Trigonometric Identities and Equations: math practice, Grade 12 – download the PDF

1 Simplify with Pythagorean identities ★★★

Simplify each expression: a. \(1-\cos^2x\)   b. \(\sec^2x-1\)   c. \(\csc^2x-\cot^2x\).

2 Cosine given, sine and tangent wanted ★★★

Angle \(\theta\) is in Quadrant IV and \(\cos\theta=\dfrac{5}{13}\). Find \(\sin\theta\) and \(\tan\theta\).

3 Exact value of sin 15 degrees ★★★

Use \(15^\circ=45^\circ-30^\circ\) to find the exact value of \(\sin15^\circ\).

4 Cosine of a double angle ★★★

If \(\sin\theta=\dfrac13\), find \(\cos2\theta\).

5 True or false: sine of a sum ★★★

Is it true that \(\sin(A+B)=\sin A+\sin B\) for all angles? Justify with a counterexample.

6 Exact value of sin 105 degrees ★★★

Write \(105^\circ=60^\circ+45^\circ\) and find the exact value of \(\sin105^\circ\).

7 Tangent of a double angle ★★★

If \(\tan\theta=\dfrac12\), find \(\tan2\theta\).

8 A basic equation on one turn ★★★

Solve \(\sin x=\dfrac{\sqrt3}{2}\) on \([0,2\pi)\). Use the graph to see why there are two answers.

9 Cosine of 15 degrees from a difference ★★★

Use \(\dfrac{\pi}{12}=\dfrac{\pi}{3}-\dfrac{\pi}{4}\) to find the exact value of \(\cos\dfrac{\pi}{12}\).

10 Tangent of 15 degrees by half-angle ★★★

Use the half-angle formula \(\tan\dfrac\theta2=\dfrac{1-\cos\theta}{\sin\theta}\) with \(\theta=30^\circ\) to find \(\tan15^\circ\).

11 Half-angle with a given cosine ★★★

Let \(\cos\theta=-\dfrac{7}{25}\) with \(90^\circ<\theta<180^\circ\). Find \(\sin\dfrac\theta2\) and \(\cos\dfrac\theta2\).

12 Product-to-sum practice ★★★

a. Write \(2\sin5x\cos3x\) as a sum.   b. Find the exact value of \(\cos75^\circ\cos15^\circ\).

13 Sum-to-product practice ★★★

a. Write \(\sin7x+\sin3x\) as a product.   b. Find the exact value of \(\cos75^\circ+\cos15^\circ\).

14 Quadratic in sine ★★★

Solve \(2\sin^2x-\sin x-1=0\) on \([0,2\pi)\).

15 Verify tan + cot ★★★

Verify the identity \(\tan x+\cot x=\sec x\csc x\).

16 Verify a half-angle quotient ★★★

Verify that \(\dfrac{\sin2x}{1+\cos2x}=\tan x\).

17 Equation with cos 2x and cos x ★★★

Solve \(\cos2x=\cos x\) on \([0,2\pi)\).

18 Do not divide by cosine ★★★

Solve \(\sin2x=\sqrt3\cos x\) on \([0,2\pi)\).

19 Launch angle of a soccer ball ★★★

A ball kicked at \(v=20\text{ m/s}\) (about 65.6 ft/s) at angle \(\theta\) travels the horizontal distance \(R=\dfrac{v^2\sin2\theta}{g}\) with \(g=9.8\text{ m/s}^2\). At what angles between \(0^\circ\) and \(90^\circ\) does the ball land 30 m (about 98 ft) away? Round to the nearest tenth of a degree.

20 Two electrical signals ★★★

Two signals are \(\sin\left(x+\dfrac\pi3\right)\) and \(\sin\left(x-\dfrac\pi3\right)\). Prove that their sum is simply \(\sin x\).

21 Triple-angle formula ★★★

Prove that \(\cos3x=4\cos^3x-3\cos x\). Hint: write \(3x=2x+x\).

22 Extraneous solutions ★★★

Solve \(\sin x+\cos x=1\) on \([0,2\pi)\) by squaring both sides, then check every candidate.