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TRIG FUNCTIONS OF ANY ANGLE
Section 4.4
Precalculus PreAP/Dual, Revised ©2017
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 1
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 2
Determine the Trigonometric Functions for 𝜽
REVIEW
54
3
𝐬𝐢𝐧 𝜽 = 𝐜𝐨𝐬 𝜽 = 𝐭𝐚𝐧 𝜽 =
𝐜𝐬𝐜 𝜽 = 𝐬𝐞𝐜 𝜽 = 𝐜𝐨𝐭 𝜽 =
4
5
3
5
4
3
5
4
5
3
3
4
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 3
A. For 𝜽 be an angle in standard position with any point 𝒙, 𝒚
1. 𝐬𝐢𝐧 𝜽 = y/r
2. 𝐜𝐨𝐬 𝜽 = x/r
3. 𝐭𝐚𝐧 𝜽= y/x
4. 𝐜𝐬𝐜 𝜽 = r/y
5. 𝐬𝐞𝐜 𝜽 = r/x
6. 𝐜𝐨𝐭 𝜽= x/y
B. To establish the radius, the equation is 𝒓 = 𝒙𝟐 + 𝒚𝟐
C. Think of “ASTC: All Students Take Calculus”
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 𝜽 be an angle in standard position with any point 𝒙, 𝒚 :
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 4
Quadrant I (+, +)Quadrant II (– , +)
Quadrant IV (+, –)Quadrant III (–, –)
“ALL STUDENTS TAKE CALCULUS”
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
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 5
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 6
Let 𝟑, 𝟒 be a point on the terminal side of 𝜽. Determine the value of the six trigonometric functions for 𝜽.
EXAMPLE 1
𝐬𝐢𝐧 𝜽 = 𝐜𝐨𝐬 𝜽 = 𝐭𝐚𝐧 𝜽 =
𝐜𝐬𝐜 𝜽 = 𝐬𝐞𝐜 𝜽 = 𝐜𝐨𝐭 𝜽 =
Let 𝟑, 𝟒 be a point on the terminal side of 𝜽. Determine the value of the six trigonometric functions for 𝜽.
8/1/2018 12:30 AM §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
𝐬𝐢𝐧 𝜽 = 𝐜𝐨𝐬 𝜽 = 𝐭𝐚𝐧 𝜽 =
𝐜𝐬𝐜 𝜽 = 𝐬𝐞𝐜 𝜽 = 𝐜𝐨𝐭 𝜽 =
Let 𝟏𝟎
𝟏𝟎, −
𝟑 𝟏𝟎
𝟏𝟎be a point on the terminal side of 𝜽. Determine the
value of the six trigonometric functions for 𝜽.
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 8
EXAMPLE 2
3 10
10−
10
103−
10
3− 10
1
3−
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 9
Let 𝟏,−𝟏 be a point on the terminal side of 𝜽. Determine the value of the six trigonometric functions for 𝜽.
YOUR TURN
𝐬𝐢𝐧 𝜽 = 𝐜𝐨𝐬 𝜽 = 𝐭𝐚𝐧 𝜽 =
𝐜𝐬𝐜 𝜽 = 𝐬𝐞𝐜 𝜽 = 𝐜𝐨𝐭 𝜽 =
2
2−
2
21−
2− 2 1−
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 10
Let 𝜽 be in Quadrant II. Given 𝐬𝐢𝐧 𝜽 =𝟏
𝟑, determine the value of the six
trigonometric functions for 𝜽.
EXAMPLE 3
siny
r = ( ) II ,Quadrant = − +
( )
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 = −
2 2−
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 11
Let 𝜽 be in Quadrant II. Given 𝐬𝐢𝐧 𝜽 =𝟏
𝟑, determine the value of the six
trigonometric functions for 𝜽.
EXAMPLE 3
2 2−
𝐬𝐢𝐧 𝜽 = 𝐜𝐨𝐬 𝜽 = 𝐭𝐚𝐧 𝜽 =
𝐜𝐬𝐜 𝜽 = 𝐬𝐞𝐜 𝜽 = 𝐜𝐨𝐭 𝜽 =
1
3
3
2 2
3−
2 2−3 2
4−
2
4−
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 12
Let 𝐜𝐨𝐭 𝜽 < 𝟎. Given 𝐜𝐬𝐜 𝜽 = 𝟒 , determine the value of the six trigonometric functions for 𝜽.
EXAMPLE 4
𝐬𝐢𝐧 𝜽 = 𝐜𝐨𝐬 𝜽 = 𝐭𝐚𝐧 𝜽 =
𝐜𝐬𝐜 𝜽 = 𝐬𝐞𝐜 𝜽 = 𝐜𝐨𝐭 𝜽 =
1
4
4
15
4−
15
15−
4 15
15− 15−
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 13
Given 𝒚 =𝟏
𝟑𝒙 and 𝜽 is in Quadrant III, determine the value of the six
trigonometric functions for 𝜽.
EXAMPLE 5
( ) ( )
2 2
2 23 1
9 1
10
r x y
r
r
r
= +
= − + −
= +
=
𝐬𝐢𝐧 𝜽 = 𝐜𝐨𝐬 𝜽 = 𝐭𝐚𝐧 𝜽 =
𝐜𝐬𝐜 𝜽 = 𝐬𝐞𝐜 𝜽 = 𝐜𝐨𝐭 𝜽 =
10
10−
10−
2
2−
1
3
2− 3
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 14
Let 𝐭𝐚𝐧 𝜽 > 𝟎. Given 𝐜𝐨𝐬 𝜽 = −𝟐
𝟑, determine the value of the six
trigonometric functions for 𝜽.
YOUR TURN
𝐬𝐢𝐧 𝜽 = 𝐜𝐨𝐬 𝜽 = 𝐭𝐚𝐧 𝜽 =
𝐜𝐬𝐜 𝜽 = 𝐬𝐞𝐜 𝜽 = 𝐜𝐨𝐭 𝜽 =
5
3−
3 5
5−
2
3−
5
2
3
2− 2 5
5
ASSIGNMENT
Page 296
11, 15-23 odd, 27-35 odd (omit 29)
8/1/2018 12:30 AM §4.4: Trig Functions of Any Angle 15