Transcript Angles, Degrees, and Special Triangles

Reference Angle
Trigonometry
MATH 103
S. Rook
Overview
• Section 3.1 in the textbook:
– Reference angle
– Reference angle theorem
– Approximating with the calculator
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Reference Angle
Reference Angle
• One of the most important definitions in this
class is the reference angle
– Allows us to calculate ANY angle θ using an
equivalent positive acute angle
• We can now work in all four quadrants of the Cartesian
Plane instead of just Quadrant I!
• Reference angle: the positive acute angle
that lies between the terminal side of θ and
the x-axis
θ MUST be in standard position
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Reference Angle Examples –
Quadrant I
Note that both θ
and the reference
angle are 60°
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Reference Angle Examples –
Quadrant II
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Reference Angle Examples –
Quadrant III
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Reference Angle Examples –
Quadrant IV
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Reference Angle Summary
• Depending in which quadrant θ terminates, we can
formulate a general rule for finding reference angles:
– For any positive angle θ, 0° ≤ θ ≤ 360°:
• If θ Є QI:
Ref angle = θ
• If θ Є QII:
Ref angle = 180° – θ
• If θ Є QIII:
Ref angle = θ – 180°
• If θ Є QIV:
Ref angle = 360° – θ
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Reference Angle Summary
(Continued)
– If θ > 360°:
• Keep subtracting 360° from θ until 0° ≤ θ ≤ 360°
• Go back to the first step on the previous slide
– If θ < 0°:
• Keep adding 360° to θ until 0° ≤ θ ≤ 360°
• Go back to the first step on the previous slide
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Reference Angle (Example)
Ex 1: Draw each angle in standard position and
then name the reference angle:
a)
b)
c)
d)
e)
210°
101°
543°
-342°
-371°
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Reference Angle Theorem
Relationship Between Trigonometric
Functions with Equivalent Values
• Consider the value of cos 60° and the value of
cos 120°:
cos 60° = ½ (Should have this MEMORIZED!)
cos 120° = -½ (From Definition I with 1, 3  and
30° – 60° – 90° triangle)
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Relationship Between Trigonometric
Functions with Equivalent Values
(Continued)
• What is the reference angle of 120°?
60°
• Need to adjust the final answer depending on
which quadrant θ terminates in:
120° terminates in QII AND cos θ is negative in QII
• Therefore, cos 120° = -cos 60° = -½
– The VALUES are the same – just the signs are
different!
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Reference Angle Theorem
• Reference Angle Theorem: the value of a
trigonometric function of an angle θ is
EQUIVALENT to the VALUE of the
trigonometric function of its reference angle
– The ONLY thing that may be different is the sign
• Determine the sign based on the trigonometric
function and which quadrant θ terminates in
– The Reference Angle Theorem is the reason why
we need to memorize the exact values of 30°, 45°,
and 60° only in Quadrant I!
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Reference Angle Summary
• Recall:
– For any positive angle θ, 0° ≤ θ ≤ 360°
• If θ Є QI:
Ref angle = θ
• If θ Є QII:
Ref angle = 180° – θ
• If θ Є QIII:
Ref angle = θ – 180°
• If θ Є QIV:
Ref angle = 360° – θ
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Reference Angle Summary
(Continued)
– If θ > 360°:
• Keep subtracting 360° from θ until 0° ≤ θ ≤ 360°
• Go back to the first step
– If θ < 0°:
• Keep adding 360° to θ until 0° ≤ θ ≤ 360°
• Go back to the the first step
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Reference Angle Theorem
(Example)
Ex 2: Use reference angles to find the exact
value of the following:
a) cos 135°
b) tan 315°
c) sec(-60°)
d) cot 390°
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Approximating with the
Calculator
Approximating Angles
• Recall in Section 2.2 that we used sin-1, cos-1,
and tan-1 to derive acute angles in the first
quadrant
– The Inverse Trigonometric Functions
• Unlike the trigonometric functions, the Inverse
Trigonometric Functions CANNOT be used to
approximate every angle
– We will see why when we cover the Inverse
Trigonometric Functions in detail later
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Approximating Angles (Continued)
• To circumvent this problem, we can use
reference angles:
– Find the reference angle that corresponds to the
given value of a trigonometric function:
• Recall that a reference angle is a positive acute angle
which terminates in QI
• Because cos θ and sin θ are both positive in QI, always
use the POSITIVE value of the trigonometric function
– Apply the reference angle by utilizing the
quadrant in which θ terminates
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Approximating Angles (Example)
Ex 3: Use a calculator to approximate θ if
0° < θ < 360° and:
a)
b)
c)
d)
cos θ = 0.0644, θ Є QIV
tan θ = 0.5890, θ Є QI
sec θ = -3.4159, θ Є QII
csc θ = -1.7876, θ Є QIII
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Summary
• After studying these slides, you should be able to:
– Calculate the correct reference angle for any angle θ
– Evaluate trigonometric functions using reference
angles
– Use a calculator and reference angles to approximate
an angle θ given the quadrant in which it terminates
• Additional Practice
– See the list of suggested problems for 3.1
• Next lesson
– Radians and Degrees (Section 3.2)
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