Question

How do you rewrite $\cos 83\cos 53 + \sin 83\sin 53$ as a function of a single angle and then evaluate ?

Answer 1

To write the given trigonometric expression as a function of single angle, use the difference of angle formula for cosine function and put this expression on that formula and then simplify or rewrite it as a function of single angle and evaluate its value.Cosine formula for difference of angle is given as follows: $\cos (a - b) = \cos a\cos b + \sin a\sin b$.

Formula used:
Cosine formula for difference of angle: $\cos (a - b) = \cos a\cos b + \sin a\sin b$

Complete step by step solution:
In order to express the given trigonometric expression $\cos 83\cos 53 + \sin 83\sin 53$ as a function of single angle, we will use the compound angle trigonometric identities. In which there is a cosine formula for difference of angle which is given as the cosine of difference of two angles is equal to the sum of the product of cosine of both angles and the product of sine of both angles, mathematically it can be understood as follows
$\cos (a - b) = \cos a\cos b + \sin a\sin b$
Comparing this with the given trigonometric expression, we get
$a = 83\;{\text{and}}\;b = 53$
So using this formula for the given trigonometric expression, we will get
$\cos 83\cos 53 + \sin 83\sin 53 = \cos (83 - 53) \\\ \therefore\cos 83\cos 53 + \sin 83\sin 53 = \cos 30$
Considering that the unit of the given angle is in degrees.So from the standard trigonometric values, we know that $\cos 30 = \dfrac{{\sqrt 3 }}{2}$.

Therefore $\cos 30$ is the simplified form of the given expression as a function of single angle and $\dfrac{{\sqrt 3 }}{2}$ is its value.

Note: Generally students use the wrong cosine formula in order to apply for either addition or subtraction, because as you can see that for cosine of subtraction of two angles there’s addition in the formula whereas vice versa for the cosine of addition of two angles.