Transcript Finding the Trig functions of any angle

Chapter 4 Trigonometry
Section 4 Trigonometric
Functions of Any Angle
to infinity and beyond . . . !
Math Analysis
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Objectives

Evaluate trigonometric functions of any
angle to ± infinity

Use reference angles to evaluate
trigonometric functions.

Use trig models to solve real world
problems
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Definitions of trigonometric functions
3
Find the trig functions using a point
on the terminal side
syr cxr tyx
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Quadrant angles
x
θ
y
cos θ sin θ
0
π/2
π
3π/2
2π
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Trigonometric function signs
Star
Sin/csc
Trig
Tan/cot
All
All 6 trig functions are
positive in quadrant 1
Class
Cos/sec
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Find the quadrant
1.
sinθ > 0; cosθ <0
2.
sinθ < 0; cosθ > 0
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Reference angle
Let θ be an angle in standard position. Its reference angle is the acute
angle θ‘ formed by the terminal side of θ and the horizontal axis.
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
Reference
angles are
always
measured
between the x
axis and the
terminal side of
the angle (always
+!)

Notice the
butterfly shape
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Drawing Reference Angles
Find the reference angle θ' , and sketch θ and θ' in
standard position.
1)
θ = -145°
2)
11
 
3
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Finding the Trig functions of any
tan135 
angle
1.
2.
Find the reference angle ref   180  135
 45
Determine the value of
the given trig function of tan 45  1
the reference angle.
135  QII
3.
Determine the sign—
tan
is

in
Q
II
based on the quadrant of
the given (original) angle
tan135  1
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Now:
Find all 6 trig functions of an angle
Find the six trigonometric functions of θ with the given constraint.
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tan    ;
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sin   0
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Now, let’s make it tougher
The terminal side of θ lies on the line y = (1/3)x in quadrant I. Find
the values of the 6 trig functions of θ by finding a point on the line.
sin θ =
cos θ =
csc θ =
sec θ =
tan θ =
cot θ =
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Try These:
Find the values of the six trig functions of θ with the given constraints.
sin θ = 0;
where sec θ = -1
Find the reference angle θ‘ and sketch it and the angle θ in standard
position.
a) θ = 3.5
b) θ = 750°
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Find the trig functions of an angle
Evaluate the sine, cosine, and tangent of the angle without using a
calculator.
-405°
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Do 88(a)&(c)
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Do 91
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Objectives

Evaluate trigonometric functions of any angle to
± infinity



Use reference angles to evaluate trigonometric
functions.
1.
2.
3.

Syr Cxr Tyx
Quadrantal Angles
Find ref <
Find trig value of ref <
Determine sign of trig value for given <
Use trig models to solve real world problems

Draw, identify, solve
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Pg. 284: 3-52 (multiples of 3-- i.e. 3, 6, 9, etc.)
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