Question

If the value of $2\sin {{x}^{o}}-1=0$ and ${{x}^{o}}$ is an acute angle. If $\cos {{x}^{o}}$ is $\sqrt{m}/2$, m is?

Answer 1

The values of trigonometric functions ($\sin \theta$ and $\cos \theta$) at different angles are

$\theta$	${{0}^{o}}$	${{30}^{o}}$	${{45}^{o}}$	${{60}^{o}}$	${{90}^{o}}$
Sin$\theta$	0	$\dfrac{1}{2}$	$1/\sqrt{2}$	$\sqrt{3}/2$	1
Cos$\theta$	1	$\sqrt{3}/2$	$1/\sqrt{2}$	$\dfrac{1}{2}$	0

Solve the equation $2\sin {{x}^{o}}-1=0$ and find the value of $\sin {{x}^{o}}$. After this, compare it with the above given values and find the value of ${{x}^{o}}$.Formula Used: Here we have used the values of $\sin \theta$ and $\cos \theta$ at \theta ={{30}^{o}}$$$\sin {{30}^{o}}=1/2$ $\cos {{30}^{o}}=\sqrt{3}/2$ _**Complete step-by-step answer:**_ 2\sin {{x}^{o}}-1=0 $\Rightarrow$ $2\sin {{x}^{o}}=1$ $\Rightarrow$ $\sin {{x}^{o}}=1/2$ (1) From the table given above (in the hint) We know, $\sin {{30}^{o}}=1/2$ From (1) $\Rightarrow$ $\sin {{30}^{o}}=1/2=\sin {{x}^{o}}$ \Rightarrow \sin {{x}^{o}}=\sin {{30}^{o}} $$$$ $\Rightarrow {{x}^{o}}={{30}^{o}}$ Now, $\cos {{x}^{o}}=\sqrt{m}/2$ We know, $\cos {{30}^{o}}=\sqrt{3}/2$ {from table} \cos {{x}^{o}}=$$$\cos {{30}^{o}}=\sqrt{3}/2=\sqrt{m}/2$$\Rightarrow$$\sqrt{3}/2=\sqrt{m}/2$$\Rightarrow$$\sqrt{m}=\sqrt{3}$Squaring both sides$\Rightarrow$$m=3$

Additional Information
Values of trigonometric functions ($\sin \theta ,\cos \theta$ and $\tan \theta$) at different values of angles (${{0}^{o}},{{30}^{o}},{{45}^{o}},{{60}^{o}}$ and ${{90}^{o}}$).

$\theta$	${{0}^{o}}$	${{30}^{o}}$	${{45}^{o}}$	${{60}^{o}}$	${{90}^{o}}$
Sin$\theta$	0	$1/\sqrt{2}$	$1/\sqrt{2}$	$\sqrt{3}/2$	1
Cos$\theta$	1	$\sqrt{3}/2$	$1/\sqrt{2}$	$1/\sqrt{2}$	0

Tan$\theta$	0	1/$\sqrt{3}$	1	$\sqrt{3}$	$\infty$

Relation between trigonometric functions
$\begin{aligned} & 1/\sin \theta =\cos ec\theta \\\ & 1/\cos \theta =\sec \theta \\\ & 1/\tan \theta =\cot \theta \\\ \end{aligned}$
Note: The knowledge of the values of trigonometric functions at different angles is important for students to solve this question. The knowledge of algebra is also required to solve this question.