Transcript Angles, Degrees, and Special Triangles

Trigonometric Functions of Any
Angle
MATH 109 - Precalculus
S. Rook
Overview
• Section 4.4 in the textbook:
– Trigonometric functions of any angle
– Reference angles
– Trigonometric functions of real numbers
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Trigonometric Functions of any
Angle
Trigonometric Functions of Any
Angle
• Given an angle θ in standard position and a
point (x, y) on the terminal side of θ, then the
six trigonometric functions of ANY ANGLE θ
are can be defined in terms of x, y, and the
length of the line connecting the origin and
(x, y) denoted as r
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Trigonometric Functions of any
Angle (Continued)
Function
Abbreviation
Definition
y
r
x
The cosine of θ
cos θ
r
y
The tangent of θ
tan θ
,x  0
x
x
The cotangent of θ
cot θ
,y0
y
r
The secant of θ
sec θ
,x  0
x
r
The cosecant of θ
csc θ
,y0
y
Where r  x 2  y 2 and x and y retain their
signs from (x, y)
The sine of θ
sin θ
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Trigonometric Functions of any
Angle (Continued)
Function
Abbreviation
Definition
y
r
x
The cosine of θ
cos θ
r
y
The tangent of θ
tan θ
,x  0
x
x
The cotangent of θ
cot θ
,y0
y
r
The secant of θ
sec θ
,x  0
x
r
The cosecant of θ
csc θ
,y0
y
Where r  x 2  y 2 and x and y retain their
signs from (x, y)
The sine of θ
sin θ
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Algebraic Signs of Trigonometric
Functions
• The sign of the six trigonometric functions
depends on which quadrant θ terminates in:
r is the distance from the origin to (x, y) so it is
ALWAYS positive
– The signs of x and y depend on which quadrant
(x, y) lies
– Remember the shorthand notation involving “the
element of” symbol:
• i.e.   QIV means theta is a standard angle which
terminates in Q IV
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Algebraic Signs of Trigonometric
Functions (Continued)
Functions
sin  
y
r
cos  
x
r
y
tan  
x
θ Є QI
θ Є QII
θ Є QIII
θ Є QIV
and
csc 
r
y
+
+
–
–
and
sec  
r
x
+
–
–
+
and
cot 
x
y
+
–
+
–
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Trigonometric Functions of any
Angle (Example)
Ex 1: Find the value of all six trigonometric
functions if:
a) (-1, 2) lies on the terminal side of θ
b) (-7, -1) lies on the terminal side of θ
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Trigonometric Functions of any
Angle (Example)
Ex 2: Given sec θ = -3⁄2 where cos θ < 0, find the
exact value of tan θ and csc θ
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Reference Angles
Reference Angles
• An important definition 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: denoted θ’, 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 Angles Examples –
Quadrant I
Note that both θ
and θ’ are 60°
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Reference Angles Examples –
Quadrant II
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Reference Angles Examples –
Quadrant III
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Reference Angles Examples –
Quadrant IV
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Reference Angles 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:
θ’ = θ
• If θ Є QII:
θ‘ = 180° – θ
• If θ Є QIII:
θ‘ = θ – 180°
• If θ Є QIV:
θ’ = 360° – θ
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Reference Angles 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
– If θ is in radians:
• Either replace 180° with π and 360° with 2π OR
• Convert θ to degrees
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Reference Angles (Example)
Ex 3: i) draw θ in standard position ii) draw θ’,
the reference angle of θ:
a) 312°
c) 4π⁄5
e) 11π⁄3
b) π⁄8
d) -127°
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Trigonometric Functions of Real
Numbers
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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Evaluating a Trigonometric
Function Exactly
• To evaluate a trigonometric function of θ:
– Ensure that 0 < θ < 2π when using radians or
0° < θ < 360° when using degrees
– Find θ’ the reference angle of θ
– Evaluate the function using the EXACT values of
the reference angle and the quadrant in which θ
terminates
• Write the function in terms of sine or cosine if
necessary
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Evaluating a Trigonometric
Function (Exactly)
Ex 4: Give the exact value:
a) sin 225°
b) cos 750°
c) tan 120°
d) sec -11π⁄4
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Summary
• After studying these slides, you should be able to:
– Calculate the trigonometric function of ANY angle θ
– State the reference angle of an angle θ in standard position
– Evaluate a trigonometric function using reference angles
and exact values
• Additional Practice
– See the list of suggested problems for 4.4
• Next lesson
– Graphs of Sine & Cosine Functions (Section 4.5)
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