Question

Find the exact value of the given expression. sin(2cos^-1(5/13))

Answer 1

Answer:

+120/169 or -120/169

Step-by-step explanation:

• let $cos^{-1}[\frac{5}{13} ] = \alpha$

where, alpha is some angle that satisfies the assumed condition.

• so, $cos\alpha= 5/13$

[ taking cos to the other side  or applying cos on both sides]

• now, substitute this in the given expression

• $sin(2cos^-1(5/13)) = sin(2\alpha )$

as sin$\beta$ =  $+ or -\sqrt{1-(cos\beta) ^{2}$

[by general trigonometry formula: $cos^{2}[\beta] +sin^{2}[\beta]=1$]

so if  $cos\alpha= 5/13$, we can get sin$\alpha$ from the above formula as + or - 12/13

(because, after taking square root on both sides we keep + or -]

• as, sin$2\beta = 2*sin[\beta ]*cos[\beta ]$

[by general trigonometry formula]

• here, now $sin[2\alpha ]=2*(+or- 12/13)*5/13$

so, the final value can be 120/169 or -120/169.