Examples

Trigonometric Formulae

Contents

Formulae

Examples

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Quiz

Examples

Example 3:

If A, B and C are angles of a triangle, prove that cos A + cos (B + C) = 0. Hence prove that cos A + cos (B - C) = 2 sinB sinC.

Solution:

A = 180^o - (B + C)

cos A + cos (B + C) = cos [180^o - (B + C)] + cos (B + C)

= cos180^o cos (B + C) - sin180^o sin (B + C) + cos (B + C)

= - cos (B + C) + cos (B + C)

= 0 (shown)

cos A + cos (B - C) = cos [ 180^o - (B + C)] + cos (B - C)

= cos180^o cos (B + C) + cos (B - C)

= -cos (B + C) + cos (B - C)

= -cosB cosC + sinB sinC + cosB cosC + sinB sinC

= 2 sinB sinC

Example 4:

Prove the identity

sin q + sin 3q + sin 5q + sin 7q = tan 4q (cos q + cos 3q + cos 5q + cos 7q)

Solution:

L.H.S.= sin q + sin 3q + sin 5q + sin 7q

= 2 sin 4q cos 3q + 2 sin 4q cos q factor (sum) formulae

= 2 sin 4q (cos 3q + cos q)

= 2 sin 4q (cos 4q / cos 4q) (cos 3q + cos q)

= tan 4q (2 cos 4q cos 3q + 2 cos 4q cos q)

= tan 4q (cos q + cos 3q + cos 5q + cos 7q) factor (product) formulae

= R.H.S.