Question

Evaluate the given sum of trigonometric functions: $\cos {{48}^{\circ }}-\sin {{42}^{\circ }}$?

Answer 1

Hint: We start solving the problem by assigning the variable for the result of the given subtraction of the trigonometric functions. We then change the given angle ${{42}^{\circ }}$ as ${{90}^{\circ }}-{{48}^{\circ }}$ in order to convert the given sine function. We then use the fact that $\sin \left( 90-\alpha \right)=\cos \alpha $ for the sine function obtained. We then make the necessary calculations to get the required value of the given subtraction.

Complete step-by-step solution:
According to the problem, we need to find the value of $\cos {{48}^{\circ }}-\sin {{42}^{\circ }}$.
Let us assume the value of $\cos {{48}^{\circ }}-\sin {{42}^{\circ }}$ be ‘x’.
So, we get $x=\cos {{48}^{\circ }}-\sin {{42}^{\circ }}$.
$\Rightarrow x=\cos {{48}^{\circ }}-\sin \left( {{90}^{\circ }}-{{48}^{\circ }} \right)$ ---(1).
We know that $\sin \left( 90-\alpha \right)=\cos \alpha $. We use this result in equation (1).
So, we get $x=\cos {{48}^{\circ }}-\cos {{48}^{\circ }}$.
$\Rightarrow x=0$.
So, we have found the value of $\cos {{48}^{\circ }}-\sin {{42}^{\circ }}$ as 0.
$\therefore$ The value of $\cos {{48}^{\circ }}-\sin {{42}^{\circ }}$ is 0.

Note: Whenever we get this type of problem, we should try to convert both the trigonometric functions into the same trigonometric functions in order to reduce the calculation time. We need not always know the values of the trigonometric functions at a given value to solve this type of problems, just applying correct trigonometric identity will make the problem solvable. We can also solve the using the fact $\cos \left( 90-\alpha \right)=\sin \alpha $ in the given problem as shown below:
So, we have $x=\cos {{48}^{\circ }}-\sin {{42}^{\circ }}$.
$\Rightarrow x=\cos \left( {{90}^{\circ }}-{{42}^{\circ }} \right)-\sin {{42}^{\circ }}$.
Now, we use the trigonometric identity $\cos \left( 90-\alpha \right)=\sin \alpha $.
So, we get $x=\sin {{42}^{\circ }}-\sin {{42}^{\circ }}$.
$\Rightarrow x=0$, which is the required answer.
Similarly, we can expect problems which involve tangent and cotangent functions with similar types of relations.