If A and B are obtuse angles and $sinA = \frac{5}{13}$ and $cosB = -\frac{4}{5}$ then find sin(A+B)

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written 5.6 years ago by

teamques10 ★ 69k

Solution:

$\begin{aligned} \cos ^{2} A &=1-\sin ^{2} A \ &=1-\left(\frac{5}{13}\right)^{2} \ &=1-\frac{25}{169}=\frac{144}{169}\&. \cos A=\pm \frac{12}{13}\ & \therefore \cos A=-\frac{12}{13} \quad(\because A \text { is obtuse angle }) \sin ^{2} B=1-\cos ^{2} B\ &=1-\left(-\frac{4}{5}\right)^{2}\& \sin ^{2} B=1-\frac{16}{25}=\frac{9}{25} \ & \sin B=\pm \frac{3}{5}\end{aligned}$

$\begin{aligned} \therefore \sin (A+B) &=\sin A \cdot \cos B+\cos A \cdot \sin B \ &=\left(\frac{5}{13}\right) \times\left(-\frac{4}{5}\right)+\left(-\frac{12}{13}\right) \times\left(\frac{3}{5}\right) \ &=-\frac{56}{65} \end{aligned}$

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