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TRIG FUNCTIONS OF ANY ANGLE
Section 4.4
Precalculus PreAP/Dual, Revised Β©2017
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Determine the Trigonometric Functions for π½
REVIEW
54
3
π¬π’π§ π½ = ππ¨π¬ π½ = πππ§ π½ =
ππ¬π π½ = π¬ππ π½ = ππ¨π π½ =
4
5
3
5
4
3
5
4
5
3
3
4
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A. For π½ be an angle in standard position with any point π, π
1. π¬π’π§ π½ = y/r
2. ππ¨π¬ π½ = x/r
3. πππ§ π½= y/x
4. ππ¬π π½ = r/y
5. π¬ππ π½ = r/x
6. ππ¨π π½= x/y
B. To establish the radius, the equation is π = ππ + ππ
C. Think of βASTC: All Students Take Calculusβ
1. A: All points are always positive in Quadrant I
2. S: Sine points are positive in Quadrant II
3. T: Tan points are positive in Quadrant III
4. C: Cosine points are positive in Quadrant IV
EQUATION IN STANDARD FORM
EQUATION IN STANDARD FORM
For π½ be an angle in standard position with any point π, π :
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Quadrant I (+, +)Quadrant II (β , +)
Quadrant IV (+, β)Quadrant III (β, β)
βALL STUDENTS TAKE CALCULUSβ
When ALL trig functions are
positive
When SIN is positive
When TAN is positive
When COS is positive
STEPS IN EVALUATING FUNCTIONS GIVEN A POINT
A. Draw a picture from a coordinate plane
B. Identify and plot the point onto the coordinate plane
C. Determine the missing side using the radius equation
D. Use Trigonometric Functions to solve
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Let π, π be a point on the terminal side of π½. Determine the value of the six trigonometric functions for π½.
EXAMPLE 1
π¬π’π§ π½ = ππ¨π¬ π½ = πππ§ π½ =
ππ¬π π½ = π¬ππ π½ = ππ¨π π½ =
Let π, π be a point on the terminal side of π½. Determine the value of the six trigonometric functions for π½.
8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 7
EXAMPLE 1
2 2r x y= +
( ) ( )2 2
3 4r = +
25r =4
5
3
5
4
3
5
4
5
3
3
4
π¬π’π§ π½ = ππ¨π¬ π½ = πππ§ π½ =
ππ¬π π½ = π¬ππ π½ = ππ¨π π½ =
Let ππ
ππ, β
π ππ
ππbe a point on the terminal side of π½. Determine the
value of the six trigonometric functions for π½.
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EXAMPLE 2
3 10
10β
10
103β
10
3β 10
1
3β
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Let π,βπ be a point on the terminal side of π½. Determine the value of the six trigonometric functions for π½.
YOUR TURN
π¬π’π§ π½ = ππ¨π¬ π½ = πππ§ π½ =
ππ¬π π½ = π¬ππ π½ = ππ¨π π½ =
2
2β
2
21β
2β 2 1β
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Let π½ be in Quadrant II. Given π¬π’π§ π½ =π
π, determine the value of the six
trigonometric functions for π½.
EXAMPLE 3
siny
r = ( ) II ,Quadrant = β +
( )
2 2
22
2
3 1
3 1
r x y
x
x
= +
= +
= +
( ) ( )2
2 2
2
2
3 1
9 1
8
x
x
x
= +
= +
=
2 2x = β
2 2β
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Let π½ be in Quadrant II. Given π¬π’π§ π½ =π
π, determine the value of the six
trigonometric functions for π½.
EXAMPLE 3
2 2β
π¬π’π§ π½ = ππ¨π¬ π½ = πππ§ π½ =
ππ¬π π½ = π¬ππ π½ = ππ¨π π½ =
1
3
3
2 2
3β
2 2β3 2
4β
2
4β
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Let ππ¨π π½ < π. Given ππ¬π π½ = π , determine the value of the six trigonometric functions for π½.
EXAMPLE 4
π¬π’π§ π½ = ππ¨π¬ π½ = πππ§ π½ =
ππ¬π π½ = π¬ππ π½ = ππ¨π π½ =
1
4
4
15
4β
15
15β
4 15
15β 15β
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Given π =π
ππ and π½ is in Quadrant III, determine the value of the six
trigonometric functions for π½.
EXAMPLE 5
( ) ( )
2 2
2 23 1
9 1
10
r x y
r
r
r
= +
= β + β
= +
=
π¬π’π§ π½ = ππ¨π¬ π½ = πππ§ π½ =
ππ¬π π½ = π¬ππ π½ = ππ¨π π½ =
10
10β
10β
2
2β
1
3
2β 3
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Let πππ§ π½ > π. Given ππ¨π¬ π½ = βπ
π, determine the value of the six
trigonometric functions for π½.
YOUR TURN
π¬π’π§ π½ = ππ¨π¬ π½ = πππ§ π½ =
ππ¬π π½ = π¬ππ π½ = ππ¨π π½ =
5
3β
3 5
5β
2
3β
5
2
3
2β 2 5
5
ASSIGNMENT
Page 296
11, 15-23 odd, 27-35 odd (omit 29)
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