Cambridge International AS and A Level Mathematics Pure Mathematics 1

Trigonometry

238

P1^

7

ExAmPlE 7.11 Solve tan^2 θ = 2 for 0  θ  π.

SOlUTION

Here the range 0  θ  π indicates that

radians are required.

Since there is no request for multiples of π,

set your calculator to radians.

tan^2 θ = 2

⇒ tan^ θ = 2 or tan^ θ = − 2.

tan^ θ = 2 ⇒ θ = 0.955 radians

tan^ θ = − 2 ⇒ θ = −0.955 (not in range)

or θ = −0.955 + π = 2.186 radians.

The values of θ are 0.955 radians and 2.186 radians.

ExERCISE 7D  1  Express the following angles in radians, leaving your answers in terms of π

where appropriate.

(i) 45° (ii) 90° (iii) 120° (iv) 75°

(v) 300° (vi) 23° (vii) 450° (viii) 209°

(ix) 150° (x) 7.2°

2  Express the following angles in degrees, using a suitable approximation where

necessary.

(i) π

10

(ii) 3

5

π (iii) 2 radians (iv) 4

9

π

(v)  3 π (vi) 5

3

π

(vii)^ 0.4 radians^ (viii)^

3

4

π

(ix)  7

3

π (x) 3

7

π

3  Write the following as fractions, or using square roots.

You should not need your calculator.

(i) sinπ 4 (ii) tanπ

3

(iii) cosπ

6

(iv) cos π

(v) tan^3

4

π (vi) sin 2

3

π (vii) tan 4

3

π (viii) cos 3

4

π

(ix) sin^5

6

π (x) cos 5

3

π

4  Solve the following equation for 0  θ  2 π, giving your answers as multiples

of π.

(i) (^) cosθ=^3
2
(ii) tan θ = 1 (iii) (^) sinθ=^1
2
(iv) (^) sin–θ=^1
2
(v) (^) cos–θ=^1
(vi) (^) tanθ= 3
θ
tan θ
π 2
O
π 2
2.186
θ =
0.955
√ 2
π

• √ 2

Figure 7.26