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Example 12 Prove that


Example 13

Verify that


Example 14 If x = r cos A cos B, y= r cos A sin B and z = r sin A prove that X2 + Y2 + z2 = r2


x2 + y2+ z2= r2 ......(given to be proved.)

x2 + y2+ z2= r2

\ (r cos A cos B)2 + (r cos A sin B)2+
(r sinA) 2 = r 2 ....substituting the given relations.

\ r2cos2A cos2 B + r2cos2 A sin2 B + r2sin2A = r2 ....(algebraic manipulation)

\ r2cos 2 A (cos2B+ sin2B) + r2 sin2A = r2 .....(combining terms)

\ r2cos2A (1) + r2sin2 A = r 2 ....Pythagorean identity

\ r2(cos2 A + sin2 A) = r2

\ r2(1) .....Pythagorean identity.

Example 15


Example 16 Show that


We know that

since (x-y)2is a complete square, it can't be negative.

\ (x -y)2= 0 is the only possibility.

\ (x -y) = 0 i.e. x = y.

Example 17 If a and b are two positive real numbers. Can

ii) tan x = b/a? Give reasons.


Example 18 Express each of other trigonometric ratios of in terms of sin .


Example 19 If is in Quadrant IV, express the other trigonometric functions of in terms of sec .


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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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