Math 1060
Test 2 Review
1. Simplify the following trigonometric function using trigonometric identities.
(a) (1 − cos2 x)(csc x)
(b) cot2 x − csc2 x
(c) sin x sec x
(d) sec4 x − tan4 x
tan2 θ
sec2 θ
1 + sin x
cos x
+
(f)
1 + sin x
cos x
(e)
2. Use substitution to write the algebraic expression as a trigonometric function of θ.
√
(a) 64 − 16x2 , x = 2 cos θ
√
(b) x2 − 4, x = 2 sec θ
3. Solve the following equations
(a) tan2 3x = 3
(b) sin 3x(tan x − 1) = 0
4. Solve the following equations on the interval [0, 2π).
(a) 2 sin2 θ + 3 sin θ + 1 = 0
(b) sec x + tan x = 1
(c) cos3 x = cos x
(d) 2 sin2 x − 7 sin x + 3 = 0
5. A sharpshooter intends to hit a target at a distance of 1000 yards with
a gun that has a muzzle velocity of 1200 feet per second. Neglecting air
resistance, determine the gun’s minimum angle of elevation θ if the range
1 2
v0 sin 2θ.
r is give by r = 32
6. Find the exact value of each expression using a sum or difference formula.
(a) sin − 7π
12
(b) cos 285◦
(c) tan 13π
12
7. Write sin(arctan 2x − arccos x) as an algebraic expression.
8. Find the exact solutions of the equation in the interval [0, 2π).
1
(a) sin 2x sin x = cos x
(b) (sin 2x + cos 2x)2 = 1
9. Use a power-reducing formula to rewrite sin2 x cos2 x in terms of the first
power of the cosine.
10. Use half-angle formulas to determine the following exact values.
(a) sin 165◦
(b) cos π8
(c) tan 7π
12
11. Write 3 sin 2α sin 3α as a sum or difference.
12. Write cos 6x + cos 2x as a product.
cos t + cos 3t
= cot t.
sin 3t − sin t
Also review all your homework problems.
13. Verify the identity
2