Question

Find the exact value of the following under the given conditions:

(a) sin (α + β)              (b) cos (α + β)                  (c) sin (α - β)                  (d) tan (α - β)

Answer 1

Step 1 : 

Sum and difference formulas :

   

Step 2 :
The values are and .

Since then and

The angle is lies in first quadrant.

 

The point is lies on the curve of radius

Thus,

Since the point  is lies in first quadrant consider

The value of cosine function is

Step 3 :

Since then and

The angle is lies in fourth quadrant.

The point is lies on the curve of radius Thus,

Since the point  is lies in first quadrant consider
The value of cosine function is

Step 3 :

(a)

The expression is

From the sum and difference formulas for the sine function.

Substitute and in above equation.

Thus,

Solution :

(a) 

Answer 2

Step 1 :

(b)

The expression is

From the sum and difference formulas for the cosine function.

Substitute and in above equation.

Thus,

Step 2 :

(c)

The expression is

From the sum and difference formulas for the sine function.

Substitute and in above equation.

Thus,

Solution :

(b)

(c)

Answer 3

Step 1 :

(d)

The expression is

Rewrite the expression :

Evaluation of :

From the sum and difference formulas for the cosine function.

Substitute and in above equation.

Thus,

From (c):

 

Step 2 : 

Substitute and in

Thus,

Solution :

(d)