Finding Angles of Rotation

4.1 Radian and Degree Measure

1- Describe Angels

​The word trigonometry means “measurement of triangles.”


An angle is determined by rotating a ray (half-line) about its endpoint.

Standard Position: An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis.

positive angles --> Counterclockwise rotation

negative angles --> clockwise rotation

coterminal --> angles α and β have the same initial and terminal sides. Such angles are coterminal.

Multiple Choice

What does the word "trigonometry" literally mean?

• Rotation of lines

• Calculation of angles

• Measurement of triangles

• Study of circles

Multiple Choice

In standard position, an angle's vertex is located:

• At any point on the coordinate plane

• At the origin (0, 0)

• On the x-axis

• On the y-axis

Multiple Choice

Which statement about angle rotation is correct?

• Positive angles rotate clockwise

• Negative angles rotate counterclockwise

• Positive angles rotate counterclockwise

• All angles rotate in the same direction

Multiple Choice

Coterminal angles are angles that:

• Have the same length

• Form a right angle

• Rotate in opposite directions

• Have the same initial and terminal sides

Multiple Choice

What describes the initial side of an angle in standard position?

• Located on the negative x-axis

• Located on the positive y-axis

• Located on the positive x-axis

• Can be anywhere on the coordinate plane

Radian Measure

Radian: An angle measurement based on the radius

ways of measuring angles

1- Degree Measure:

• ​Full circle = 360 degrees

• Right angle = 90 degrees

• Straight angle = 180 degrees

2- Radian Measure:

• ​Full circle = 2π radians

• Half circle = π radians

A radian is the angle where the arc length equals the radius length

Example 1: Finding Coterminal Angles

​Add or subtract 2π to get coterminal angles Formula: θ ± 2π

Solve: Determine two coterminal angles (one positive and one negative) for each angle.

a) θ = π/4

Multiple Select

Determine two coterminal angles (one positive and one negative) for each angle. (Choose 2 answers)

• $$\frac{8\pi}{3}$$

• $$-\frac{7\pi}{3}$$

• $$\frac{5\pi}{3}$$

• $$-\frac{5\pi}{3}$$

Multiple Select

Determine two coterminal angles (one positive and one negative) for each angle. (Choose 2 answers)

• $$\frac{\pi}{4}$$

• $$-\frac{7\pi}{4}$$

• $$\frac{5\pi}{4}$$

• $$-\frac{5\pi}{4}$$

Multiple Choice

Find (if possible) the complement of $$\frac{\pi}{6}$$

• $$\frac{\pi}{5}$$

• $$\frac{\pi}{3}$$

• $$\frac{5\pi}{6}$$

• none

Multiple Choice

Find (if possible) the supplement of $$\frac{\pi}{6}$$

• $$\frac{\pi}{5}$$

• $$\frac{2\pi}{3}$$

• $$\frac{5\pi}{6}$$

• none

Multiple Choice

Find (if possible) the complement of $$\frac{5\pi}{6}$$

• $$\frac{\pi}{5}$$

• $$\frac{2\pi}{3}$$

• $$\frac{5\pi}{6}$$

• none

Multiple Choice

Find (if possible) the supplement of $$\frac{5\pi}{6}$$

• $$\frac{\pi}{6}$$

• $$\frac{\pi}{4}$$

• $$\frac{\pi}{3}$$

• none

Degree Measure

​A second way to measure angles is in degrees, denoted by the symbol °.

Multiple Choice

Convert each degree measure to radian measure as a multiple of π. Do not use a calculator.

a) 60⁰

• π/4

• π/3

• π/6

• π/9

Multiple Choice

Convert each degree measure to radian measure as a multiple of π. Do not use a calculator.

a) 320⁰

• 14π/9

• 16π/9

• 11π/9

• 17π/9

Multiple Choice

Convert each radian measure to degree measure. Do not use a calculator.

a) π / 6

• 30°

• 60°

• 45°

• 55°

Multiple Choice

Convert each radian measure to degree measure. Do not use a calculator.

a) 5π / 3

• 270°

• 320°

• 280°

• 300°