Question

If the formula for the sine of a sum of angles C and D is given by
$\sin (C+D)=\sin C.\cos D+\cos C.\sin D$, then the value of $\sin {{75}^{\circ }}$ is?
a)$\dfrac{1}{2\sqrt{2}}\left( \sqrt{3}+1 \right)$
b) $\dfrac{1}{2}\left( \sqrt{3}-1 \right)$
c) $\dfrac{\sqrt{3}}{2}$
d)$\dfrac{1}{2}$

Answer 1

Hint: We should try to write ${{75}^{\circ }}$as a sum of angles whose sine and cosine values are known. Then, we can use the formula given in the question to find $\sin {{75}^{\circ }}$ in terms of the sine and cosine values of the other two angles.

Complete Complete step by step answer:

We can write ${{75}^{\circ }}$ as the sum of ${{45}^{\circ }}\text{ and 3}{{0}^{\circ }}$, i.e.
${{75}^{\circ }}={{45}^{\circ }}+{{30}^{\circ }}$
Now, from the definition of sine cosine function, we know that, in a right angled triangle having sides a and b and hypotenuse h as shown in figure 1,
$\sin (\theta )=\dfrac{\text{length of side facing the angle theta}}{\text{length of the hypotanuse}}=\dfrac{a}{h}$
and $\cos (\theta )=\dfrac{\text{length of side adjacent to the angle theta}}{\text{length of the hypotanuse}}=\dfrac{b}{h}$
To find the sine and cosine of ${{45}^{\circ }}\text{ and 3}{{0}^{\circ }}$, we can draw the triangles shown in figure 2 and figure 3.
So, $\sin ({{30}^{\circ }})=\dfrac{1}{2}\text{ and }\cos ({{30}^{\circ }})=\dfrac{\sqrt{3}}{2}$ (from fig2)
And $\sin ({{45}^{\circ }})=\dfrac{1}{\sqrt{2}}\text{ and }\cos ({{45}^{\circ }})=\dfrac{1}{\sqrt{2}}$ (from fig3)

Now, using the formula given in the question and using the values as derived above,
$\begin{aligned} & \sin (75{}^\circ )=\sin (45{}^\circ +30{}^\circ )=\sin (45{}^\circ )\cos (30{}^\circ )+\sin (30{}^\circ )\cos (45{}^\circ ) \\\ & =\left( \dfrac{1}{\sqrt{2}}.\dfrac{\sqrt{3}}{2} \right)+\left( \dfrac{1}{2}.\dfrac{1}{\sqrt{2}} \right)=\dfrac{\sqrt{3}+1}{2\sqrt{2}} \\\ \end{aligned}$
This answer matches option (a) of the question and thus option (a) is the correct answer.

Note: While finding the sines and cosines of various angles, we should try to write them in terms of ${{30}^{\circ }}$, ${{60}^{\circ }}$ and ${{45}^{\circ }}$ as their trigonometric ratios can be easily found out from figures 1 and 2.
In many cases, the angles have a value more than ${{360}^{\circ }}$. In that case, we can subtract a multiple of ${{360}^{\circ }}$ from the original angle and find the sine and cosine of it as adding or subtracting ${{360}^{\circ }}$ does not change the value of sine or cosine.