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

1 Pythagorean identities / 4 pts

Angle \(\theta\) is in Quadrant III and \(\tan\theta=\dfrac{8}{15}\).

1. Find \(\sec\theta\) and then \(\cos\theta\).
2. Find \(\sin\theta\).
3. Find \(\sin2\theta\).
4. Find \(\cos2\theta\).

2 Exact values / 4 pts

a. Find the exact value of \(\sin75^\circ\) using a sum formula.

b. Find the exact value of \(\cos22.5^\circ\) using a half-angle formula.

3 Verifying identities / 4 pts

Verify each identity.

1. \(\dfrac{\cos x}{1-\sin x}=\dfrac{1+\sin x}{\cos x}\)
2. \(\dfrac{\cos2x}{\cos x+\sin x}=\cos x-\sin x\)

4 A trigonometric equation / 4 pts

Solve \(2\sin^2x+3\cos x=0\) on \([0,2\pi)\).

5 Products and sums / 4 pts

a. Write \(\sin3x\cos x\) as a sum.

b. Solve \(\sin5x-\sin x=0\) on \([0,\pi]\) using a sum-to-product formula.