Q1. Ferris Wheel Problem

Background

Topic: Trigonometric Modeling of Periodic Motion

This question tests your ability to model the height of a rider on a Ferris wheel using trigonometric functions, and to identify amplitude and midline in the context of real-world periodic motion.

Key Terms and Formulas:

• Amplitude: The maximum deviation from the midline.

• Midline: The horizontal line halfway between the maximum and minimum values.

• General form: or

• Period:

Step-by-Step Guidance

1. Identify the radius ( m) and the center height ($22$ m above ground).

2. Determine the amplitude: The rider moves $18$ m above and below the center.

3. The midline is at the height of the center ( m).

4. Write the function for height , considering the period ($40t=0$).

5. Set up the equation using sine or cosine, adjusting for phase shift if needed.

Try solving on your own before revealing the answer!

Q2. Arc Length and Sector Area

Background

Topic: Radian Measure, Arc Length, and Sector Area

This question tests your understanding of the relationship between arc length, central angle, and sector area in a circle.

Key Terms and Formulas:

• Arc length: (where is in radians)

• Degrees to radians:

• Sector area:

Step-by-Step Guidance

1. Given m and m, use to solve for in radians.

2. Convert from radians to degrees using the conversion formula.

3. Use the sector area formula with and to find the area.

4. Set up each calculation, but stop before plugging in the final values.

Try solving on your own before revealing the answer!

Q3. Ladder Against a Building

Background

Topic: Right Triangle Trigonometry

This question tests your ability to use trigonometric ratios to solve for angles and side lengths in a right triangle.

Key Terms and Formulas:

• Pythagorean theorem:

• Sine, cosine, tangent ratios

• Angle of elevation: or

Step-by-Step Guidance

1. Draw the triangle: ladder is hypotenuse ($20 ft), height is unknown.

2. Use the Pythagorean theorem to solve for the height.

3. Use or to find the angle of elevation.

4. Set up the equations for both height and angle, but do not compute the final values.

Try solving on your own before revealing the answer!

Q4. Trigonometric Ratios Given tan θ and sec θ

Background

Topic: Trigonometric Ratios and Quadrants

This question tests your understanding of how the signs of trigonometric functions relate to the quadrant, and how to find all six ratios given one.

Key Terms and Formulas:

• Signs of trigonometric functions in each quadrant

Step-by-Step Guidance

1. Given and , determine which quadrant is in.

2. Draw a reference triangle with opposite and adjacent sides reflecting the tangent ratio.

3. Use the Pythagorean theorem to find the hypotenuse.

4. Determine the signs of all ratios based on the quadrant.

5. Set up the expressions for all six trigonometric ratios, but stop before simplifying fully.

Try solving on your own before revealing the answer!

Q5. Exact Value of tan(sin⁻¹(−√3/3))

Background

Topic: Inverse Trigonometric Functions and Composite Functions

This question tests your ability to evaluate composite trigonometric expressions involving inverse functions.

Key Terms and Formulas:

• gives an angle whose sine is

• for a given

• Reference triangle construction

Step-by-Step Guidance

1. Let , so .

2. Draw a triangle with opposite side and hypotenuse $3$.

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

4. Set up .

Try solving on your own before revealing the answer!

Q6. Solve Triangle ABC Given a = 8, b = 11, c = 14

Background

Topic: Law of Cosines and Law of Sines

This question tests your ability to solve a triangle given all three sides (SSS), including finding all angles and sketching the triangle.

Key Terms and Formulas:

• Law of Cosines:

• Law of Sines:

Step-by-Step Guidance

1. Use the Law of Cosines to find one angle (e.g., angle ).

2. Use the Law of Sines to find the other angles.

3. Set up the equations for each angle, but stop before calculating the numeric values.

4. Sketch the triangle based on the side lengths.

Try solving on your own before revealing the answer!

Q7. Exact Values and Comparison

Background

Topic: Trigonometric Function Values and Properties

This question tests your understanding of how trigonometric functions behave under scaling and multiplication.

Key Terms and Formulas:

• Double-angle and half-angle identities

Step-by-Step Guidance

1. Given , calculate .

2. Calculate .

3. Compare the two values to see if they are equal.

4. Set up the expressions, but stop before evaluating the numeric values.

Try solving on your own before revealing the answer!

Q8. Write cos(sin⁻¹ x − tan⁻¹ y) in terms of x and y only

Background

Topic: Trigonometric Expressions and Inverse Functions

This question tests your ability to rewrite composite trigonometric expressions in terms of variables using identities and reference triangles.

Key Terms and Formulas:

• Let ,

• Use

• Reference triangle relationships

Step-by-Step Guidance

1. Let , so , .

2. Let , so , , .

3. Use the cosine difference identity to write in terms of and .

4. Set up the expression, but stop before simplifying fully.

Try solving on your own before revealing the answer!

Q9. Write sin(2 tan⁻¹ x) as an algebraic expression in x

Background

Topic: Double-Angle Identities and Inverse Functions

This question tests your ability to use double-angle identities and reference triangles to rewrite trigonometric expressions in terms of .

Key Terms and Formulas:

• Let

• Double-angle identity:

• Reference triangle relationships

Step-by-Step Guidance

1. Let , so .

2. From a reference triangle, , .

3. Use the double-angle identity to write in terms of .

4. Set up the expression, but stop before simplifying fully.

Try solving on your own before revealing the answer!

Q10. Prove Trigonometric Identities

Background

Topic: Trigonometric Identities and Proofs

This question tests your ability to manipulate and prove trigonometric identities using algebraic and trigonometric properties.

Key Terms and Formulas:

• Basic trigonometric identities

• Reciprocal identities: ,

• Sum and difference formulas

Step-by-Step Guidance

1. For each identity, start by expressing all terms in sine and cosine.

2. Simplify the numerator and denominator as appropriate.

3. Look for opportunities to factor or combine terms.

4. Set up the proof steps, but stop before completing the final simplification.

Try solving on your own before revealing the answer!

Q11. Solve Trigonometric Equations

Background

Topic: Solving Trigonometric Equations

This question tests your ability to solve trigonometric equations for all solutions in a given interval and to state the general solution.

Key Terms and Formulas:

• Trigonometric equations: , , ,

• General solution:

• Quadratic equations in trigonometric functions

Step-by-Step Guidance

1. For each equation, isolate the trigonometric function.

2. Solve for the function value (e.g., , ).

3. Find all solutions in .

4. Set up the general solution, but stop before listing all values.

Try solving on your own before revealing the answer!