trig functions of any angle - dangmath.comΒ Β· 8/1/2018 12:30 am Β§4.4: trig functions of any angle...
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TRIG FUNCTIONS OF ANY ANGLE
Section 4.4
Precalculus PreAP/Dual, Revised Β©2017
8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 1
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8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 2
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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8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 3
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
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EQUATION IN STANDARD FORM
For π½ be an angle in standard position with any point π, π :
8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 4
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
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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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8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 6
Let π, π be a point on the terminal side of π½. Determine the value of the six trigonometric functions for π½.
EXAMPLE 1
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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
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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.
8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 8
EXAMPLE 2
3 10
10β
10
103β
10
3β 10
1
3β
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8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 9
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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8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 10
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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8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 11
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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8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 12
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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8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 13
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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8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 14
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
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ASSIGNMENT
Page 296
11, 15-23 odd, 27-35 odd (omit 29)
8/1/2018 12:30 AM Β§4.4: Trig Functions of Any Angle 15