Question

Use the information given about the angle θ, 0 ≤ θ < 2π, to find the exact value of

(a) sin (2θ)                   (b) cos (2θ)                              (c) sin (θ/2)                                  (d) cos (θ/2)

sin θ = 3/5,              0 < θ < π/2

Answer 1

Step 1:

The trigonometric function is in the interval .

.

From Pythagorean theorem,

The square of the hypotenuse is equal to sum the squares of the other two sides.

.

Where lies in quadrant I.

In quadrant I, the six trigonometric functions are positive.

.

Step 2:

(a)

Find .

Use double-angle formula : .

Substitute and in above expression.

.

Step 3:

(b)

Find .

Use double-angle formula : .

Substitute and in above expression.

.

Answer 2

Contd....

Step 4:

(c)

Find .

Use half-angle formula : .

Substitute in above expression.

Where lies in quadrant I, since lies in quadrant I.

.

Step 5:

(d)

Find .

Use half-angle formula : .

Substitute in above expression.

Where lies in quadrant I, since lies in quadrant I.

.

Solution :

(a) .

(b) .

(c) .

(d) .