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Mathematics Question on Trigonometric Functions

If cos⁡ T=35\cos \, T = \frac{3}{5} and if sin⁡ R=817\sin \, R = \frac{8}{17}, where T is in the fourth quadrant and R is in the second quadrant, then cos (T - R) is equal to:

A

7785\frac{77}{85}

B

1385\frac{13}{85}

C

−1385 - \frac{13}{85}

D

−7785 - \frac{77}{85}

Answer

−7785 - \frac{77}{85}

Explanation

Solution

As given, cos T = 3/5 sin⁡2T=1−3252=1−925\sin^2 T = 1 - \frac{3^2}{5^2} = 1 - \frac{9}{25} So, sin⁡ T=1625=−45\sin \, T = \frac{16}{25} = - \frac{4}{5} [since T is in IV quad. + ve value is ignored.] Also, given, sin R = 817\frac{8}{17} ⇒ cos⁡2=1−82172=1−64289=225289\Rightarrow \, \cos^2 = 1 - \frac{8^2}{17^2} = 1 - \frac{64}{289} = \frac{225}{289} So, cos⁡ R=−1517\cos \, R = - \frac{15}{17} [Since R is in II quad. + ve value is ignored] Now, cos (T - R) = cos T cos R + sin T sin R =35×−1517−45×817=−4585−3285= \frac{3}{5} \times\frac{-15}{17} - \frac{4}{5} \times\frac{8}{17} = - \frac{45}{85} - \frac{32}{85} =−[45+3285]=−7785= - \left[\frac{45+32}{85}\right] = - \frac{77}{85}