel Lines
• Parametric Differentiation
• Parametric Equations
• Parametric Hyperbolas
• Parametric Integration
• Parametric Parabolas
• Partial Fractions
• Pascal's Triangle
• Percentage Increase and Decrease
• Perimeter of a Triangle
• Permutations and Combinations
• Perpendicular Lines
• Points Lines and Planes
• Polynomial Graphs
• Polynomials
• Powers and Exponents
• Powers and Roots
• Prime Factorization
• Prime Numbers
• Problem-solving Models and Strategies
• Product Rule
• Proof and Mathematical Induction
• Proof by Deduction
• Proof by Exhaustion
• Proof by Induction
• Properties of Exponents
• Proving an Identity
• Pythagorean Identities
• Quotient Rule
• Rates of Change
• Ratio Fractions
• Rational Exponents
• Rational Expressions
• Rational Functions
• Rational Numbers and Fractions
• Ratios as Fractions
• Real Numbers
• Reciprocal Graphs
• Recurrence Relation
• Recursion and Special Sequences
• Reducible Differential Equations
• Remainder and Factor Theorems
• Representation of Complex Numbers
• Rewriting Formulas and Equations
• Roots of Complex Numbers
• Roots of Polynomials
• Roots of Unity
• SAS Theorem
• SSS Theorem
• Scalar Products
• Scalar Triple Product
• Scale Drawings and Maps
• Scale Factors
• Scientific Notation
• Second Order Recurrence Relation
• Second-order Differential Equations
• Sector of a Circle
• Segment of a Circle
• Sequences and Series
• Series Maths
• Similar Triangles
• Similar and Congruent Shapes
• Simple Interest
• Simplifying Fractions
• Simultaneous Equations
• Sine and Cosine Rules
• Small Angle Approximation
• Solving Linear Equations
• Solving Linear Systems
• Solving Rational Equations
• Solving Simultaneous Equations Using Matrices
• Solving Systems of Inequalities
• Solving and Graphing Quadratic Equations
• Solving and Graphing Quadratic Inequalities
• Special Products
• Standard Form
• Standard Integrals
• Standard Unit
• Straight Line Graphs
• Substraction and addition of fractions
• Sum and Difference of Angles Formulas
• Sum of Natural Numbers
• Surjective functions
• Tables and Graphs
• Tangent of a Circle
• The Quadratic Formula and the Discriminant
• Transformations
• Transformations of Graphs
• Transformations of Roots
• Translations of Trigonometric Functions
• Triangle Rules
• Triangle trigonometry
• Trigonometric Functions
• Trigonometric Functions of General Angles
• Trigonometric Identities
• Trigonometric Ratios
• Trigonometry
• Turning Points
• Types of Functions
• Types of Numbers
• Types of Triangles
• Unit Circle
• Variables in Algebra
• Verifying Trigonometric Identities
• Volumes of Revolution
• Writing Equations
• Writing Linear Equations
• Bias in Experiments
• Binomial Distribution
• Binomial Hypothesis Test
• Bivariate Data
• Categorical Variables
• Central Limit Theorem
• Chi Square Test for Goodness of Fit
• Chi Square Test for Homogeneity
• Chi Square Test for Independence
• Chi-Square Distribution
• Combining Random Variables
• Comparing Data
• Comparing Two Means Hypothesis Testing
• Conditional Probability
• Conducting a Study
• Conducting a Survey
• Conducting an Experiment
• Confidence Interval for Population Mean
• Confidence Interval for Population Proportion
• Confidence Interval for Slope of Regression Line
• Confidence Interval for the Difference of Two Means
• Confidence Intervals
• Correlation Math
• Cumulative Distribution Function
• Cumulative Frequency
• Data Analysis
• Data Interpretation
• Degrees of Freedom
• Discrete Random Variable
• Distributions
• Empirical Rule
• Errors in Hypothesis Testing
• Estimator Bias
• Events (Probability)
• Frequency Polygons
• Generalization and Conclusions
• Geometric Distribution
• Hypothesis Test for Correlation
• Hypothesis Test for Regression Slope
• Hypothesis Test of Two Population Proportions
• Hypothesis Testing
• Inference for Distributions of Categorical Data
• Inferences in Statistics
• Large Data Set
• Least Squares Linear Regression
• Linear Interpolation
• Linear Regression
• Mean and Variance of Poisson Distributions
• Measures of Central Tendency
• Methods of Data Collection
• Normal Distribution
• Normal Distribution Hypothesis Test
• Normal Distribution Percentile
• Paired T-Test
• Point Estimation
• Probability
• Probability Calculations
• Probability Density Function
• Probability Distribution
• Probability Generating Function
• Product Moment Correlation Coefficient
• Quantitative Variables
• Random Variables
• Randomized Block Design
• Residual Sum of Squares
• Sample Mean
• Sample Proportion
• Sampling Distribution
• Scatter Graphs
• Single Variable Data
• Spearman's Rank Correlation Coefficient
• Standard Deviation
• Standard Error
• Standard Normal Distribution
• Statistical Graphs
• Statistical Measures
• Stem and Leaf Graph
• Sum of Independent Random Variables
• Survey Bias
• T-distribution
• The Power Function
• Transforming Random Variables
• Tree Diagram
• Two Categorical Variables
• Two Quantitative Variables
• Type I Error
• Type II Error
• Types of Data in Statistics
• Variance for Binomial Distribution
• Venn Diagrams

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Nie wieder prokastinieren mit unseren Lernerinnerungen.

A trigonometric equation is an equation that consists of a trigonometric function. These functions include sine, cosine, tangent, cotangent, secant and cosecant. Depending on the type of trigonometric equation, they can be solved using a CAST diagram, the quadratic formula, one of the various trigonometric identities available, or the unit circle.

## How do we use a CAST diagram when solving trigonometric equations?

A CAST diagram is used to solve trigonometric equations. It helps us remember the signs of the trigonometric functions in each quadrant and what happens to the angle that needs to be calculated, depending on the trigonometric function used. • All trig functions are positive in the first quadrant.
• Only sine is positive in the second quadrant.
• Only tangent is positive in the third quadrant.
• Only cosine is positive in the fourth quadrant.

When using the CAST diagram, you will first isolate the trig function, calculate your acute angle, and then use the diagram to solve for the solutions. You can use this method to solve linear trig equations, trig equations involving a single function, and use your calculator.

4 sin x ° + 3 = 0 0 ≤ x ≤ 360 °

Step 1: Rearrange the equation to have the trig function be on its own.

4 sin x ° + 3 = 0 sin x ° = - 3 4

Step 2 : Calculate the value of your acute angle using the inverse of your trig function. Note that the negative will always be ignored when calculating the acute angle.

sin x ° = - 3 4 x ° = sin - 1 ( - 3 4 ) x ° = - 48 . 59 °

Step 3: Based on the sign of the function, determine the quadrants of the solutions and use the information from this to solve the equation.

In our example, sine is negative. Therefore our solutions are in the 3rd (180 ° + x °) and 4th (360 ° -x °) quadrants.

3 r d q u a d r a t n t : x ° = 180 ° + 48 . 59 ° = 228 . 59 ° 4 t h q u a d r a n t : x ° = 360 ° - 48 . 59 ° = 311 . 41 °

## What is the unit circle in trigonometry?

• A unit circle is a circle that has a radius of 1 and is used to illustrate particular common angles.

## How do we solve quadratic trigonometric equations?

Quadratic trigonometric equations are second degree trigonometric equations. They can be solved by using the quadratic formula: x = - b ± b 2 - 4 a c 2 a

2 sin 2 a + 3 sin a - 1 = 0

Step 1: Replace your trig function with a variable of your choice.

In our example, we will say let sin (a) = x

2 x 2 + 3 x - 1 = 0

a = 2 b = 3 c = - 1 x = - ( 3 ) ± 3 2 - 4 ( 2 ) ( - 1 ) 2 ( 2 ) = - 3 ± 17 4

Step 3: Replace your variable back as the function and take the inverse of the function to solve for the + equation. ( ± , m e a n s t h e r e a r e 2 s o l u t i o n s )

sin - 1 ( - 3 + 17 4 ) = 18 . 11 ° x = sin ( 18 . 11 ) = 0 . 28

Step 4: Use the unit circle to determine the solution to the - equation as the domain of the inverse function is - 1 , 1 .

Due to sine being positive in the first and second quadrants, the second solution would be:

x = π - 0 . 28 = 2 . 86

## How do we use identities to solve trigonometric equations?

Identities are used to solve trigonometric functions by simplifying the equation and then solving mainly by using the unit circle.

Here are a few important trigonometric formulas you should know:

sin 2 x + cos 2 x = 1 cos x cos y + sin x sin y = cos ( x - y ) tan 2 x + 1 = s e c 2 x cos 2 x = cos 2 x - sin 2 x = 2 cos 2 x - 1 = 1 - 2 sin 2 x sin 2 x = 2 sin x cos x tan x = sin x cos x

cos x cos ( 2 x ) + sin x sin ( 2 x ) = 3 2

Step 1: Simplify your equation with a known identity.

In this example, it is the difference formula for cosine: cos a cos b + sin a sin b = cos ( a - b )

cos x cos ( 2 x ) + sin x sin ( 2 x ) = 3 2 c os ( x - 2 x ) = 3 2 cos ( - x ) = 3 2 cos ( x ) = 3 2 , i n t h i s p a r t w e u s e d t h e n e g a t i v e a n g l e t r i g i d e n t i t y : cos ( - x ) = cos ( x )

Step 2: Use the unit circle to determine the values of your angle (x).

In our example, we will focus on the 4th and 1st quadrants as cosine is positive in those quadrants.

Therefore, x = π 6 a n d x = 11 π 6

## How do we solve trigonometric equations with multiple angles?

Trigonometric equations with multiple angles are solved by first rewriting the equation as an inverse, determining which angles satisfy the equation and then dividing these angles by the number of angles. In solving these, you will most likely have more than two solutions as when you have a function in this form: cos (nx) = c, you will need to go around the circle n times.

Trigonometric equations with multiple angles look like this: sin 2 x , tan x 2 , cos 3 x , e t c The variables all have coefficients.

cos 2 x = 1 2 o n [ 0 , 2 π )

Step 1: Determine the quadrants of your initial solutions and the possible angles by using the unit circle.

cos _ 1 ( 1 2 ) = 60 ° ∴ p o s s i b l e a n g l e s a r e 2 x = π 3 a n d 2 x = 5 π 3

Step 2: Calculate the value of your initial solutions by dividing the possible angle by the number of angles.

2 x = π 3 x = π 6 2 x = 5 π 3 x = 5 π 6

Step 3: Determine your other solutions by revolving around the circle by the number of angles and only selecting the answers within your range.

F i r s t q u a d r a n t 2 x = π 3 First rotation : 2 x = π 3 + 2 π = 7 π 3 x = 7 π 6 This value is between 0 and 2 π and is therefore a solution Second Rotation : 2 x = π 3 + 4 π = 13 π 3 x = 13 π 6 This value is greater than 2 π and is therefore not a solution . Fourth q u a d r a n t : 2 x = 5 π 3 F i r s t R o t a t i o n : 2 x = 5 π 3 + 2 π = 11 π 3 x = 11 π 6 This value is between the range and is therefore a solution . Second Rotation : 2 x = 5 π 3 + 4 π 2 x = 17 π 3 x = 17 π 6 This value is not within your range and thefore cant be a solution .

## Solving Trigonometric Equations - Key takeaways

• When using the CAST diagram, you will first isolate the trig function, calculate your acute angle and then use the diagram to solve for the solutions.
• Quadratic trig equations can be solved with the quadratic formula: x = - b ± ± b 2 - 4 a c 2 a
• Identities are used to solve trigonometric functions by simplifying the equation and then solving using the unit circle.
• In solving trig functions with multiple angles, you will most likely have more than two solutions as when you have a function in this form: cos (nx) = c, you will need to go around the circle n times.

--> how do we solve trigonometeric equations.

Trigonometric equations can be solved using a CAST diagram, the quadratic formula, trigonometric identities and the unit circle.

## --> How do we solve trigonomteric equations in radians?

Step 1: Determine the quadrants of your initial solutions and the possible angle, by using the unit circle.

Step 2:  Calculate the value of your initial solutions by dividing the possible angle by the number of angles.

## --> How to solve trigonometric equations algebraically?

Step 1:   Rearrange the equation to have the trig function on its own.

Step 2:  Calculate the value of your acute angle, using the inverse of your trig function. Note that the negative will always be ignored when calculating the acute angle.

Step 3:  Based on the sign of the function, determine the quadrants of the solutions and use the information from this to solve the equation.

## --> How to solve trigonometric equations using identities ?

Step 1:  Simplify your equation with  a known identity.

Step 2: Use the unit of circle to determine the values of your angle.

## --> How to solve for trigonometric equations  for the given domain?

Step 1:  Determine the quadrants of your initial solutions and the possible angles, by using the unit circle.

Step 2: Calculate the value of your initial solutions by dividing the possible angle by the number of angles.

Step 3:  Determine your other solutions by revolving around the circle by the number of angles and only selecting the answers within your range.

## Final Solving Trigonometric Equations Quiz

Solving trigonometric equations quiz - teste dein wissen.

What is a trigonometric equation?

A trigonometric equation is one that includes a trigonometric function which could be: sine, cosine, tangent, cotangent, secant and cosecant.

Show question

What is a CAST diagram?

A CAST diagram is one that helps us remember the signs of the trigonometric functions in each of the quadrants and what happens to the angle that needs to be calculated, depending on the trigonometric function used.

In which quadrants is tangent positive?

Tangent is positive in the first and third quadrants.

Flashcards in Solving Trigonometric Equations 3 ## Learn with 3 Solving Trigonometric Equations flashcards in the free StudySmarter app

• Mechanics Maths

of the users don't pass the Solving Trigonometric Equations quiz! Will you pass the quiz?

How would you like to learn this content?

Free math cheat sheet!

Everything you need to know on . A perfect summary so you can easily remember everything.

## Join over 22 million students in learning with our StudySmarter App

The first learning app that truly has everything you need to ace your exams in one place

• Flashcards & Quizzes
• AI Study Assistant
• Study Planner
• Smart Note-Taking ## More explanations about Pure Maths

Discover the right content for your subjects, engineering.

## This is still free to read, it's not a paywall.

You need to register to keep reading, start learning with studysmarter, the only learning app you need..  Save explanations to your personalised space and access them anytime, anywhere!

By signing up, you agree to the Terms and Conditions and the Privacy Policy of StudySmarter.

## StudySmarter bietet alles, was du für deinen Lernerfolg brauchst - in einer App!

Privacy overview.

• Topic Specification

## Trig Equations Revision

Try it yourself.

• Example Questions
• Worksheets 1

## You May Also Like

Mme learning portal, trig equations.

Solving the basic trig equations is pretty easy, but what happens if we’ve got a function which has been stretched or translated ?

Make sure you are happy with the following topics before continuing.

## Trig Graphs

First, we need to find an initial solution.

So, for example, let’s say we want to find the values of x when \textcolor{red}{\tan x} = \textcolor{purple}{\dfrac{1}{\sqrt{3}}} .

We want to draw the graph, and mark on a horizontal line where the condition is met, i.e. where \textcolor{red}{\tan x} = \textcolor{purple}{\dfrac{1}{\sqrt{3}}} .

By inspection, we know that the initial solution is at 30° , and we can see it repeats at every 180° .

We’ll denote this 30° ± 180°n .

## A Level Maths Predicted Papers

The MME A level maths predicted papers are an excellent way to practise, using authentic exam style questions that are unique to our papers. Our examiners have studied A level maths past papers to develop predicted A level maths exam questions in an authentic exam format. The profit from every pack is reinvested into making free content on MME, which benefits millions of learners across the country. ## A Level Maths Revision Cards

The best A level maths revision cards for AQA, Edexcel, OCR, MEI and WJEC. Maths Made Easy is here to help you prepare effectively for your A Level maths exams. The profit from every pack is reinvested into making free content on MME, which benefits millions of learners across the country. ## CAST Diagrams

The CAST diagram is a handy tool to show us where values of the standard trig functions are positive . So, here’s the breakdown:

• For 0° < x < 90° ,  ALL of \textcolor{blue}{\sin x}, \textcolor{limegreen}{\cos x} and \textcolor{red}{\tan x} are positive
• For 90° < x < 180° , ONLY \textcolor{blue}{\sin x} is positive
• For 180° < x < 270° , ONLY \textcolor{red}{\tan x} is positive
• For 270° < x < 360° , ONLY \textcolor{limegreen}{\cos x} is positive

Think back to plotting the Unit Circle, in the Trig Basics section.

Let’s say we want to find the values of x such that \textcolor{red}{\tan x} = \textcolor{purple}{\dfrac{1}{\sqrt{3}}} , as before.

We know that there is a solution when x = 30° .

First, we plot the point on the diagram, then find the corresponding angles: From here, we can see that \textcolor{red}{\tan x} = \textcolor{purple}{\dfrac{1}{\sqrt{3}}} when x = 30° ± 360°n or 210° ± 360°n , or, more concisely, 30° ± 180°n .

We ignored the two solutions where \textcolor{red}{\tan x} is not positive, i.e. 90° < x < 180° and 270° < x < 360° .

## Dealing With Trig Transformations

Transformations pose a little bit of a problem… See, CAST diagrams become much harder to navigate now. You’re much better off sketching out the function and solving using the horizontal line technique.

So, let’s just begin with an example.

We have f(x) = \cos 3x . Find the values of x \in \lbrack 0°, 360°\rbrack * such that f(x) = \textcolor{purple}{\dfrac{1}{2}} .

* This is just set notation, meaning 0° \leq x \leq 360°

Well, we have a series of solutions, but they’re not immediately obvious.

What we can do instead is plot the regular \textcolor{limegreen}{\cos x} graph on an interval three times as large as the proposed interval, and divide our solutions there by 3 .

\textcolor{limegreen}{\cos x} = \textcolor{purple}{\dfrac{1}{2}} has solutions 60°, 300°, 420°, 660°, 780°, 1020° .

Therefore, \cos 3x = \textcolor{purple}{\dfrac{1}{2}} has solutions 20°, 100°, 140°, 220°, 260°, 340° .

## Trig Equations Example Questions

Question 1:  Sketch the \sin x graph and by inspection, find the solutions of \sin x = \dfrac{-\sqrt{3}}{2} , in the interval -360° \leq x \leq 360° .

x = \sin ^{-1} \left( \dfrac{\sqrt{3}}{2}\right) = -120°, -60°, 240°, 300°

Question 2:  Using a CAST diagram, find the values of x such that \cos x = 0.2588 .

\cos x = 0.2588 gives x = 75° . Question 3: For f(x) = 2\sin \dfrac{3x}{2} , find the values of x \in \lbrack -2\pi, 2\pi \rbrack where f(x) = 1 .

For f(x) = 1 , we want \sin \dfrac{3x}{2} = \dfrac{1}{2} .

If we look for values of -3\pi \leq x \leq 3\pi where \sin x = \dfrac{1}{2} , and multiply our values of x by a scale factor of \dfrac{2}{3} , we have our new set of solutions.

\sin x = \dfrac{1}{2} occurs at \dfrac{-11\pi}{6}, \dfrac{-7\pi}{6}, \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{13\pi}{6}, \dfrac{17\pi}{6}

Therefore the solution of f(x) = 1 is x = \dfrac{-11\pi}{9}, \dfrac{-7\pi}{9}, \dfrac{\pi}{9}, \dfrac{5\pi}{9}, \dfrac{13\pi}{9}, \dfrac{17\pi}{9} .

Exam tips cheat sheet, formula booklet, trig equations worksheet and example questions. ## Sin cos and tan

You may also like.... Online exams, practice questions and revision videos for every GCSE level 9-1 topic! No fees, no trial period, just totally free access to the UK’s best GCSE maths revision platform.

## Related Topics

By clicking continue and using our website you are consenting to our use of cookies in accordance with our Cookie Policy

## Report a Question

You must be logged in to vote for this question..

## Nav view search

• Testimonials
• Rates & FAQS
• Maths/Physics Notes
• Online Tests
• 10 Lesson Offers ## The CAST Diagram

• " onclick="window.open(this.href,'win2','status=no,toolbar=no,scrollbars=yes,titlebar=no,menubar=no,resizable=yes,width=640,height=480,directories=no,location=no'); return false;" rel="nofollow"> Print

Name (required)

E-mail (required, but will not display) ## Latest Blog Posts

04 december 2023.

• How Injuries Breed
• Leg Still Broken  #### IMAGES

1. Solving Trig Equations 2. C2 Trigonometry 1 CAST diagram 3. Solving trig equations using a CAST diagram 4. PPT 5. Trig. Equations Examples using CAST Diagrams (solutions, examples, worksheets, videos, activities) 6. Solving Trig equations using cast diagram #### VIDEO

1. 142: Solving Trig Equations

2. Solving trig equations lesson part 6 11 14 23

3. Unit 5

4. Solving Trig Equations

5. Solving trig equations part 4 11 10 23

6. 5.3B Solving Trig Equations Using Inverse of cosine

1. What Are the Four Steps for Solving an Equation?

The four steps for solving an equation include the combination of like terms, the isolation of terms containing variables, the isolation of the variable and the substitution of the answer into the original equation to check the answer.

Whether you love math or suffer through every single problem, there are plenty of resources to help you solve math equations. Skip the tutor and log on to load these awesome websites for a fantastic free equation solver or simply to find an...

3. What Is a Diagram of an Automotive Air Conditioning System Used For?

A diagram of an air conditioning system for a car can be useful for car owners, who may be able to use it in solving some of the basic problems that the system may incur. Diagrams for general AC systems are especially useful for students.

4. An Introduction to Solving Trigonometric Equations

This video explains how the CAST diagram can be used to solve trig equations. Solving Trig Equations 1:https://youtu.be/cILaBqbmPX0 Practice

5. Trig. Equations Examples using CAST Diagrams

The Cast diagram helps us to remember the signs of the trigonometric functions in each of the quadrants. The CAST diagram is also called the Quadrant Rule or

6. The CAST diagram

This is a quick demonstration of how to use the CAST diagram to find all the potential solutions to a trigonometric equation.

7. Trigonometry CAST Diagrams and Solving Equations

Trigonometry at AS/A Level includes solving trigonometric equations, and finding multiple solutions using a CAST diagram (or graph).

8. Solving Trig Equations (CAST Diagram)

Solving Trig Equations (CAST Diagram) | AS-Level Maths (Temporary Video). 1.5K views · 3 years ago ...more

9. Trigonometric equations

Solving equations using the CAST diagram. 1. Find the inverse trigonometric value of the positive value to ensure the angle is between and.

10. Solving Trigonometric Equations: Graphs & Quadratic

When using the CAST diagram, you will first isolate the trig function, calculate your acute angle, and then use the diagram to solve for the solutions. You can

11. Trig Equations

Think back to plotting the

12. Quadrant rule to solve trig equations

The quadrant rule or CAST diagram is very useful to know as it saves ... Using the quadrant rule to solve simple trigonometric equations.

13. The CAST Diagram

The CASt diagram provides a method to find multiple solutions to trigonometric equations. It indicates the quadrants in which the elementary

14. Solving Trig Equations 1

The cast diagram shows you the quadrants in which the named trig functions are positive. In the example below we want the quadrants where sine is negative. Sine