1 Marks Question

Question 11 Mark

Write $\cos 7 \theta \cos 3 \theta$ as sum and difference of angles.

Answer

$\begin{aligned} \cos 7 \theta \cos 3 \theta & =\frac{1}{2}[2 \cos 7 \theta \cos 3 \theta] \\ & =\frac{1}{2}[\cos (7 \theta+3 \theta)+\cos (7 \theta-3 \theta)] \\ & =\frac{1}{2}[\cos 10 \theta+\cos 4 \theta]\end{aligned}$

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Question 21 Mark

Write the value of $\sin \left(-330^{\circ}\right)$.

Answer

$\begin{aligned} \sin \left(-330^{\circ}\right) & =-\sin 330^{\circ} \quad \because \sin (-\theta)=-\sin \theta \\ & =-\sin \left(360^{\circ}-30^{\circ}\right) \\ & =-\left[-\sin 30^{\circ}\right] \\ {[\because \sin (2 \pi} & -\theta)=-\sin \theta]=\sin 30^{\circ}=\frac{1}{2}\end{aligned}$

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Question 31 Mark

Find the value of expression $\left\{\cos A \cdot \cos \left(90^{\circ}- A \right)\right\}$.

Answer

$\begin{aligned} \text {Given expression } & =\cos A \cdot \cos \left(90^{\circ}- A \right) \\ & =\cos A \cdot \sin A \\ & =\sin A \cdot \cos A =\frac{1}{2} \times 2 \sin A \cdot \cos A \\ & =\frac{1}{2} \sin 2 A\end{aligned}$

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Question 41 Mark

If $\tan \alpha=\frac{3}{4}$, then find the value of $\tan 3 \alpha$.

Answer

We know that $: \tan 3 \alpha=\frac{3 \tan \alpha-\tan ^3 \alpha}{1-3 \tan ^2 \alpha}$
$ \tan 3 \alpha=\frac{3 \times \frac{3}{4}-\left(\frac{3}{4}\right)^3}{1-3 \times\left(\frac{3}{4}\right)^2}=\frac{\frac{9}{4}-\frac{27}{64}}{1-\frac{27}{16}} $
$=-\frac{117}{44}$

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Question 51 Mark

Write the expression $\frac{2 \tan A}{\tan 2 A}$ in the form of $\tan A$.

Answer

$\begin{aligned}\text{Given expression } & =\frac{2 \tan A}{\tan 2 A} \\ & =\frac{2 \tan A}{\frac{2 \tan A}{1-\tan ^2 A}}=2 \tan A \times \frac{1-\tan ^2 A}{2 \tan A} \\ & =1-\tan ^2 A\end{aligned}$

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Question 61 Mark

Write the value of $\sin ^2 24^{\circ}-\sin ^2 6^{\circ}$.

Answer

$\begin{array}{l}\Rightarrow \quad \sin ^2 24^{\circ}-\sin ^2 6^{\circ} \\ \Rightarrow \quad \sin \left(24^{\circ}+6^{\circ}\right) \sin \left(24^{\circ}-6^{\circ}\right) \\ \Rightarrow \quad \sin 30^{\circ} \sin 18^{\circ}=\frac{1}{2} \times\left(\frac{\sqrt{5}-1}{4}\right) \\ \Rightarrow \quad \frac{1}{8}(\sqrt{5}-1) \end{array}$

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Question 71 Mark

Write the value of $\sin \frac{5 \pi}{12} \sin \frac{\pi}{12}$.

Answer

$\begin{aligned} \sin \frac{5 \pi}{12} \sin \frac{\pi}{12}= & \frac{1}{2}\left(2 \sin \frac{5 \pi}{12} \sin \frac{\pi}{12}\right) \\ & =\frac{1}{2}\left[\cos \left(\frac{5 \pi}{12}-\frac{\pi}{12}\right)-\cos \left(\frac{5 \pi}{12}+\frac{\pi}{12}\right)\right] \\ & =\frac{1}{2}\left(\cos \frac{\pi}{3}-\cos \frac{\pi}{2}\right) \\ & =\frac{1}{2}\left[\frac{1}{2}-0\right]=\frac{1}{2} \times \frac{1}{2}=\frac{1}{4}\end{aligned}$

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