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 πΒ° 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°
Pre-Calc 12
Example 2: The point A lies on the terminal arm of the rotation angle πΒ°. Draw each angle πΒ°.
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 Β± (360Β°)π, π β π.
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 isnβt 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
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
4 b. β
7π
3 c. 1.57
Arc Length
ππππ π’ππ ππ ππ πππππ ππ πππππππ = πππ πππππ‘β
πππππ’π , or π =
π
π
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
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