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Pre-Calc 12

4.1 Angles and Angle Measure

Big Idea:

Using inverses is the foundation of solving equations and can be extended to relationships between

functions

Curricular Competencies:

Explore, analyze and apply mathematical ideas

Use inquiry and problem solving to gain understanding

Angles

can be measured in where is one full rotation.

Rotation Angles (in standard position)

A rotation angle is formed by rotating an initial arm through an angle 𝜃° about a fixed point (

)

The angle formed between the arm and the arm is the rotation angle.

A rotation angle in standard position:

Angles in Standard Position

Example 1: Sketch each angle in standard position. State the quadrant in which the angle terminates.

a) 110° b) -150°

c) 400° d) -500°

Pre-Calc 12

Example 2: The point A lies on the terminal arm of the rotation angle 𝜃°. Draw each angle 𝜃°.

a. A(-3,4) b. A(-7,-2)

Co-terminal Angles

Angles in position with the same terminal arm are called co-terminal angles.

Example 3:

a. 150° b. -210° c. 590° d. 230°

Principal Angles

The smallest positive rotation angle with the same terminal arm is called the principal angle. It is

always between and . The principal angle for 590° and 230° is .

The measure of any co-terminal angle with its principal angle can be expresses by 𝜃 ± (360°)𝑛, 𝑛 ∈ 𝑁.

Pre-Calc 12

Reference Angles

A reference angle is the angle formed between the terminal arm of the rotation angle

and the x-axis.

Example 4:

a. 150° b. 285° c. 22° d. -269°

Radian Measure of an Angle

The radian measure of an angle is a ratio that compares the length of an arc of a circle to the radius

of the circle. It is an exact measure.

One radian is equal to

How many radians in 180°?

How many radians in 360°?

The symbol ° following a number means that the angle is measured in . If

there isn’t a unit or symbol after the number, the angle is measured in .

Conversions

Since π radians = 180°

From radian to degree, multiply by From degree to radian, multiply by

Pre-Calc 12

Example 5: Convert from degrees to radians. Give answer in exact values.

a. 120° b. -315° c. 205°

What would a look like sketched?

Example 6: Convert from radians to degrees. Round to nearest hundredth if needed.

a. 𝜋

4 b. −

7𝜋

3 c. 1.57

Arc Length

𝑀𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑎𝑛 𝑎𝑛𝑔𝑙𝑒 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ

𝑟𝑎𝑑𝑖𝑢𝑠, or 𝜃 =

𝑎

𝑟

Example 7: Calculate the arc length (to the nearest tenth of a metre) of a sector of a circle with a

diameter of 9.2m if the sector angle is 150°.

Example 8: A pendulum 30 cm long swings through an arc of 45cm. Through

what angle does the pendulum swing? Answer in both degrees and radians to

the nearest tenth.

Assignment: p 175 1-3, 5-9, 11, 13, 14ac, 17, 18 (do ace where appropriate) I did 9ab, 13 abd