Q1a. What is the supplement of the angle of measure 27°?

Background

Topic: Supplementary Angles

This question tests your understanding of supplementary angles, which are two angles whose measures add up to 180°.

Key Terms and Formula:

• Supplementary angles:

• Given angle:

Step-by-Step Guidance

1. Recall that the supplement of an angle is what you add to it to get .

2. Set up the equation: .

3. Isolate to find the supplement: .

Try solving on your own before revealing the answer!

Q1b. What is the complement of the angle of measure 27°?

Background

Topic: Complementary Angles

This question tests your understanding of complementary angles, which are two angles whose measures add up to 90°.

Key Terms and Formula:

• Complementary angles:

• Given angle:

Step-by-Step Guidance

1. Recall that the complement of an angle is what you add to it to get .

2. Set up the equation: .

3. Isolate to find the complement: .

Try solving on your own before revealing the answer!

Q2a. In which quadrant is the terminal side of a 1182° angle in standard position?

Background

Topic: Angles in Standard Position and Quadrants

This question tests your ability to determine the quadrant of an angle greater than 360° by reducing it to an equivalent angle between 0° and 360°.

Key Terms and Formula:

• Standard position: Angle measured from the positive x-axis

• Quadrants: I (0°–90°), II (90°–180°), III (180°–270°), IV (270°–360°)

• Find coterminal angle:

Step-by-Step Guidance

1. Divide by to find how many full rotations.

2. Calculate the remainder: .

3. The remainder gives the coterminal angle between and .

4. Determine which quadrant this angle falls into based on its value.

Try solving on your own before revealing the answer!

Q2b. Which angle is coterminal to −1363°?

Background

Topic: Coterminal Angles

This question tests your ability to find angles coterminal with a given angle by adding or subtracting multiples of 360°.

Key Terms and Formula:

• Coterminal angles:

• Given angle:

Step-by-Step Guidance

1. Add repeatedly to until the result is between and $360^\circ$.

2. Calculate for integer .

3. Compare the resulting angle to the choices given: a) , b) , c) , d) .

4. Identify which option is coterminal with .

Try solving on your own before revealing the answer!

Q3. The terminal side of angle β passes through the point (−3, −5). Calculate the exact value of the six trigonometric functions of the angle β.

Background

Topic: Trigonometric Functions from Coordinates

This question tests your ability to find sine, cosine, tangent, and their reciprocals given a point on the terminal side of an angle.

Key Terms and Formula:

• Given point:

• Radius:

Step-by-Step Guidance

1. Calculate .

2. Find and .

3. Find .

4. Set up , , .

5. Leave your answers in exact form using radicals.

Try solving on your own before revealing the answer!

Q4. What is the exact value of sin β and cos β if tan β = −7 and the terminal side of β lies in quadrant II?

Background

Topic: Trigonometric Functions and Quadrants

This question tests your ability to find sine and cosine given tangent and quadrant information.

Key Terms and Formula:

• Quadrant II: ,

• ,

Step-by-Step Guidance

1. Let , (since tangent is negative and in quadrant II).

2. Calculate .

3. Find and .

4. Leave your answers in exact form using radicals.

Try solving on your own before revealing the answer!

Q5. Exactly how many radians will the restaurant at the top of the Stratosphere Tower have rotated in 56 minutes?

Background

Topic: Radian Measure and Angular Rotation

This question tests your ability to convert time to radians based on a full revolution.

Key Terms and Formula:

• Full revolution: radians

• Time for full revolution: 60 minutes

• Rotation in minutes:

Step-by-Step Guidance

1. Set up the proportion: of a full revolution.

2. Multiply by to find the radians rotated.

3. Simplify the fraction before multiplying.

Try solving on your own before revealing the answer!