5.5.1a Proving Trigonometric Identities Using Addition Formula And Double Angle Formulae (Part 2)

Posted on May 22, 2020 by Myhometuition

Example 3:

(a) Given that

$\sin P = \frac{3}{5} and \sin Q = \frac{5}{13},$ such that P is an acute angle and Q is an obtuse angle, without using tables or a calculator, find the value of cos (P + Q).

(b) Given that

$\sin A = - \frac{3}{5} and \sin B = \frac{12}{13},$ such that A and B are angles in the third and fourth quadrants respectively, without using tables or a calculator, find the value of sin (A – B).

Solution:

(a)

$\begin{array}{l} \sin P = \frac{3}{5}, \cos P = \frac{4}{5} \\ \sin Q = \frac{5}{13}, \cos Q = - \frac{12}{13} \\ \cos (P + Q) \\ = \cos A \cos B - \sin A \sin B \\ = (\frac{4}{5}) (- \frac{12}{13}) - (\frac{3}{5}) (\frac{5}{13}) \\ = - \frac{48}{65} - \frac{15}{65} \\ = - \frac{63}{65} \end{array}$

(b)

$\begin{array}{l} \sin A = - \frac{3}{5}, \cos A = - \frac{4}{5} \\ \sin B = - \frac{5}{13}, \cos B = \frac{12}{13} \\ s i n (A - B) \\ = s i n A \cos B - c o s A \sin B \\ = (- \frac{3}{5}) (\frac{12}{13}) - (- \frac{4}{5}) (- \frac{5}{13}) \\ = - \frac{36}{65} - \frac{20}{65} \\ = - \frac{56}{65} \end{array}$