Trigonometric identities and trigonometric equations — practice questions

Question 1 · difficulty L2 · understanding

Consider the curve $C_1$ given by $$y=e^{-x} \quad \text { for } x \in[0,10 \pi],$$ and the curve $C_2$ given by $$y=e^{-x}(\sin x+\cos x) \quad \text { for } x \in[0,10 \pi] .$$ Let $n$ be the total number of points of intersection of the curves $C_1$ and $C_2$. Suppose that $\alpha_1, \alpha_2, \ldots, \alpha_n \in[0,10 \pi]$ are the $x$-coordinates of the points of intersection of the curves $C_1$ and $C_2$ such that $$\alpha_1<\alpha_2<\cdots<\alpha_n .$$ The value of $n$ is $\_\_\_\_$ .

Question 2 · difficulty L2 · understanding

If $\theta \in[-2 \pi, 2 \pi]$, then the number of solutions of $2 \sqrt{2} \cos ^2 \theta+(2-\sqrt{6}) \cos \theta-\sqrt{3}=0$, is equal to:

• A. 8
• B. 6
• C. 10
• D. 12

Question 3 · difficulty L2 · understanding

The number of solutions of the equation $4 \sin ^2 x-4 \cos ^3 x+9-4 \cos x=0 ; x \in[-2 \pi, 2 \pi]$ is :

• A. 0
• B. 3
• C. 1
• D. 2

Question 4 · difficulty L2 · understanding

Let $$P = \{ \theta :\sin \theta - \cos \theta = \sqrt 2 \cos \theta \}$$ and $$Q = \{ \theta :\sin \theta + \cos \theta = \sqrt 2 \sin \theta \}$$ be two sets. Then

• A. $$P \subset Q$$ and $$Q - P \ne \emptyset$$
• B. $$Q \not\subset P$$
• C. $$P \not\subset Q$$
• D. $$P = Q$$

Question 5 · difficulty L2 · understanding

The value of $\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ} \cot 70^{\circ}-1\right)$ is

• A. 0
• B. 2/3
• C. 1
• D. 3/2

Question 6 · difficulty L2 · understanding

If $$\sin x=-\frac{3}{5}$$, where $$\pi< x <\frac{3 \pi}{2}$$, then $$80\left(\tan ^2 x-\cos x\right)$$ is equal to

• A. 109
• B. 108
• C. 19
• D. 18

Question 7 · difficulty L2 · understanding

$$96\cos {\pi \over {33}}\cos {{2\pi } \over {33}}\cos {{4\pi } \over {33}}\cos {{8\pi } \over {33}}\cos {{16\pi } \over {33}}$$ is equal to :

• A. 4
• B. 2
• C. 1
• D. 3

Question 8 · difficulty L3 · understanding

If $S=\left\{\theta \in[-\pi, \pi]: \cos \theta \cos \frac{5 \theta}{2}=\cos 7 \theta \cos \frac{7 \theta}{2}\right\}$, then $n(S)$ is equal to $\_\_\_\_$ .

Question 9 · difficulty L3 · understanding

The number of solutions of $$\sin ^2 x+\left(2+2 x-x^2\right) \sin x-3(x-1)^2=0$$, where $$-\pi \leq x \leq \pi$$, is ________.

Question 10 · difficulty L3 · understanding

Let $$S=\left\{\sin ^2 2 \theta:\left(\sin ^4 \theta+\cos ^4 \theta\right) x^2+(\sin 2 \theta) x+\left(\sin ^6 \theta+\cos ^6 \theta\right)=0\right.$$ has real roots $$\}$$. If $$\alpha$$ and $$\beta$$ be the smallest and largest elements of the set $$S$$, respectively, then $$3\left((\alpha-2)^2+(\beta-1)^2\right)$$ equals __________.