Case Study / Scenario-Based MCQs for Sub-Topics of Topic 8: Trigonometry

Question 1. A carpenter is building a ramp from the ground to a doorway that is $1.2$ meters high. The ramp makes a right angle with the vertical line from the doorway to the ground. If the angle of elevation of the ramp is $30^\circ$, answer the following questions.

In the right triangle formed by the ramp, the ground, and the vertical line from the doorway, the height of the doorway represents which side relative to the $30^\circ$ angle?

(A) Hypotenuse

(B) Adjacent side

(C) Opposite side

(D) Base

Question 2. Referring to Question 1, which trigonometric ratio relates the height of the doorway ($1.2\ \text{m}$) and the length of the ramp (hypotenuse)?

(A) $\sin 30^\circ$

(B) $\cos 30^\circ$

(C) $\tan 30^\circ$

(D) $\cot 30^\circ$

Question 3. Referring to Question 1, calculate the length of the ramp.

(A) $2.4\ \text{m}$

(B) $1.2\sqrt{3}\ \text{m}$

(C) $1.2/\sqrt{3}\ \text{m}$

(D) $1.2\ \text{m}$

Question 4. Referring to Question 1, calculate the horizontal distance from the foot of the ramp to the point directly below the doorway.

(A) $2.4\ \text{m}$

(B) $1.2\sqrt{3}\ \text{m}$

(C) $1.2/\sqrt{3}\ \text{m}$

(D) $1.2\ \text{m}$

Question 5. Referring to Question 1, what is the sine of the angle between the ramp and the doorway (the angle opposite the horizontal distance)?

(A) $\sin 30^\circ$

(B) $\cos 30^\circ$

(C) $\tan 30^\circ$

(D) $\sin 60^\circ$

Question 1. A square ABCD has a diagonal AC. Consider the triangle ABC. Answer the following questions.

What is the measure of $\angle BAC$?

(A) $30^\circ$

(B) $45^\circ$

(C) $60^\circ$

(D) $90^\circ$

Question 2. Referring to Question 1, if the side length of the square is 'a', what is the length of the diagonal AC in terms of 'a'?

(A) $a$

(B) $a\sqrt{2}$

(C) $a\sqrt{3}$

(D) $2a$

Question 3. Referring to Question 1, what is the value of $\sin (\angle BAC)$?

(A) $\frac{1}{2}$

(B) $\frac{1}{\sqrt{2}}$

(C) $\frac{\sqrt{3}}{2}$

(D) 1

Question 4. Referring to Question 1, consider the angles $\angle BAC$ and $\angle BCA$. What is their relationship?

(A) They are equal.

(B) They are complementary.

(C) They are supplementary.

(D) Their sum is $90^\circ$.

Question 5. Referring to Question 1, based on the complementary angle relationship, what is the relationship between $\sin(\angle BAC)$ and $\cos(\angle BCA)$?

(A) $\sin(\angle BAC) = \cos(\angle BCA)$

(B) $\sin(\angle BAC) = -\cos(\angle BCA)$

(C) $\sin(\angle BAC) = \frac{1}{\cos(\angle BCA)}$

(D) $\sin(\angle BAC) + \cos(\angle BCA) = 1$

Question 1. Consider the expression $E = (\sin \theta + \cos \theta)^2 - 2 \tan \theta \cot \theta$. Simplify this expression using fundamental identities. Answer the following questions.

Expand $(\sin \theta + \cos \theta)^2$.

(A) $\sin^2 \theta + \cos^2 \theta$

(B) $\sin^2 \theta + \cos^2 \theta + \sin \theta \cos \theta$

(C) $\sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta$

(D) $\sin^2 \theta - \cos^2 \theta$

Question 2. Referring to Question 1, what is the value of $\tan \theta \cot \theta$ (where defined)?

(A) $\tan^2 \theta$

(B) $\cot^2 \theta$

(C) 1

(D) 0

Question 3. Referring to Question 1, what is the value of $\sin^2 \theta + \cos^2 \theta$?

(A) $\sin 2\theta$

(B) $\cos 2\theta$

(C) 1

(D) 0

Question 4. Referring to Question 1, substitute the simplified parts back into the original expression $E = (\sin \theta + \cos \theta)^2 - 2 \tan \theta \cot \theta$. What is the simplified form of E?

(A) $1 + 2\sin\theta\cos\theta - 2$

(B) $1 - 2\sin\theta\cos\theta - 2$

(C) $1 + \sin 2\theta - 2$

(D) $2 + \sin 2\theta - 2$

Question 5. Referring to Question 1, what is the final simplified value of E?

(A) 0

(B) -1

(C) $1 + \sin 2\theta$

(D) $\sin 2\theta - 1$

Question 1. A fan blade of length $35\ \text{cm}$ rotates at 300 revolutions per minute. Consider the motion of the tip of the blade. Answer the following questions.

How many revolutions does the blade make in one second?

(A) 5 revolutions

(B) 30 revolutions

(C) 60 revolutions

(D) 300 revolutions

Question 2. Referring to Question 1, how many radians does the blade turn through in one revolution?

(A) $\pi$ radians

(B) $2\pi$ radians

(C) $180$ radians

(D) $360$ radians

Question 3. Referring to Question 1, what is the angular speed of the blade in radians per second?

(A) $5\pi$ rad/s

(B) $10\pi$ rad/s

(C) $300\pi$ rad/s

(D) $600\pi$ rad/s

Question 4. Referring to Question 1, how far does the tip of the blade travel in one second?

(A) $35 \times 10\pi$ cm

(B) $350\pi$ cm

(C) $35$ cm

(D) $10\pi$ cm

Question 5. Referring to Question 1, what is the angle turned by the blade in $0.1$ seconds?

(A) 1 radian

(B) $\pi$ radians

(C) $0.5\pi$ radians

(D) $2\pi$ radians

Question 1. Consider the angle $\theta = \frac{5\pi}{6}$ radians. Answer the following questions based on the unit circle approach.

In which quadrant does the terminal side of the angle $\theta = \frac{5\pi}{6}$ lie?

(A) Quadrant I

(B) Quadrant II

(C) Quadrant III

(D) Quadrant IV

Question 2. Referring to Question 1, what is the reference angle for $\theta = \frac{5\pi}{6}$?

(A) $\frac{5\pi}{6}$

(B) $\frac{\pi}{6}$

(C) $\frac{\pi}{3}$

(D) $\frac{\pi}{2}$

Question 3. Referring to Question 1, what is the value of $\sin (\frac{5\pi}{6})$?

(A) $\frac{1}{2}$

(B) $-\frac{1}{2}$

(C) $\frac{\sqrt{3}}{2}$

(D) $-\frac{\sqrt{3}}{2}$

Question 4. Referring to Question 1, what is the value of $\cos (\frac{5\pi}{6})$?

(A) $\frac{1}{2}$

(B) $-\frac{1}{2}$

(C) $\frac{\sqrt{3}}{2}$

(D) $-\frac{\sqrt{3}}{2}$

Question 5. Referring to Question 1, what is the value of $\tan (\frac{5\pi}{6})$?

(A) $\frac{1}{\sqrt{3}}$

(B) $-\frac{1}{\sqrt{3}}$

(C) $\sqrt{3}$

(D) $-\sqrt{3}$

Question 1. A buoy in the ocean oscillates up and down due to waves. Its displacement from its equilibrium position (in meters) is modeled by the function $y = 1.5 \sin(\frac{\pi}{2} t)$, where $t$ is the time in seconds. Answer the following questions based on this model.

What is the amplitude of the oscillation?

(A) $1\ \text{meter}$

(B) $1.5\ \text{meters}$

(C) $\pi/2\ \text{meters}$

(D) $2\pi\ \text{meters}$

Question 2. Referring to Question 1, what is the period of the oscillation?

(A) $\pi/2\ \text{seconds}$

(B) $\pi\ \text{seconds}$

(C) $2\ \text{seconds}$

(D) $4\ \text{seconds}$

Question 3. Referring to Question 1, what is the displacement of the buoy at $t=1$ second?

(A) $0\ \text{meters}$

(B) $1\ \text{meter}$

(C) $1.5\ \text{meters}$

(D) $-1.5\ \text{meters}$

Question 4. Referring to Question 1, what is the displacement of the buoy at $t=2$ seconds?

(A) $0\ \text{meters}$

(B) $1\ \text{meter}$

(C) $1.5\ \text{meters}$

(D) $-1.5\ \text{meters}$

Question 5. Referring to Question 1, the graph of the function $y = 1.5 \sin(\frac{\pi}{2} t)$ starts at its equilibrium position and moves upwards. Which characteristic of the sine graph does this correspond to?

(A) Passing through $(0,0)$ with increasing slope.

(B) Passing through $(0,0)$ with decreasing slope.

(C) Starting at a maximum value.

(D) Starting at a minimum value.

Question 1. You are asked to calculate the exact value of $\cos 75^\circ$. You can use the identity $\cos(A+B) = \cos A \cos B - \sin A \sin B$. Answer the following questions.

Which pair of standard angles A and B can be used to write $75^\circ = A+B$?

(A) $30^\circ$ and $45^\circ$

(B) $45^\circ$ and $30^\circ$

(C) $60^\circ$ and $15^\circ$

(D) Both A and B

Question 2. Referring to Question 1, if you use $A = 30^\circ$ and $B = 45^\circ$, what are the values of $\cos A, \cos B, \sin A, \sin B$?

(A) $\cos 30^\circ = \frac{\sqrt{3}}{2}, \cos 45^\circ = \frac{1}{\sqrt{2}}, \sin 30^\circ = \frac{1}{2}, \sin 45^\circ = \frac{1}{\sqrt{2}}$

(B) $\cos 30^\circ = \frac{1}{2}, \cos 45^\circ = \frac{1}{\sqrt{2}}, \sin 30^\circ = \frac{\sqrt{3}}{2}, \sin 45^\circ = \frac{1}{\sqrt{2}}$

(C) $\cos 30^\circ = \frac{\sqrt{3}}{2}, \cos 45^\circ = \frac{\sqrt{3}}{2}, \sin 30^\circ = \frac{1}{2}, \sin 45^\circ = \frac{1}{2}$

(D) $\cos 30^\circ = \frac{1}{2}, \cos 45^\circ = \frac{1}{2}, \sin 30^\circ = \frac{\sqrt{3}}{2}, \sin 45^\circ = \frac{\sqrt{3}}{2}$

Question 3. Referring to Question 1, substitute the values from Question 2 into the formula $\cos(A+B) = \cos A \cos B - \sin A \sin B$ to find $\cos 75^\circ$.

(A) $\frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}} - \frac{1}{2} \cdot \frac{1}{\sqrt{2}} = \frac{\sqrt{3}-1}{2\sqrt{2}}$

(B) $\frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}} + \frac{1}{2} \cdot \frac{1}{\sqrt{2}} = \frac{\sqrt{3}+1}{2\sqrt{2}}$

(C) $\frac{1}{2} \cdot \frac{1}{\sqrt{2}} - \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}} = \frac{1-\sqrt{3}}{2\sqrt{2}}$

(D) $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{1}{2} \cdot \frac{1}{2} = \frac{3}{4} - \frac{1}{4} = \frac{1}{2}$

Question 4. You want to simplify the expression $\sin \theta \cos \theta \cos 2\theta$. Which identities can be used for this?

(A) $\sin 2\theta = 2 \sin \theta \cos \theta$

(B) $\cos 2\theta = 2 \cos^2 \theta - 1$

(C) $\sin 4\theta = 2 \sin 2\theta \cos 2\theta$

(D) $\sin^2 \theta + \cos^2 \theta = 1$

Question 5. Referring to Question 4, simplify the expression $\sin \theta \cos \theta \cos 2\theta$.

(A) $\frac{1}{2} \sin 2\theta \cos 2\theta$

(B) $\frac{1}{4} \sin 4\theta$

(C) $\sin 4\theta$

(D) $\frac{1}{2} \sin 4\theta$

Question 1. You are asked to simplify the expression $\sin 80^\circ - \sin 20^\circ$. You can use a sum-to-product formula. Answer the following questions.

Which sum-to-product formula is appropriate for $\sin C - \sin D$?

(A) $2 \sin \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)$

(B) $2 \cos \left(\frac{C+D}{2}\right) \sin \left(\frac{C-D}{2}\right)$

(C) $2 \cos \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)$

(D) $-2 \sin \left(\frac{C+D}{2}\right) \sin \left(\frac{C-D}{2}\right)$

Question 2. Referring to Question 1, apply the chosen formula to $\sin 80^\circ - \sin 20^\circ$. What do you get?

(A) $2 \sin 50^\circ \cos 30^\circ$

(B) $2 \cos 50^\circ \sin 30^\circ$

(C) $2 \cos 50^\circ \cos 30^\circ$

(D) $2 \sin 50^\circ \sin 30^\circ$

Question 3. Referring to Question 1 and 2, substitute the value of $\cos 30^\circ$ and simplify.

(A) $2 \sin 50^\circ \cdot \frac{1}{2} = \sin 50^\circ$

(B) $2 \cos 50^\circ \cdot \frac{1}{2} = \cos 50^\circ$

(C) $2 \cos 50^\circ \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \cos 50^\circ$

(D) $2 \sin 50^\circ \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \sin 50^\circ$

Question 4. You want to express the product $\cos 3x \cos x$ as a sum or difference. Which product-to-sum formula is appropriate?

(A) $2 \sin A \cos B = \sin(A+B) + \sin(A-B)$

(B) $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$

(C) $2 \sin A \sin B = \cos(A-B) - \cos(A+B)$

(D) $2 \cos A \sin B = \sin(A+B) - \sin(A-B)$

Question 5. Referring to Question 4, apply the formula to $\cos 3x \cos x$.

(A) $\frac{1}{2} (\sin 4x + \sin 2x)$

(B) $\frac{1}{2} (\cos 4x + \cos 2x)$

(C) $\frac{1}{2} (\cos 2x - \cos 4x)$

(D) $\frac{1}{2} (\sin 4x - \sin 2x)$

Question 1. A simple pendulum swings such that its horizontal displacement is modeled by $x(t) = A \cos(\omega t)$, where $A$ and $\omega$ are constants. You want to find the times when the displacement is half of its maximum value (i.e., $x(t) = A/2$). This leads to the equation $\cos(\omega t) = 1/2$. Assume $\omega t = \theta$. Answer the following questions about the equation $\cos \theta = 1/2$.

What is the principal value of $\theta$ for $\cos \theta = 1/2$?

(A) $\frac{\pi}{6}$

(B) $\frac{\pi}{4}$

(C) $\frac{\pi}{3}$

(D) $\frac{2\pi}{3}$

Question 2. Referring to Question 1, what is the general solution for $\cos \theta = 1/2$?

(A) $\theta = n\pi + \frac{\pi}{3}, n \in \mathbb{Z}$

(B) $\theta = 2n\pi + \frac{\pi}{3}, n \in \mathbb{Z}$

(C) $\theta = n\pi \pm \frac{\pi}{3}, n \in \mathbb{Z}$

(D) $\theta = 2n\pi \pm \frac{\pi}{3}, n \in \mathbb{Z}$

Question 3. Consider the equation $2 \sin^2 x - 3 \sin x + 1 = 0$. You need to solve this equation for $x$. Answer the following questions.

If you let $y = \sin x$, what is the resulting quadratic equation?

(A) $2y^2 - 3y + 1 = 0$

(B) $2y^2 + 3y + 1 = 0$

(C) $2y^2 - 3y - 1 = 0$

(D) $y^2 - 3y + 2 = 0$

Question 4. Referring to Question 3, solve the quadratic equation for y. What are the values of y?

(A) $y = 1, \frac{1}{2}$

(B) $y = -1, -\frac{1}{2}$

(C) $y = 1, -1$

(D) $y = \frac{1}{2}, -1$

Question 5. Referring to Question 3 and 4, you need to solve for $x$ where $\sin x = 1$ or $\sin x = 1/2$. Which of the following represents the general solutions for $x$?

(A) $x = n\pi + (-1)^n \frac{\pi}{2}$ or $x = n\pi + (-1)^n \frac{\pi}{6}, n \in \mathbb{Z}$

(B) $x = 2n\pi + \frac{\pi}{2}$ or $x = n\pi + (-1)^n \frac{\pi}{6}, n \in \mathbb{Z}$

(C) $x = (2n+1)\frac{\pi}{2}$ or $x = n\pi + (-1)^n \frac{\pi}{6}, n \in \mathbb{Z}$

(D) $x = n\pi + \frac{\pi}{2}$ or $x = 2n\pi + \frac{\pi}{6}$ or $x = (2n+1)\pi - \frac{\pi}{6}, n \in \mathbb{Z}$

Question 1. You are asked to find the principal value of $\cos^{-1} (-\frac{\sqrt{3}}{2})$. Answer the following questions.

What is the principal value branch (range) of $\cos^{-1} x$?

(A) $[-\frac{\pi}{2}, \frac{\pi}{2}]$

(B) $(-\frac{\pi}{2}, \frac{\pi}{2})$

(C) $[0, \pi]$

(D) $(0, \pi)$

Question 2. Referring to Question 1, for what acute angle $\alpha$ is $\cos \alpha = \frac{\sqrt{3}}{2}$?

(A) $\frac{\pi}{6}$

(B) $\frac{\pi}{4}$

(C) $\frac{\pi}{3}$

(D) $0$

Question 3. Referring to Question 1, in which quadrant does the angle whose cosine is negative lie, considering the principal value branch?

(A) Quadrant I

(B) Quadrant II

(C) Quadrant III

(D) Quadrant IV

Question 4. Referring to Question 1, use the relationship between $\cos^{-1} (-x)$ and $\cos^{-1} x$ to find the principal value of $\cos^{-1} (-\frac{\sqrt{3}}{2})$.

(A) $\frac{\pi}{6}$

(B) $\pi - \frac{\pi}{6} = \frac{5\pi}{6}$

(C) $-\frac{\pi}{6}$

(D) $\pi + \frac{\pi}{6} = \frac{7\pi}{6}$

Question 5. You are asked to simplify the expression $\tan^{-1} (1) + \tan^{-1} (2) + \tan^{-1} (3)$. Answer the following questions.

What is the principal value of $\tan^{-1} (1)$?

(A) $0$

(B) $\frac{\pi}{4}$

(C) $\frac{\pi}{2}$

(D) $\pi$

Question 6. Referring to Question 5, use the formula $\tan^{-1} x + \tan^{-1} y = \pi + \tan^{-1} \left( \frac{x+y}{1-xy} \right)$ for $xy > 1, x>0, y>0$ to combine $\tan^{-1} (2) + \tan^{-1} (3)$.

(A) $\pi + \tan^{-1} \left( \frac{2+3}{1-2 \times 3} \right) = \pi + \tan^{-1} \left( \frac{5}{-5} \right) = \pi + \tan^{-1} (-1)$

(B) $\tan^{-1} \left( \frac{2+3}{1-2 \times 3} \right) = \tan^{-1} (-1)$

(C) $\pi - \tan^{-1} (1)$

(D) $\tan^{-1} \left( \frac{2-3}{1+2 \times 3} \right)$

Question 7. Referring to Question 5 and 6, what is the principal value of $\tan^{-1} (-1)$?

(A) $\frac{\pi}{4}$

(B) $-\frac{\pi}{4}$

(C) $\frac{3\pi}{4}$

(D) $-\frac{3\pi}{4}$

Question 8. Referring to Question 5, 6 and 7, what is the final value of $\tan^{-1} (1) + \tan^{-1} (2) + \tan^{-1} (3)$?

(A) $\frac{\pi}{4} + \pi - \frac{\pi}{4} = \pi$

(B) $\frac{\pi}{4} - \frac{\pi}{4} = 0$

(C) $\pi + \frac{\pi}{4} = \frac{5\pi}{4}$

(D) $\frac{\pi}{4} + \pi + \frac{\pi}{4} = \frac{3\pi}{2}$

Question 1. An observer is standing $50\ \text{meters}$ away from the base of a building. He measures the angle of elevation to the top of the building to be $60^\circ$. His eye level is $1.5\ \text{meters}$ above the ground. Answer the following questions.

What is the height from the observer's eye level to the top of the building?

(A) $50 \tan 60^\circ\ \text{m}$

(B) $50 \cos 60^\circ\ \text{m}$

(C) $50 \sin 60^\circ\ \text{m}$

(D) $50 \cot 60^\circ\ \text{m}$

Question 2. Referring to Question 1, calculate the height from the observer's eye level to the top of the building.

(A) $50\ \text{m}$

(B) $50\sqrt{3}\ \text{m}$

(C) $50/\sqrt{3}\ \text{m}$

(D) $25\ \text{m}$

Question 3. Referring to Question 1, what is the total height of the building?

(A) $50\sqrt{3}\ \text{m}$

(B) $50\sqrt{3} + 1.5\ \text{m}$

(C) $50\sqrt{3} - 1.5\ \text{m}$

(D) $50\ \text{m}$

Question 4. A pole stands vertically on the ground. From a point P on the ground, the angle of elevation to the top of the pole is $30^\circ$. From a point Q on the ground 20m closer to the pole than P, the angle of elevation to the top of the pole is $45^\circ$. Let h be the height of the pole and x be the distance of P from the foot of the pole. Answer the following questions.

Express the height h in terms of x using the angle of elevation from point P.

(A) $h = x \sin 30^\circ$

(B) $h = x \cos 30^\circ$

(C) $h = x \tan 30^\circ$

(D) $h = x \cot 30^\circ$

Question 5. Referring to Question 4, express the height h in terms of x and the distance PQ using the angle of elevation from point Q.

(A) $h = (x-20) \sin 45^\circ$

(B) $h = (x-20) \cos 45^\circ$

(C) $h = (x-20) \tan 45^\circ$

(D) $h = (x-20) \cot 45^\circ$

Question 6. Referring to Question 4, use the equations from Question 4 and 5 to find the height h of the pole.

(A) $10\sqrt{3}\ \text{m}$

(B) $20\ \text{m}$

(C) $10(\sqrt{3}+1)\ \text{m}$

(D) $10(\sqrt{3}-1)\ \text{m}$