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How to use double-angle identities

Trigonometry: Master Angles, Functions & Applications

Imaginary and Complex Numbers

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Introduction to imaginary numbers

Complex numbers and complex planes

Adding and subtracting complex numbers

Complex conjugates

Complex Conjugates

Multiplying and dividing complex numbers

Distance and midpoint of complex numbers

Angle and absolute value of complex numbers

Polar form of complex numbers

Operations on complex numbers in polar form

Inverse Trigonometric Functions

Finding inverse trigonometric function from its graph

Evaluating inverse trigonometric functions

Finding inverse reciprocal trigonometric function from its graph

Finding exact value of inverse reciprocal trig functions

Inverse Reciprocal Trigonometric Function: Finding the Exact Value

Inverse reciprocal trigonometric function: finding the exact value

Bearings

Introduction to bearings

Bearings and direction word problems

Bearings and Direction Word Problems

Angle of elevation and depression

Trigonometric Ratios and Angle Measure

Unit circle

Law of sines

Ambiguous case: SSA triangles

Law of cosines

Applications of the sine law and cosine law

Angle in standard position

Coterminal angles

Coterminal Angles

Reference angle

Find the exact value of trigonometric ratios

ASTC rule in trigonometry (All Students Take Calculus)

ASTC rule in trigonometry

ASTC rule in trigonometry (All Students Take Calculus)

Converting between degrees and radians

Trigonometric ratios of angles in radians

Radian measure and arc length

Graphing Trigonometric Functions

Sine graph: y = sin x

Cosine graph: y = cos x

Tangent graph: y = tan x

Cotangent graph: y = cot x

Secant graph: y = sec x

Cosecant graph: y = csc x

Graphing transformations of trigonometric functions

Determining trigonometric functions given their graphs

Applications of Trigonometric Functions

Ferris wheel trig problems

Tides and water depth trig problems

Spring (simple harmonic motion) trig problems

Trigonometric Identities

Quotient identities and reciprocal identities

Pythagorean identities

Sum and difference identities

Sum and Difference Identities

Double-angle identities

Cofunction identities

Solving Trigonometric Equations

Solving first degree trigonometric equations

Solving second degree trigonometric equations

Solving trigonometric equations involving multiple angles

Determining non-permissible values for trig expressions

Solving trigonometric equations using pythagorean identities

Solving trigonometric equations using Pythagorean identities

Solving trigonometric equations using sum and difference identities

Solving trigonometric equations using double-angle identities

Right Triangle Trigonometry

Use sine ratio to calculate angles and sides (Sin = \(  \frac{o}{h}\) )

Use cosine ratio to calculate angles and sides (Cos =  \(  \frac{a}{h}\)  )

Combination of SohCahToa questions

Word problems relating ladder in trigonometry

Word problems relating guy wire in trigonometry

Solving expressions using 45-45-90 special right triangles

Solving expressions using 30-60-90 special right triangles 

Use tangent ratio to calculate angles and sides (Tan =  \(  \frac{o}{a}\)  )

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Ex. 1:

Ex. 2:

Ex. 3:

Finding sin 2A, sec 2A, and tan 2A given tangent and quadrant

Simplifying 14 sin(6x)cos(6x) using the sine double-angle identity

Simplifying cos(9/2 x) sin(9/2 x) using the sine double-angle identity

Simplifying cos²(5a) − sin²(5a) using the double-angle identity

Simplifying 3 minus 6 cosine squared using the double-angle identity

Simplifying 6tan(5x) over tan²(5x)−1 using the double-angle identity

Proving sin(6x)/(1+cos(6x)) = tan(3x) using double-angle identities

Proving a double-angle identity by starting from the right side

Proving a double-angle identity by simplifying sin4x and cos4x to tan x

Proving a double-angle identity by simplifying both sides to cotangent

In this lesson, we will learn how to make use of the double-angle identities, a.k.a. double-angle formulas to find the sine and cosine of a double angle. It's hard to simplify complex trigonometric functions without these formulas. 

Double-angle identities

What You'll Learn

Apply sine, cosine, and tangent double-angle formulas to simplify trigonometric expressions

Identify which cosine 2θ formula to use based on expression structure

Convert complex trigonometric expressions using double-angle identities

Verify factored trigonometric results by applying quotient and reciprocal identities

What You'll Practice

Simplifying expressions using sin(2θ) = 2sin(θ)cos(θ)

Selecting among three cos(2θ) formulas to cancel terms

Proving trigonometric identities with double-angle formulas

Finding exact values of double-angle expressions from given ratios

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

10 Video lessons

Practice exercises

Learning resources

Skills

Double-Angle Identities

Trigonometric Identities

Simplification

Factoring

Quotient Identities

Reference Angles

Proof Techniques