Download 9.8: Reference Angles

Section 9.8: Reference Angles
9.8:
Reference Angles
9.8: Reference Angles
•
Associated with every angle drawn in the standard position
(except quadrantal angles) there is another angle called the
reference angle.
angle
•
The reference angle is the positive acute
angle formed by the terminal side of the
given angle and the xx-axis.
axis. Reference
angles may appear in all four quadrants.
Angles in quadrant I are their own reference
angles.
• Remember:
Remember: The reference
angle is measured from the
terminal side to the xx-axis.
(NOT THE Y-AXIS!)
Y AXIS!)
Chapter 9: Trig Part I
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Section 9.8: Reference Angles
Remember:
Remember: The reference angle is measured
from the terminal side to the x-axis.
x axis.
(NOT THE Y-AXIS!)
Y AXIS!)
Remember:
Remember: The reference angle is measured
from the terminal side to the x-axis.
x axis.
(NOT THE Y-AXIS!)
Y AXIS!)
Chapter 9: Trig Part I
2
Section 9.8: Reference Angles
Remember:
Remember: The reference angle is measured
from the terminal side to the x-axis.
x axis.
(NOT THE Y-AXIS!)
Y AXIS!)
Remember:
Remember: The reference angle is measured
from the terminal side to the x-axis.
x axis.
(NOT THE Y-AXIS!)
Y AXIS!)
Chapter 9: Trig Part I
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Section 9.8: Reference Angles
Examples of
Reference Angles
Examples: Draw each angle in standard
position and identify the reference angle.
1. 120°
Chapter 9: Trig Part I
2. 330°
4
Section 9.8: Reference Angles
Examples: Draw each angle in standard
position and identify the reference angle.
3. 150°
4. −45°
5. Write cos 280º as a function of a
positive acute angle. (Multiple Choice)
a) cos 10° b) cos 80° c) cos 85°
Chapter 9: Trig Part I
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Section 9.8: Reference Angles
6. The value of cos 390º is equal to
which of the following?
a) sin 30° b) sin 60° c) -sin 30°
7. Write tan (-110º) as a function of a
positive acute angle. (Multiple Choice)
a) tan 10° b) tan 20° c) tan 70°
Chapter 9: Trig Part I
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Section 9.8: Reference Angles
More Examples of
Reference Angles
8. For each indicated value and
quadrant, find the angle measure to the
nearest degree, on the interval [0,360).
(Always use the positive ratio measure when putting into calculator.)
tanθ = −4.7047 Quadrant II
102°
cos θ = 0.9816 Quadrant IV
349°
sinθ = −0.2756 Quadrant IV
344°
Chapter 9: Trig Part I
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Section 9.8: Reference Angles
Finding Exact Values of Trigonometric Functions
2. Find the exact value of each expression:
sin 300°
Step 1: Sketch the angle and determine
the measure of the reference angle.
sin 60°
Finding Exact Values of Trigonometric Functions
2. Find the exact value of each expression:
sin 300°
Step 2: Determine the sign of the
function (positive or negative).
sin 300°
− sin 60°
Chapter 9: Trig Part I
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Section 9.8: Reference Angles
Finding Exact Values of Trigonometric Functions
2. Find the exact value of each expression:
sin 300°
Step 3: Evaluate the function.
[Use special right triangles or the
unit circle]
− sin 60°
3
=−
2
3. Find the exact value of each expression:
cos 135°
− cos 45°
=−
Chapter 9: Trig Part I
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2
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Section 9.8: Reference Angles
4. Find the exact value of each expression:
tan 240 °
+ tan 60°
= 3
5. Find the exact value of each expression:
sin( −135)°
− sin 45°
=−
Chapter 9: Trig Part I
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2
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Section 9.8: Reference Angles
6. Find the exact value of each expression:
cos(−210°)
− cos 30°
=−
3
2
7. Find the exact value of each expression:
tan 330°
− tan 30°
=−
Chapter 9: Trig Part I
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3
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