Question

Express $\sin 67{}^\circ +\cos 75{}^\circ$ in terms of trigonometric ratios of angle between 0 degrees and 45 degrees.

Answer 1

Hint: For solving this problem, we individually consider sin and cos terms given in the expression. Now, by using the trigonometric equation $\sin \theta =\cos \left( 90-\theta \right)\text{ and }\cos \theta =\sin \left( 90-\theta \right)$, we can easily obtain the required terms in the range of 0 degrees and 45 degrees.

Complete step-by-step answer:
According to the problem statement, we are given an expression $\sin 67{}^\circ +\cos 75{}^\circ$. Now, we have to use trigonometric relations to express the angles between 0 and 45 degrees. The most fundamental relations in trigonometric properties include the conversion of angle given in sine function to cos function. Both of the angles are related by complementary relationships. This can be mathematically expressed as: $\sin \theta =\cos \left( 90-\theta \right)$. The converse of our relationship is also true. So, it can also be mathematically expressed as: $\cos \theta =\sin \left( 90-\theta \right)$.
Now, converting $\sin 67{}^\circ$ in terms of cos to get the angle between 0 and 45 degree:

& \sin 67{}^\circ =\cos \left( 90-67 \right){}^\circ \\\ & \cos \left( 90-67 \right){}^\circ =\cos 23{}^\circ \\\ & \therefore \sin 67{}^\circ =\cos 23{}^\circ \ldots \left( 1 \right) \\\ \end{aligned}$$ Now, converting $\cos 75{}^\circ $ in terms of sin to get the angle between 0 and 45 degrees: $\begin{aligned} & \cos 75{}^\circ =\sin \left( 90-75 \right){}^\circ \\\ & \sin \left( 90-75 \right){}^\circ =\sin 15{}^\circ \\\ & \therefore \cos 75{}^\circ =\sin 15{}^\circ \ldots \left( 1 \right) \\\ \end{aligned}$ Adding equation (1) and equation (2), we obtain $$\sin 67{}^\circ +\cos 75{}^\circ =\cos 23{}^\circ +\sin 15{}^\circ $$. Hence, the expression of $\sin 67{}^\circ +\cos 75{}^\circ $ in terms of trigonometric ratios of angles between 0 degree and 45 degree is $$\cos 23{}^\circ +\sin 15{}^\circ $$. Note: Students must remember the trigonometric formulas associated with different functions. The complementary relationship is essential for obtaining the potential answer to such problems where angles are to be converted.