Q1. Find the exact value for each trigonometric function:

• sin()

• sec()

• csc()

• tan()

• cos()

• cos()

• tan()

• csc()

• sec()

• sin()

Background

Topic: Exact Trigonometric Values

This question tests your ability to find exact values of trigonometric functions for standard angles, often using the unit circle or reference triangles.

Key Terms and Formulas:

• Unit Circle: Used to determine sine, cosine, tangent, secant, cosecant, and cotangent values for common angles.

• Reference Angle: The acute angle formed with the x-axis, used to find function values for angles outside the first quadrant.

• Trigonometric Function Definitions:

Step-by-Step Guidance

1. For each angle, identify its location on the unit circle and determine its reference angle.

2. Recall the exact values for sine, cosine, and tangent for common angles such as , , , etc.

3. Use the sign of the function based on the quadrant in which the angle lies.

4. For reciprocal functions (csc, sec, cot), use the reciprocal of the corresponding sine, cosine, or tangent value.

5. For angles greater than or negative angles, reduce them to an equivalent angle between $0 by subtracting or adding as needed.

Try solving on your own before revealing the answer!

Q2. Circle Motion: Find angle, arc length, and linear speed for a point P on a circle.

• Given: radius , angular speed , time .

• Find: (a) angle generated in time , (b) distance traveled along the circle, (c) linear speed.

• Example: cm, radians/sec, sec

Background

Topic: Circular Motion and Angular/Linear Speed

This question tests your ability to relate angular speed, radius, and time to find the angle swept, arc length, and linear speed.

Key Terms and Formulas:

• Angle generated:

• Arc length:

• Linear speed:

Step-by-Step Guidance

1. Calculate the angle generated by multiplying angular speed by time: .

2. Find the arc length by multiplying the radius by the angle (in radians): .

3. Determine the linear speed by multiplying the radius by the angular speed: .

4. Check units to ensure consistency (radius in cm, time in sec, angle in radians).

5. Set up the calculations for each part, but stop before plugging in the final values.

Try solving on your own before revealing the answer!

Q3. Find angular speed for various objects.

• (1) Hour hand of a clock

• (2) Gear revolving 300 times per minute

• (3) Minute hand of a clock

• (4) Gear revolving 120 times per half-hour

Background

Topic: Angular Speed

This question tests your ability to convert revolutions or time intervals into angular speed in radians per unit time.

Key Terms and Formulas:

• Angular speed:

• One revolution = radians

• For revolutions per unit time: radians per unit time

Step-by-Step Guidance

1. Identify the number of revolutions and the time interval for each scenario.

2. Convert revolutions to radians using radians per revolution.

3. Divide the total angle swept by the time interval to find angular speed.

4. Set up the formula for each case, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q4. Pulley and Bicycle Motion Problems

• (1) Two pulleys: Large radius 15 cm, small radius 8 cm, large rotates 24 times/sec. Find angular speed of each.

• (2) Bicycle tire: radius 13.0 in, 275 rev/min. Find speed in mph and rev/min needed for 30 mph.

Background

Topic: Rotational Motion and Linear Speed

This question tests your ability to relate angular speed, radius, and revolutions to linear speed and conversions between units.

Key Terms and Formulas:

• Angular speed: radians/sec

• Linear speed:

• Unit conversions: inches to miles, minutes to hours

Step-by-Step Guidance

1. For pulleys, calculate angular speed for the large pulley using radians/sec.

2. Use the relationship between the pulleys (belt speed is equal) to find the angular speed of the smaller pulley.

3. For the bicycle, calculate the linear speed using and convert units to mph.

4. Set up the equation to solve for the required revolutions per minute for 30 mph.

5. Stop before plugging in the final values.

Try solving on your own before revealing the answer!

Q5. Arc Length and Sector Area Applications

• (1) Rope wound around drum: radius 0.764 ft, angle 37.2°

• (2) Gears: smaller gear rotates 230°, find larger gear's rotation

• (3) Irrigation system: radius 251 m, angle 15°, find area

• (4) Bicycle: wheel radius 15.0 in, pedal gear radius 4.80 in, rear gear radius 1.50 in, pedal rotates 270°, find distance moved

Background

Topic: Arc Length and Sector Area

This question tests your ability to use arc length and sector area formulas, and apply them to real-world scenarios.

Key Terms and Formulas:

• Arc length: (with in radians)

• Area of sector:

• Convert degrees to radians:

Step-by-Step Guidance

1. Convert the given angle from degrees to radians using .

2. For arc length, multiply the radius by the angle in radians.

3. For sector area, use with the angle in radians.

4. For gear and bicycle problems, relate the rotation angle to the distance traveled using arc length.

5. Set up the calculations, but stop before plugging in the final values.

Try solving on your own before revealing the answer!

Q6. Graph the following trigonometric functions over one period.

• Examples: sin(2x), cos(), 2sin(2πx), -sin(3x), csc(2x), sec(3x), tan(2x), cot(3x), etc.

Background

Topic: Graphs of Trigonometric Functions

This question tests your ability to graph trigonometric functions, including amplitude, period, phase shift, and vertical shift.

Key Terms and Formulas:

• Period of sine/cosine: for

• Amplitude:

• Phase shift:

• Vertical shift:

Step-by-Step Guidance

1. Identify the amplitude, period, phase shift, and vertical shift for each function.

2. Calculate the period using for sine/cosine, for tangent/cotangent.

3. Determine the phase shift by solving .

4. Sketch the base graph and apply the transformations step by step.

5. Set up the axes and mark key points, but stop before drawing the full graph.

Try sketching the graphs on your own before revealing the answer!