TRIG FUNCTIONS OF ANY ANGLE

Section 4.4

Precalculus PreAP/Dual, Revised ©2017

viet.dang@humbleisd.net

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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

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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 :

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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

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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.

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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.

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EXAMPLE 2

3 10

10−

10

103−

10

3− 10

1

3−

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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−

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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−

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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−

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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−

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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

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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)

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