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

Hint: 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
$\begin{align}
  & 1/\sin \theta =\cos ec\theta \\
 & 1/\cos \theta =\sec \theta \\
 & 1/\tan \theta =\cot \theta \\
\end{align}$
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.