Exercise 6.5 - Chapter 6 Trigonometry 9th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $6.5$
Multiple Choice Questions:
Question 1 .
If $\sin 30^{\circ}=x$ and $\cos 60^{\circ}=y$, then $x^{2}+y^{2}$ is
(1) $\frac{1}{2}$
(2) 0
(3) $\sin 90^{\circ}$
(4) $\cos 90^{\circ}$
Hint:
$\begin{aligned}
x &=\sin 30^{\circ}=\frac{1}{2} \\
y &=\cos 60^{\circ}=\frac{1}{2} \\
x^{2}+y^{2} &=\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}=\frac{1}{4}+\frac{1}{4}=\frac{1}{2}
\end{aligned}$
Solution:
(1) $\frac{1}{2}$

Question 2 .
If $\tan \theta=\cot 37^{\circ}$, then the value of $\theta$ is
(1) $37^{\circ}$
(2) $53^{\circ}$
(3) $90^{\circ}$
(4) $1^{\circ}$
Hint:
$\tan \theta=\cot 37^{\circ}$
$\tan \left(90^{\circ}-37^{\circ}\right)=\cot 37^{\circ}$

$\theta=90^{\circ}-37^{\circ}=53^{\circ}$
Solution:
(2) $53^{\circ}$

Question $3 .$
The value of $\tan 72^{\circ} \tan 18^{\circ}$ is
(1) 0
(2) 1
(3) $18^{\circ}$
(4) $72^{\circ}$

Solution:
(2) 1

Question $4 .$
The value of $\frac{2 \tan 30^{\circ}}{1-\tan ^{2} 30^{\circ}}$ is equal to
(1) $\cos 60^{\circ}$
(2) $\sin 60^{\circ}$
(3) $\tan 60^{\circ}$
(4) $\sin 30^{\circ}$

Solution:
(3) $\tan 60^{\circ}$

Question $5 .$
If $2 \sin 2 \theta=\sqrt{3}$, then the value of $\theta$ is
(1) $90^{\circ}$
(2) $30^{\circ}$
(3) $45^{\circ}$
(4) $60^{\circ}$
Hint:
$\begin{aligned}
2 \sin 2 \theta &=\sqrt{3} \\
\sin 2 \theta &=\frac{\sqrt{3}}{2}=\sin 60^{\circ} \\
2 \theta &=60^{\circ} \Rightarrow \theta=30^{\circ}
\end{aligned}$
Solution:
(2) $30^{\circ}$

Question $6 .$
The value of $3 \sin 70^{\circ} \sec 20^{\circ}+2 \sin 49^{\circ} \sec 51^{\circ}$ is
(1) 2
(2) 3
(3) 5
(4) $60^{\circ}$

Solution:
(3) 5

Question 7.
The value of $\frac{1-\tan ^{2} 45^{\circ}}{1+\tan ^{2} 45^{\circ}}$ is
(1) 2
(2) 1
(3) 0
(4) $\frac{1}{2}$
Hint:
$\frac{1-\tan ^{2} 45^{\circ}}{1+\tan ^{2} 45^{\circ}}=\frac{1-1^{2}}{1+1^{2}}=\frac{0}{2}=0$
Solution:
(3) 0

Question 8 .
The value of $\operatorname{cosec}\left(70^{\circ}+\theta\right)-\sec \left(20^{\circ}-\theta\right)+\tan \left(65^{\circ}+\theta\right)-\cot \left(25^{\circ}-\theta\right)$ is
(1) 0
(2) 1
(3) 2
(4) 3

Solution:
(1) 0

Question $9 .$
The value of $\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}$ is
(1) 0
(2) 1
(3) 2
(4) $\frac{\sqrt{3}}{2}$

Solution:
(2) 1

Question $10 .$
Given that $\sin \alpha=\frac{1}{2}$ and $\cos \beta=\frac{1}{2}$, then the value of $\alpha+\beta$ is
(1) $0^{\circ}$
(2) $90^{\circ}$
(3) $30^{\circ}$
(4) $60^{\circ}$
Hint:
$\sin \alpha=\frac{1}{2}, \cos \beta=\frac{1}{2}, \alpha+\beta=90^{\circ}$
Solution:

(2)$90^{\circ}$