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4.3 The multiple - angle (Double & Half Angle) Formulas

With the help of the sum and difference (compound angle formulas studied in the previous article, we will express the trigonometric functions of angle in terms of /2 ).

For any angle

(1) \ sin 2 = 2 sin cos then

\ sin = 2 sin ( /2) cos (/2)

(2) \ cos 2 = cos2 - sin2

\ cos = cos2( /2) - sin2( /2)

\ cos 2 = 2 cos2 -1

\ cos = 2 cos2( /2) - 1 and

\ cos 2 = 1 - 2 sin2

\ cos = 1 - 2 sin2 (/2)

From the above formulas, we derive the following formulas

EXAMPLE 1 Find the value of cos 150, using the ratios of 300only.

Solution:

EXAMPLE 2 Find the exact value for sin 1050using the half- angle identity.

Solution :

EXAMPLE 3 Find the exact value of sin (292.50 ) using the half angle formulas.

Solution : /2 = 292.50 is in Quad. IV in which sine ratio is negative.

EXAMPLE 4

Find the values of sine and cosine of . Given that sin = 5 / 13 , is in Quad II

Solution:

EXAMPLE 5 Find the exact value for cos 1650 using half- angle identity.

Solution:

EXAMPLE 6 Find sin 22.50, cos 22.50 and tan 22. 5 0using the half angle formulas.

Solution:

EXAMPLE 7 in terms of sin x and cos x

Solution:

EXAMPLE 8 Prove that

Solution

EXAMPLE 9

Solution:

EXAMPLE 10

Solution:

EXAMPLE 11

Solution:

EXAMPLE 12

Solution:

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Index

Trignometric Identities

4.1 Fundamental Identities
4.2 The addition formulas
4.3 The multiple-angle (Double & Half angle) formulas
4.4 Tangent Identities
4.5 Factorization & Defactorization

Supplementary Problems


Chapter 5

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