Precalculus Trigonometric Identities Worksheet (PDF)

Precalculus math worksheet: Trigonometric Identities – Maths PDF

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This trig identities worksheet for Precalculus covers trigonometric identities (Common Core F-TF.C.8, F-TF.C.9), with a free PDF to print.

These 20 problems practice trigonometric identities (Common Core F-TF.C.8 and F-TF.C.9). You will simplify expressions, use sum, difference and double-angle formulas for exact values, verify identities and solve equations.

Precalculus · Common Core State Standards: F-TF.C.8, F-TF.C.9

Trig identities worksheet: exercises 1 to 10

Exercise 1 – Simplify a product

Simplify \(\tan x\cos x\).

Exercise 2 – Pythagorean simplification

Simplify \(\frac{1-\cos^{2}x}{\sin x}\).

Exercise 3 – Sum of squares

Simplify \(\sin^{2}x+\cos^{2}x+\tan^{2}x\) in terms of cosine.

Exercise 4 – Double angle values

If \(\sin\theta=\frac35\) and \(\theta\) is in quadrant II, find \(\sin2\theta\) and \(\cos2\theta\).

Exercise 5 – Exact sine of 15 degrees

Same method as sin 75 degrees.

Find the exact value of \(\sin15^{\circ}\) by writing \(15^{\circ}=45^{\circ}-30^{\circ}\).

Exercise 6 – Exact cosine of 15 degrees

Find the exact value of \(\cos15^{\circ}\).

Exercise 7 – Tangent sum

Find \(\tan\left(\frac{\pi}{4}+\frac{\pi}{6}\right)\) exactly using \(\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\).

Exercise 8 – Shift identity

Two curves that look the same.

Use the sum formula to show \(\sin\left(x+\frac{\pi}{2}\right)=\cos x\).

Exercise 9 – Verify a square

Verify \((\sin x+\cos x)^{2}=1+\sin2x\).

Exercise 10 – Verify with fractions

Verify \(\frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}=\frac{2}{\sin x}\).

Exercises 11 and up

Exercise 11 – Verify tangent plus cotangent

Verify \(\tan x+\frac{\cos x}{\sin x}=\frac{1}{\sin x\cos x}\).

Exercise 12 – Half-angle power

Use \(\cos^{2}x=\frac{1+\cos2x}{2}\) to find \(\cos^{2}\frac{\pi}{8}\) exactly.

Exercise 13 – Cosine double angle from sine

If \(\sin x=\frac13\), find \(\cos2x\).

Exercise 14 – Solve with a double angle

Graphs of sin 2x and sin x.

Solve \(\sin2x=\sin x\) for \(0\le x< 2\pi\).

Exercise 15 – Solve cos 2x = cos x

Solve \(\cos2x=\cos x\) for \(0\le x< 2\pi\).

Exercise 16 – Use Pythagorean to solve

Solve \(2\cos^{2}x+\sin x-1=0\) for \(0\le x< 2\pi\).

Exercise 17 – Projectile range

Range of a projectile as the launch angle changes.

A ball launched at \(v=88\) ft/s with angle \(\theta\) travels \(R=\frac{v^{2}\sin2\theta}{32}\) feet. Find \(R\) for \(\theta=30^{\circ}\) and \(\theta=45^{\circ}\), to the nearest foot.

Exercise 18 – Electrical amplitude

The voltage is \(V=170\sin t\cos t\). Use a double-angle formula to write it as a single sine and give the amplitude.

Exercise 19 – Sum of two angles

Two acute angles.

Acute angles \(\alpha\) and \(\beta\) satisfy \(\sin\alpha=\frac{5}{13}\) and \(\cos\beta=\frac45\). Find \(\sin(\alpha+\beta)\) and \(\cos(\alpha+\beta)\).

Exercise 20 – Prove a triple angle

Show that \(\cos3x=4\cos^{3}x-3\cos x\) by writing \(\cos3x=\cos(2x+x)\).

Key facts behind this trig identities worksheet

Key facts

sin^2 x + cos^2 x = 1 and 1 + tan^2 x = 1/cos^2 x.
tan x = sin x / cos x; csc, sec, cot are reciprocals of sin, cos, tan.
sin(A + B) = sin A cos B + cos A sin B; cos(A + B) = cos A cos B – sin A sin B.
sin 2A = 2 sin A cos A; cos 2A = cos^2 A – sin^2 A = 1 – 2 sin^2 A = 2 cos^2 A – 1.
To verify an identity, work on one side only until it matches the other.

How to use this trig identities worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the trig identities worksheet as a PDF with the button below and keep it with your notes.

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