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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