Test 3
1. Prove the identity
tan θ
1+sec θ
+
1+sec θ
tan θ
2. Prove the identity cosec(θ + φ) ≡
=
2
sin θ
cosec θ cosec φ
cot θ+cot φ
3.
a. Express sin 3θ in terms of sin θ.
b. Solve the equation sin 3θ = sin θ for 0° ≤ θ ≤ 360°
4.
a. Express 4 cos θ + 6 sin θ in the form R cos(θ − α), where R > 0 and 0° ≤ α ≤ 90°, giving the
exact value of R and the value of α.
b. Hence solve the equation 4 cos θ + 6 sin θ = 5 for 0° ≤ θ ≤ 360°.
c. Find the greatest and least possible values of [(4 cos θ + 6 sin θ)2 + 5] as θ varies.
5. Let f(x) =
3−x
(1−2x)(1+x2 )
a. Express f(x) in partial fractions.
b. Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term
in x 2.