5.4 Multiple Angle Identities


Since sin ( A + B ) = sin A cos B + cos A sin B, it follows that

sin ( 2A ) = 2 sin A cos A

which is the double angle identity for the sine function. There are several double angle identities for the cosine function.

Since cos ( A + B ) = cos A cos B - sin A sin B, it follows that

cos ( 2A ) = cos2A - sin2A

and since cos2A = 1 - sin2A and sin2A = 1 - cos2A, we get two additional identities

cos ( 2A ) = 1 - 2 sin2A

and

cos ( 2A ) = 2 cos2A - 1.

Since tan ( A + B ) = ( tan A + tan B ) / ( 1 - tan A tan B ),

tan ( 2A ) = ( 2 tan A ) / ( 1 - tan2A ).

Exercise 5.4.1

Given that A is in quadrant II and sin A = 2 / 3, find sin ( 2A ) and cos ( 2A ). In which quadrant does 2A lie?

Solution

Exercise 5.4.2

Given that tan A = -3, find tan ( 2A ).

Solution

Exercise 5.4.3

Given that A is in quadrant I and cos A = 3 / 5, find sin ( 3A ). Hint: 3A = A + 2A

Solution

The identities cos ( 2A ) = 1 - 2 sin2A and cos ( 2A ) = 2 cos2A - 1 are sometimes written in the form

sin2A = [ 1 - cos ( 2A ) ] / 2

and

cos2A = [ 1 + cos ( 2A ) ] / 2

In this form, they are called the power reduction identities, since they reduce second powers of trigonometric functions to first powers of trigonometric functions.

If A is replaced by x / 2 in the power reduction identities and the resulting identities are solved for sin ( x / 2 ) and cos ( x / 2 ), then we get the half-angle identities for the sine and cosine functions.

sine of x over 2 equals plus or minus the root of half of 1 minus cosine x
 

cosine of x over 2 equals plus or minus the root of half 1 plus cosine of x

Exercise 5.4.4

Given that x / 2 is in quadrant II and cos x = 1 / 2, find cos ( x / 2 ) and sin ( x / 2 ).

Solution

In the unit circle, construct an angle A in standard position with its terminal side in quadrant I intersecting the unit circle at the point P. Construct a second angle B with its vertex at the point
( -1, 0 ), initial side the x-axis to the right of ( -1, 0 ) and terminal side containing P. Then, according to a theorem from geometry, B = A / 2. [See the diagram for clarification.]
 

Clarification diagram for tangent half angle identity

Since the tangent of an acute angle of a right triangle equals the ratio of the side opposite the angle and the side adjacent to the angle, it can be seen that

tangent of half of A equals the sine of A divided by the sum of 1 and cosine of A

Exercise 5.4.5

Given that A is in quadrant IV and cos A = 4 / 5, find tan ( A / 2 ).

Solution

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