Question

How will you find the exact value of $\sin \left( {u + v} \right)$ given that $\sin u = \dfrac{5}{{13}}$ and $\cos v = - \dfrac{3}{5}$?

Answer 1

Hint:We know that to find the exact value of $\sin \left( {u + v} \right)$, we need to use the sum formula for sine functions. We are given the values of $\sin u$ and $\cos v$. But, for using the sum formula for sine function, we need values of $\sin v$ and $\cos u$. For this, we will use the basic trigonometric relation between sine and cosine function.

Formulas used:
$\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$
${\sin ^2}A + {\cos ^2}A = 1$

Complete step by step answer:
We will first find the value of $\cos u$. For this we will use the relation between sine and cosine function ${\sin ^2}A + {\cos ^2}A = 1$. For angle $u$, we can say that,
${\sin ^2}u + {\cos ^2}u = 1 \\
\Rightarrow {\cos ^2}u = 1 - {\sin ^2}u \\ $
We are given the value of $\sin u = \dfrac{5}{{13}}$,
\[{\cos ^2}u = 1 - {\left( {\dfrac{5}{{13}}} \right)^2} \\
\Rightarrow {\cos ^2}u = 1 - \dfrac{{25}}{{169}} \\
\Rightarrow {\cos ^2}u = \dfrac{{144}}{{169}} \\
\Rightarrow \cos u = \pm \dfrac{{12}}{{13}} \\ \]
Now, we will find the value of $\sin v$.
For angle $v$, we can say that,
${\sin ^2}v + {\cos ^2}v = 1 \\
\Rightarrow {\sin ^2}v = 1 - {\cos ^2}v \\ $
We are given the value of $\cos v = - \dfrac{3}{5}$,
\[{\sin ^2}v = 1 - {\left( { - \dfrac{3}{5}} \right)^2} \\
\Rightarrow {\sin ^2}v = 1 - \dfrac{9}{{25}} \\
\Rightarrow {\sin ^2}v = \dfrac{{16}}{{25}} \\
\Rightarrow \sin v = \pm \dfrac{4}{5} \\ \]
We will now find the value of $\sin \left( {u + v} \right)$.
We know that $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$.
Therefore, we can say that,
$\sin \left( {u + v} \right) = \sin u\cos v + \cos u\sin v$
We have, $\sin u = \dfrac{5}{{13}}$, $\cos v = - \dfrac{3}{5}$, \[\cos u = \pm \dfrac{{12}}{{13}}\], \[\sin v = \pm \dfrac{4}{5}\]
$ \Rightarrow \sin \left( {u + v} \right) = \left( {\dfrac{5}{{13}}} \right)\left( { - \dfrac{3}{5}} \right) + \left( { \pm \dfrac{{12}}{{13}}} \right)\left( { \pm \dfrac{4}{5}} \right) = - \dfrac{{15}}{{65}} \pm \dfrac{{48}}{{65}}$
\[ \Rightarrow \sin \left( {u + v} \right) = \dfrac{{ - 15 - 48}}{{65}} = - \dfrac{{63}}{{65}}\]
\[ \therefore \sin \left( {u + v} \right) = \dfrac{{ - 15 + 48}}{{65}} = \dfrac{{33}}{{65}}\]

Therefore, the exact value of $\sin \left( {u + v} \right)$ is $ - \dfrac{{63}}{{65}}$ or $\dfrac{{33}}{{65}}$.

Note:Here, we have used the formula for the sum of the sine functions. This states that the sum formula for sine function states that the sine of the sum of two angles equals the product of the sine of the first angle and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second angle. The formulas for sum and difference for different trigonometric functions can be used to find the exact values of the sine, cosine, or tangent of an angle.