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Page 1: 4.1 Angles and Angle Measurekbriggsmath.weebly.com/uploads/2/4/5/0/24504307/4.1...Pre-Calc 12 Reference Angles A reference angle is the angle formed between the terminal arm of the

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 πœƒΒ° 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) 110Β° b) -150Β°

c) 400Β° d) -500Β°

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Pre-Calc 12

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

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. 150Β° b. -210Β° c. 590Β° d. 230Β°

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 590Β° and 230Β° is .

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

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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. 150Β° b. 285Β° c. 22Β° d. -269Β°

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 180Β°?

How many radians in 360Β°?

The symbol Β° 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 Ο€ radians = 180Β°

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

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Pre-Calc 12

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

a. 120Β° b. -315Β° c. 205Β°

What would a look like sketched?

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

a. πœ‹

4 b. βˆ’

7πœ‹

3 c. 1.57

Arc Length

π‘€π‘’π‘Žπ‘ π‘’π‘Ÿπ‘’ π‘œπ‘“ π‘Žπ‘› π‘Žπ‘›π‘”π‘™π‘’ 𝑖𝑛 π‘Ÿπ‘Žπ‘‘π‘–π‘Žπ‘›π‘  = π‘Žπ‘Ÿπ‘ π‘™π‘’π‘›π‘”π‘‘β„Ž

π‘Ÿπ‘Žπ‘‘π‘–π‘’π‘ , or πœƒ =

π‘Ž

π‘Ÿ

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 150Β°.

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