Trigonometry Final Exam Review – Step-by-Step Guidance

Q1. The wheels on a truck are spinning at an angular speed of 200 rotations per minute. If the radius of the wheel is 13 inches, how fast is the truck driving (in mph)?

Background

Topic: Angular and Linear Velocity

This question tests your ability to relate angular speed (rotations per minute) to linear speed (miles per hour) using the radius of a wheel.

Key Terms and Formulas

• Angular speed ($\omega$): How fast something rotates, usually in radians per unit time.

• Linear speed ($v$): How fast a point on the edge of the wheel moves along the ground.

• Relationship: $v = r\omega$

• 1 rotation = $2\pi$ radians

Step-by-Step Guidance

1. Convert the angular speed from rotations per minute to radians per minute: $200 \text{ rotations/min} \times 2\pi \text{ radians/rotation}$.

2. Calculate the linear speed in inches per minute using $v = r\omega$, where $r = 13$ inches.

3. Convert the linear speed from inches per minute to miles per hour. (Hint: 1 mile = 63,360 inches, 1 hour = 60 minutes.)

Try solving on your own before revealing the answer!

Q2. A support cable is attached to a utility pole at a height of 10 feet above the ground. The surrounding area is flat, the pole is perpendicular to the ground, and the cable forms a 50° angle with the ground. How long is the cable?

Background

Topic: Right Triangle Trigonometry

This question tests your ability to use trigonometric ratios to solve for the length of a side in a right triangle.

Key Terms and Formulas

• Sine, cosine, and tangent ratios

• For a right triangle: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$

Step-by-Step Guidance

1. Draw a right triangle representing the situation, labeling the height (opposite side) as 10 ft and the angle with the ground as 50°.

2. Identify which trigonometric ratio relates the height and the cable length (hypotenuse).

3. Set up the equation using $\sin 50^\circ = \frac{10}{\text{cable length}}$.

4. Rearrange the equation to solve for the cable length.

Try solving on your own before revealing the answer!

Q3. Suppose $\theta$ is in Quadrant 2 such that $\sin \theta = \frac{2}{3}$. Find the value of the other five trig functions at $\theta$.

Background

Topic: Trigonometric Functions and Reference Triangles

This question tests your understanding of how to find all six trigonometric functions given one value and the quadrant.

Key Terms and Formulas

• Pythagorean Identity: $\sin^2\theta + \cos^2\theta = 1$

• Definitions of $\sin, \cos, \tan, \csc, \sec, \cot$

• Signs of trig functions in Quadrant II

Step-by-Step Guidance

1. Draw a reference triangle for $\theta$ in Quadrant II, labeling the opposite side as 2 and hypotenuse as 3.

2. Use the Pythagorean theorem to find the adjacent side (remember the sign in Quadrant II).

3. Write the values for $\cos \theta$ and $\tan \theta$ using the triangle sides.

4. Find the reciprocal functions: $\csc \theta, \sec \theta, \cot \theta$.

Try solving on your own before revealing the answer!

Q4. Calculate $\cos\left(-\frac{8\pi}{3}\right)$ by hand, using reference angles and memorized trig values.

Background

Topic: Evaluating Trig Functions Using Reference Angles

This question tests your ability to reduce an angle to a coterminal angle and use reference angles to find exact values.

Key Terms and Formulas

• Reference angle

• Coterminal angles: Add or subtract $2\pi$ as needed

• Unit circle values for cosine

Step-by-Step Guidance

1. Find a coterminal angle for $-\frac{8\pi}{3}$ between $0$and $2\pi$$2\pi$ by adding $2\pi$ as needed.

2. Determine the reference angle for the resulting angle.

3. Use the unit circle to find the exact value of cosine for that reference angle, considering the sign based on the quadrant.

Try solving on your own before revealing the answer!

Q5. Sketch the graph of $y = -2 \sin\left(\frac{2}{3}x + \frac{\pi}{2}\right) + 1$

Background

Topic: Graphing Sine Functions

This question tests your understanding of amplitude, phase shift, vertical shift, and period for sine functions.

Key Terms and Formulas

• General form: $y = a \sin(bx + c) + d$

• Amplitude: $|a|$

• Period: $\frac{2\pi}{|b|}$

• Phase shift: $-\frac{c}{b}$

• Vertical shift: $d$

Step-by-Step Guidance

1. Identify the amplitude, period, phase shift, and vertical shift from the equation.

2. Determine the key points for one period of the sine function.

3. Apply the transformations (reflection, shift, stretch/compression) to sketch the graph.

Try sketching the graph before revealing the answer!

Q6. Sketch the graph of $y = -2 \tan x$

Background

Topic: Graphing Tangent Functions

This question tests your ability to graph tangent functions with amplitude changes and reflections.

Key Terms and Formulas

• General form: $y = a \tan(bx + c) + d$

• Period of tangent: $\frac{\pi}{|b|}$

• Vertical asymptotes at $x = \frac{\pi}{2} + n\pi$ for $y = \tan x$

Step-by-Step Guidance

1. Identify the amplitude and reflection from the coefficient $-2$.

2. Determine the period and location of vertical asymptotes.

3. Plot key points and sketch one period of the graph, showing the effect of the negative amplitude.

Try sketching the graph before revealing the answer!

Q7. Calculate the exact value of $\cos\left(\sin^{-1}\left(-\frac{3}{4}\right)\right)$ by hand.

Background

Topic: Inverse Trigonometric Functions and Right Triangles

This question tests your ability to evaluate composite trig functions using right triangle relationships.

Key Terms and Formulas

• If $\theta = \sin^{-1}(x)$, then $\sin \theta = x$

• Pythagorean theorem to find the adjacent side

• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

Step-by-Step Guidance

1. Let $\theta = \sin^{-1}\left(-\frac{3}{4}\right)$, so $\sin \theta = -\frac{3}{4}$.

2. Draw a right triangle with opposite side $-3$ and hypotenuse $4$(since sine is negative,$\theta$ is in Quadrant IV).

3. Use the Pythagorean theorem to find the adjacent side.

4. Write $\cos \theta$ as $\frac{\text{adjacent}}{4}$.

Try solving on your own before revealing the answer!

Q8. Verify the identity: $\csc^2 x - \sec^2 x = \frac{4 \cos^2 x}{\sin^4 x}$

Background

Topic: Trigonometric Identities

This question tests your ability to manipulate and verify trigonometric identities using algebraic techniques.

Key Terms and Formulas

• $\csc x = \frac{1}{\sin x}$, $\sec x = \frac{1}{\cos x}$

• Pythagorean identities

Step-by-Step Guidance

1. Rewrite $\csc^2 x$ and $\sec^2 x$ in terms of sine and cosine.

2. Combine the terms over a common denominator.

3. Simplify the numerator and denominator to match the right side of the identity.

Try verifying the identity before revealing the answer!

Q9. Use a sum or difference formula to calculate the exact value of $\sin 345^\circ$

Background

Topic: Sum and Difference Formulas

This question tests your ability to use the sine sum or difference identity to find exact values.

Key Terms and Formulas

• $\sin(A - B) = \sin A \cos B - \cos A \sin B$

• Special angles: $360^\circ, 0^\circ, 45^\circ, 30^\circ, 60^\circ$

Step-by-Step Guidance

1. Express $345^\circ$ as a difference of two special angles (e.g., $345^\circ = 360^\circ - 15^\circ$ or $345^\circ = 300^\circ + 45^\circ$).

2. Apply the sine sum or difference formula.

3. Substitute the exact values for sine and cosine of the special angles.

Try solving on your own before revealing the answer!

Q10. If $\theta$ is in Quadrant 3 with $\tan \theta = 2$, find the exact value of:

• a. $\cos \theta$

• b. $\sin 2\theta$

• c. $\cos \frac{\theta}{2}$

• d. $\tan 2\theta$

Background

Topic: Trigonometric Functions, Double-Angle and Half-Angle Formulas

This question tests your ability to use right triangle relationships and double/half-angle identities.

Key Terms and Formulas

• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

• Pythagorean theorem

• Double-angle formulas: $\sin 2\theta = 2 \sin \theta \cos \theta$, $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$

• Half-angle formula for cosine: $\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$

• $\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$

Step-by-Step Guidance

1. Draw a reference triangle in Quadrant III with $\tan \theta = 2$ (choose sides accordingly, remembering signs in Quadrant III).

2. Find $\sin \theta$ and $\cos \theta$ using the triangle.

3. Use the double-angle and half-angle formulas to set up expressions for each part.

Try solving on your own before revealing the answer!

Q11. Solve the equation. Write all solutions between $0$and $2\pi$: $2 \cos^2 x + \sin x = 2$

Background

Topic: Solving Trigonometric Equations

This question tests your ability to solve trig equations by expressing all terms in terms of one function and factoring.

Key Terms and Formulas

• Pythagorean identity: $\sin^2 x + \cos^2 x = 1$

• Factoring quadratic equations

Step-by-Step Guidance

1. Express $\cos^2 x$ in terms of $\sin^2 x$ or vice versa, if needed.

2. Rearrange the equation to set it equal to zero.

3. Factor the equation and solve for $x$ in the given interval.

Try solving on your own before revealing the answer!

Q12. Solve the equation. Write all solutions between $0$and $2\pi$: $\cos 3x = -\frac{\sqrt{3}}{2}$

Background

Topic: Solving Trigonometric Equations

This question tests your ability to solve equations involving multiple angles and to find all solutions in a given interval.

Key Terms and Formulas

• Unit circle values for cosine

• General solution for $\cos \alpha = k$

Step-by-Step Guidance

1. Find all angles $\alpha$ where $\cos \alpha = -\frac{\sqrt{3}}{2}$.

2. Set $3x = \alpha$ and solve for $x$ in $[0, 2\pi)$.

3. List all solutions for $x$ in the required interval.

Try solving on your own before revealing the answer!

Q13. Solve the triangle: $A = 35^\circ$, $a = 8$, $b = 11$

Background

Topic: Law of Sines

This question tests your ability to solve for unknown sides and angles in a triangle using the Law of Sines.

Key Terms and Formulas

• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Step-by-Step Guidance

1. Set up the Law of Sines using the given values.

2. Solve for $B$ using $\frac{a}{\sin A} = \frac{b}{\sin B}$.

3. Find $C$ using the triangle angle sum.

4. Solve for $c$ using the Law of Sines.

Try solving on your own before revealing the answer!

Q14. Solve the triangle: $B = 62^\circ$, $a = 10$, $c = 12$

Background

Topic: Law of Sines or Law of Cosines

This question tests your ability to solve for unknown sides and angles in a triangle using the Law of Sines or Cosines.

Key Terms and Formulas

• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

• Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos C$

Step-by-Step Guidance

1. Determine which law to use first based on the given information.

2. Set up the appropriate equation to solve for one of the unknowns (angle or side).

3. Continue solving for the remaining sides and angles using the Law of Sines or Cosines as needed.

Try solving on your own before revealing the answer!