Question

How do you use the sum or difference identities to find the exact value of $\sin 165^\circ$?

Answer 1

Here we will use the sum of the sin identities and will split the given angle as $165 = 135 + 30$ and further $135 = 90 + 45$ to get the exact value of sine function.

Complete step by step answer:
Use the identity –
$\sin (x + y) = \sin x\cos y + \cos x\sin y$ …. (A)
Now, split the given angle as $165^\circ = 135^\circ + 30^\circ$
So, the equation (A) can be written as –
$\sin (135^\circ + 30^\circ ) = \sin 135^\circ \cos 30^\circ + \cos 135^\circ \sin 30^\circ$ …. (B)
Now,
$\sin 135^\circ = \sin (90^\circ + 45^\circ )$
By referring to the All STC rule, sine is positive in the second quadrant and by referring the trigonometric table for value.
$\sin 135^\circ = \sin (90^\circ + 45^\circ ) = \dfrac{1}{{\sqrt 2 }}$ …. (C)
And similarly, $\cos 135^\circ = \cos (90^\circ + 45^\circ )$
By referring to the All STC rule, cosine is negative in the second quadrant and by referring to the trigonometric table for value.
$\cos 135^\circ = \cos (90^\circ + 45^\circ ) = - \dfrac{1}{{\sqrt 2 }}$ ….. (D)
Placing the values of equations (C) and (D) in equation (B) and also referring to the trigonometric table for value.
$\sin (135^\circ + 30^\circ ) = \dfrac{1}{{\sqrt 2 }}.\dfrac{{\sqrt 3 }}{2} + \left( { - \dfrac{1}{{\sqrt 2 }}} \right).\dfrac{1}{2}$
Simplify the above expression
$\sin (135^\circ + 30^\circ ) = \dfrac{{\sqrt 3 }}{{2\sqrt 2 }} - \dfrac{1}{{2\sqrt 2 }}$
It can be re-written as –
$\sin (135^\circ + 30^\circ ) = \dfrac{{\sqrt 3 - 1}}{{2\sqrt 2 }}$
This is the required solution.

Additional Information:
Remember the properties of sines and cosines and apply accordingly. The odd and even trigonometric functions states that -
$\sin ( - \theta ) = - \sin \theta \\\ \cos ( - \theta ) = \cos \theta \\\$
The most important property of sines and cosines is that their values lie between minus one and plus one. Every point on the circle is unit circle from the origin. So, the coordinates of any point are within one of zero as well.

Note: Remember the All STC rule, it is also known as ASTC rule in geometry. It states that all the trigonometric ratios in the first quadrant ($0^\circ \;{\text{to 90}}^\circ$ ) are positive, sine and cosec are positive in the second quadrant ($90^\circ {\text{ to 180}}^\circ$ ), tan and cot are positive in the third quadrant ($180^\circ \;{\text{to 270}}^\circ$ ) and sin and cosec are positive in the fourth quadrant ($270^\circ {\text{ to 360}}^\circ$ ).