Question

How do you use angle sum identity to find the exact value of $\cos 105$ ?

Answer 1

Hint : In order to find the exact value of $\cos 105$ , we will use the angle sum property, $\cos \left( {a + b} \right) = \cos a\cos b - \sin a\sin b$ and we will write $105$ in terms of $0,\,30,\,45,\,60,\,90$ for easy substitution of ratios from the trigonometric table. Then, we will evaluate to determine the value of $\cos 105$ .

Complete step-by-step answer :
We need to use angle sum identity to determine the exact value of $\cos 105$ .
Now, we can rewrite $\cos 105$ as $\cos \left( {60 + 45} \right)$ ,
$\cos \left( {a + b} \right) = \cos a\cos b - \sin a\sin b$
Hence, by applying the values in the angle sum identity, we have,
$\cos \left( {60 + 45} \right) = \cos 60\cos 45 - \sin 60\sin 45$
From trigonometric ratios, we know that
$\cos 60 = \dfrac{1}{2}$
$\cos 45 = \dfrac{1}{{\sqrt 2 }} = \dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }} = \dfrac{{\sqrt 2 }}{2}$
$\sin 60 = \dfrac{{\sqrt 3 }}{2}$
$\sin 45 = \dfrac{1}{{\sqrt 2 }} = \dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }} = \dfrac{{\sqrt 2 }}{2}$
Therefore, let us apply the value of trigonometric ratios, we have,
$\cos \left( {60 + 45} \right) = \left( {\dfrac{1}{2}} \right) \times \left( {\dfrac{{\sqrt 2 }}{2}} \right) - \left( {\dfrac{{\sqrt 3 }}{2}} \right) \times \left( {\dfrac{{\sqrt 2 }}{2}} \right)$
$\cos \left( {60 + 45} \right) = \left( {\dfrac{{\sqrt 2 }}{4}} \right) - \left( {\dfrac{{\sqrt 6 }}{4}} \right)$
$\cos \left( {60 + 45} \right) = \dfrac{1}{4}\left( {\sqrt 2 - \sqrt 6 } \right)$
$\cos \left( {60 + 45} \right) = \dfrac{1}{4}\left( {\sqrt 2 - \sqrt 3 \times \sqrt 2 } \right)$
$\cos \left( {60 + 45} \right) = \dfrac{{\sqrt 2 }}{4}\left( {1 - \sqrt 3 } \right)$
The value of $\sqrt 2 = 1.414$ and $\sqrt 3 = 1.732$
Now, by applying the values, we have,
$\cos \left( {60 + 45} \right) = \dfrac{{1.414}}{4}\left( {1 - 1.732} \right)$
$\cos \left( {60 + 45} \right) = \dfrac{{1.414}}{4}\left( { - 0.732} \right)$
$\cos \left( {60 + 45} \right) = \dfrac{{ - 1.035}}{4}$
$\cos \left( {60 + 45} \right) = - 0.258$
Hence, the exact value of $\cos 105$ is $- 0.258$ .
So, the correct answer is “ $- 0.258$ ”.

Note : Three basic trigonometric identities involve the sums of angles. The functions involved in these identities are sine, cosine and tangent. We can use the angle sum identities to determine the function values of any angles. These identities are useful whenever expressions involving trigonometric functions need to be simplified.
The angle sum identities are
$\cos \left( {a + b} \right) = \cos a\cos b - \sin a\sin b$
$\cos \left( {a - b} \right) = \cos a\cos b + \sin a\sin b$
$\sin \left( {a + b} \right) = \cos a\sin b + \sin a\cos b$
$\sin \left( {a - b} \right) = \cos a\sin b - \sin a\cos b$
$\tan \left( {a + b} \right) = \dfrac{{\tan a + \tan b}}{{1 - \tan a\tan b}}$
$\tan \left( {a - b} \right) = \dfrac{{\tan a - \tan b}}{{1 + \tan a\tan b}}$