Precalculus Trigonometric Identities Worksheet Solutions (PDF)

Precalculus math worksheet solutions: Trigonometric Identities – Maths PDF

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

Use this key to check each identity or exact value. Verification solutions work on one side only, and final answers are in bold.

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

Trig identities answers: exercises 1 to 10

Exercise 1 – Simplify a product

\(\frac{\sin x}{\cos x}\cdot\cos x=\sin x\).

\(\sin x\)

Exercise 2 – Pythagorean simplification

\(1-\cos^{2}x=\sin^{2}x\), so \(\frac{\sin^{2}x}{\sin x}\).

\(\sin x\)

Exercise 3 – Sum of squares

\(1+\tan^{2}x=\frac{1}{\cos^{2}x}\).

\(\frac{1}{\cos^{2}x}\)

Exercise 4 – Double angle values

\(\cos\theta=-\frac45\). \(\sin2\theta=2\cdot\frac35\cdot\left(-\frac45\right)=-\frac{24}{25}\). \(\cos2\theta=\frac{16}{25}-\frac{9}{25}=\frac{7}{25}\).

\(\sin2\theta=-\frac{24}{25}\); \(\cos2\theta=\frac{7}{25}\)

Exercise 5 – Exact sine of 15 degrees

Use the difference formula this time.

\(\sin45^{\circ}\cos30^{\circ}-\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt6}{4}-\frac{\sqrt2}{4}\).

\(\frac{\sqrt6-\sqrt2}{4}\)

Exercise 6 – Exact cosine of 15 degrees

\(\cos(45^{\circ}-30^{\circ})=\cos45^{\circ}\cos30^{\circ}+\sin45^{\circ}\sin30^{\circ}=\frac{\sqrt6}{4}+\frac{\sqrt2}{4}\).

\(\frac{\sqrt6+\sqrt2}{4}\)

Exercise 7 – Tangent sum

\(\frac{1+\frac{\sqrt3}{3}}{1-\frac{\sqrt3}{3}}=\frac{3+\sqrt3}{3-\sqrt3}\). Multiply by \(\frac{3+\sqrt3}{3+\sqrt3}\): \(\frac{12+6\sqrt3}{6}=2+\sqrt3\).

\(2+\sqrt3\)

Exercise 8 – Shift identity

The curves coincide.

\(\sin x\cos\frac{\pi}{2}+\cos x\sin\frac{\pi}{2}=\sin x\cdot0+\cos x\cdot1=\cos x\).

The identity holds

Exercise 9 – Verify a square

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

Verified

Exercise 10 – Verify with fractions

Common denominator \(\sin x(1+\cos x)\): numerator \(\sin^{2}x+(1+\cos x)^{2}=\sin^{2}x+1+2\cos x+\cos^{2}x=2+2\cos x=2(1+\cos x)\). Cancel \(1+\cos x\).

Verified

Exercises 11 and up

Exercise 11 – Verify tangent plus cotangent

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

Verified

Exercise 12 – Half-angle power

\(\cos^{2}\frac{\pi}{8}=\frac{1+\cos\frac{\pi}{4}}{2}=\frac{1+\frac{\sqrt2}{2}}{2}=\frac{2+\sqrt2}{4}\).

\(\frac{2+\sqrt2}{4}\)

Exercise 13 – Cosine double angle from sine

\(\cos2x=1-2\sin^{2}x=1-\frac29\).

\(\frac79\)

Exercise 14 – Solve with a double angle

Four intersections.

\(2\sin x\cos x-\sin x=0\), so \(\sin x(2\cos x-1)=0\). \(\sin x=0\): \(x=0,\pi\). \(\cos x=\frac12\): \(x=\frac{\pi}{3},\frac{5\pi}{3}\).

\(x=0,\ \frac{\pi}{3},\ \pi,\ \frac{5\pi}{3}\)

Exercise 15 – Solve cos 2x = cos x

\(2\cos^{2}x-1=\cos x\), so \(2\cos^{2}x-\cos x-1=0\) and \((2\cos x+1)(\cos x-1)=0\). \(\cos x=1\): \(0\). \(\cos x=-\frac12\): \(\frac{2\pi}{3},\frac{4\pi}{3}\).

\(x=0,\ \frac{2\pi}{3},\ \frac{4\pi}{3}\)

Exercise 16 – Use Pythagorean to solve

Replace \(\cos^{2}x=1-\sin^{2}x\): \(2-2\sin^{2}x+\sin x-1=0\), so \(2\sin^{2}x-\sin x-1=0\) and \((2\sin x+1)(\sin x-1)=0\).

\(x=\frac{\pi}{2},\ \frac{7\pi}{6},\ \frac{11\pi}{6}\)

Exercise 17 – Projectile range

Maximum range at 45 degrees.

\(R(30^{\circ})=\frac{7744\sin60^{\circ}}{32}\approx 210\). \(R(45^{\circ})=\frac{7744\sin90^{\circ}}{32}=242\).

\(210\) ft and \(242\) ft

Exercise 18 – Electrical amplitude

\(\sin t\cos t=\frac12\sin2t\), so \(V=85\sin2t\).

\(V=85\sin2t\); amplitude \(85\)

Exercise 19 – Sum of two angles

Triangles 5-12-13 and 3-4-5.

\(\cos\alpha=\frac{12}{13}\), \(\sin\beta=\frac35\). \(\sin(\alpha+\beta)=\frac{5}{13}\cdot\frac45+\frac{12}{13}\cdot\frac35=\frac{56}{65}\). \(\cos(\alpha+\beta)=\frac{12}{13}\cdot\frac45-\frac{5}{13}\cdot\frac35=\frac{33}{65}\).

\(\frac{56}{65}\) and \(\frac{33}{65}\)

Exercise 20 – Prove a triple angle

\(\cos2x\cos x-\sin2x\sin x=(2\cos^{2}x-1)\cos x-2\sin^{2}x\cos x\). Replace \(\sin^{2}x=1-\cos^{2}x\): \(2\cos^{3}x-\cos x-2\cos x+2\cos^{3}x=4\cos^{3}x-3\cos x\).

Verified

Key facts behind this trig identities answers

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 answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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