Question

If and \[-2>\theta \ge {{30}^{\circ }}\], then the value of \[\theta \] is:
A. \[{{10}^{\circ }}\]
B. \[{{15}^{\circ }}\]
C. \[{{20}^{\circ }}\]
D. \[{{30}^{\circ }}\]

Answer 1

Hint: Use the trigonometric table and find the values. \[\cot {{90}^{\circ }}\] is also zero.
Thus substitute, cancel out and get the value of \[\theta \].

Complete step-by-step answer:
We have been given the expression \[\cot \left( {{60}^{\circ }}+\theta \right)=0\].
We can use the trigonometric table to solve the above expression.


The trigonometric table helps us to find the values of trigonometric standard angles such as \[{{0}^{\circ }},{{30}^{\circ }},{{45}^{\circ }},{{60}^{\circ }},{{90}^{\circ }}\]. It consists of trigonometric ratios- sine, cosine, tangent, cosecant, secant, and cotangent, i.e. they can be written as sin, cos, tan, sec, cosec, cot.

Angles (degree)	\[{{0}^{\circ }}\]	\[{{30}^{\circ }}\]	\[{{45}^{\circ }}\]	\[{{60}^{\circ }}\]	\[{{90}^{\circ }}\]
Angles (radian)	0	\[{}^{\pi }/{}_{6}\]	\[{}^{\pi }/{}_{4}\]	\[{}^{\pi }/{}_{3}\]	\[{}^{\pi }/{}_{2}\]
Sin	0	\[{}^{1}/{}_{2}\]	\[{}^{1}/{}_{\sqrt{2}}\]	\[{}^{\sqrt{3}}/{}_{2}\]	1
Cos	1	\[{}^{\sqrt{3}}/{}_{2}\]	\[{}^{1}/{}_{\sqrt{2}}\]	\[{}^{1}/{}_{2}\]	0
Tan	0	\[{}^{1}/{}_{\sqrt{3}}\]	1	\[\sqrt{3}\]	N.A.
Cot	N.A.	\[\sqrt{3}\]	1	\[{}^{1}/{}_{\sqrt{3}}\]	0
Cosec	N.A.	2	\[\sqrt{2}\]	\[{}^{2}/{}_{\sqrt{3}}\]	1
Sec	1	\[{}^{2}/{}_{\sqrt{3}}\]	\[\sqrt{2}\]	2	N.A.

Thus we have drawn the trigonometric table.
\[\cot \left( {{60}^{\circ }}+\theta \right)=0\]

From the trigonometric table, we can see that, \[\cot {{90}^{\circ }}=0\].

Thus we can change the expression as,

\[\cot \left( {{60}^{\circ }}+\theta \right)=\cot {{90}^{\circ }}\]

We can cancel our cot from both sides.

\[\begin{align}

  & \therefore 60+\theta =90 \\

 & \therefore \theta =90-60 \\

 & \therefore \theta =30 \\

\end{align}\]

Thus we got the value of \[\theta =30\].

Option D is the correct answer.

Note: The values of the trigonometric ratio are standard angles and are very important to solve the trigonometric problems. Therefore it is necessary to remember the value of the trigonometric ratios of these standard angles.