PreCalc 2 · Trigonometric Functions
Draw the angle, name the quadrant (degrees or radians), read a measure from a picture, and find a positive and a negative coterminal angle.
What standard position means
The vertex is at (0, 0). The initial ray lies on the positive x-axis. The terminal ray is where the angle stops.
Positive angle
A positive angle opens counterclockwise (up from the positive x-axis).
Negative angle
A negative angle opens clockwise (down from the positive x-axis).
More than 360°
An angle can spin more than one full turn. 400° is one extra full turn plus 40°.
Coterminal angles
Coterminal means the angles start on the same initial ray and stop on the same terminal ray. They look like the same picture, but one may have extra spins.
To find a coterminal angle, add or subtract 360° (or 2π radians). You can do that as many times as you want.
Example: 50° and 410° are coterminal because 50° + 360° = 410°.
Quadrants
I: 0° to 90°. II: 90° to 180°. III: 180° to 270°. IV: 270° to 360°. Axis angles are not in a quadrant. The same idea works in radians: 0 to π/2 is I, π/2 to π is II, and so on.
Find the measure from a picture
The marked number is often just a piece from an axis, not the whole angle. Start at the positive x-axis. Use 90°, 180°, 270°, or 360° plus or minus that piece.
Draw a 60° angle in standard position and state the quadrant.
Open 60° counterclockwise from the positive x-axis. That is between 0° and 90°.
Answer: Quadrant I
Draw a 30° angle in standard position and state the quadrant.
30° is between 0° and 90°.
Answer: Quadrant I
Draw a 150° angle in standard position and state the quadrant.
150° is between 90° and 180°.
Answer: Quadrant II
Draw a 5π/3 angle in standard position and state the quadrant.
5π/3 is the same as 300°. That is between 270° and 360°.
Answer: Quadrant IV
Draw a 3π/4 angle in standard position and state the quadrant.
3π/4 is the same as 135°. That is between 90° and 180°.
Answer: Quadrant II
Draw a −π/3 angle in standard position and state the quadrant.
−π/3 is 60° clockwise, which lands in Quadrant IV (same as 300°).
Answer: Quadrant IV
Draw a π/3 angle in standard position and state the quadrant.
π/3 is the same as 60°. That is between 0° and 90°.
Answer: Quadrant I
Draw a −π angle in standard position and state the quadrant.
−π is 180° clockwise, which lands on the negative x-axis.
Answer: on the x-axis
Find the measure of the angle in standard position.
The terminal ray is 50° past the positive y-axis.
90° + 50° = 140°.
Answer: 140°
Find the measure of the positive angle in standard position.
The terminal ray is 35° clockwise from the positive x-axis.
360° − 35° = 325°.
Answer: 325°
Find the measure of the angle in standard position.
The terminal ray is 25° before 180°.
180° − 25° = 155°.
Answer: 155°
Find the measure of the angle in standard position.
The −x-axis is π. Add π/3.
π + π/3 = 4π/3.
Answer: 4π/3
Find the measure of the angle in standard position.
The −y-axis is 270°.
270° + 20° = 290°.
Answer: 290°
Find the measure of the angle in standard position.
The +y-axis is π/2. Add π/6.
π/2 + π/6 = 2π/3.
Answer: 2π/3
Draw a 5π/6 angle in standard position and state the quadrant.
5π/6 is the same as 150°. That is between 90° and 180°.
Answer: Quadrant II
Draw a −2π/3 angle in standard position and state the quadrant.
−2π/3 is 120° clockwise, which lands in Quadrant III.
Answer: Quadrant III
Find one positive coterminal angle and one negative coterminal angle for 40°.
Add 360°: 40° + 360° = 400°.
Subtract 360°: 40° − 360° = −320°.
Answer: 400° and −320°
Find one positive and one negative coterminal angle for 80°.
80° + 360° = 440°.
80° − 360° = −280°.
Answer: 440° and −280°
Find one positive and one negative coterminal angle for −50°.
−50° + 360° = 310°.
−50° − 360° = −410°.
Answer: 310° and −410°
Find one positive and one negative coterminal angle for 5π/6.
5π/6 + 2π = 17π/6.
5π/6 − 2π = −7π/6.
Answer: 17π/6 and −7π/6
Find one positive and one negative coterminal angle for 7π/4.
7π/4 + 2π = 15π/4.
7π/4 − 2π = −π/4.
Answer: 15π/4 and −π/4
Find one positive and one negative coterminal angle for 510°.
510° − 360° = 150° (still positive).
510° − 720° = −210°.
Answer: 150° and −210°
Draw an 11π/4 angle in standard position and state the quadrant.
11π/4 = 2π + 3π/4, so one extra full turn plus 135°.
The terminal ray is in Quadrant II.
Answer: Quadrant II
Find one positive and one negative coterminal angle for 7π/6.
7π/6 + 2π = 19π/6.
7π/6 − 2π = −5π/6.
Answer: 19π/6 and −5π/6
Find one positive and one negative coterminal angle for 405°.
405° − 360° = 45° (still positive).
405° − 720° = −315°.
Answer: 45° and −315°
Find one positive and one negative coterminal angle for 16π/3.
16π/3 − 4π = 16π/3 − 12π/3 = 4π/3.
16π/3 − 6π = 16π/3 − 18π/3 = −2π/3.
Answer: 4π/3 and −2π/3
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