1. Trig Functions of Any Angle
Section 4.4
Precalculus PreAP/Dual, Revised ©2017
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§4.4: Trig Functions of Any Angle

2. §4.4: Trig Functions of Any Angle
Review
Determine the Trigonometric Functions for 𝜽 5 4 3
𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽=
𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽=
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§4.4: Trig Functions of Any Angle

3. Equation in Standard Form
For 𝜽 be an angle in standard position with any point 𝒙, 𝒚
𝐬𝐢𝐧 𝜽 = y/r
𝐜𝐨𝐬 𝜽 = x/r
𝐭𝐚𝐧 𝜽= y/x
𝐜𝐬𝐜 𝜽 = r/y
𝐬𝐞𝐜 𝜽 = r/x
𝐜𝐨𝐭 𝜽= x/y
B. To establish the radius, the equation is 𝒓= 𝒙 𝟐 + 𝒚 𝟐
C. Think of “ASTC: All Students Take Calculus”
A: All points are always positive in Quadrant I
S: Sine points are positive in Quadrant II
T: Tan points are positive in Quadrant III
C: Cosine points are positive in Quadrant IV
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§4.4: Trig Functions of Any Angle

4. Equation in Standard Form
For 𝜽 be an angle in standard position with any point 𝒙, 𝒚 : S A
When ALL trig functions are positive
When SIN is positive
Quadrant II (– , +)
Quadrant I (+, +) T C
When TAN is positive
When COS is positive
Quadrant III (–, –)
Quadrant IV (+, –)
“All Students Take Calculus”
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§4.4: Trig Functions of Any Angle

5. Steps in Evaluating Functions given a Point
Draw a picture from a coordinate plane
Identify and plot the point onto the coordinate plane
Determine the missing side using the radius equation
Use Trigonometric Functions to solve
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§4.4: Trig Functions of Any Angle

6. §4.4: Trig Functions of Any Angle
Example 1
Let 𝟑, 𝟒 be a point on the terminal side of 𝜽. Determine the value of the six trigonometric functions for 𝜽. 4 3
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§4.4: Trig Functions of Any Angle

7. §4.4: Trig Functions of Any Angle
Example 1
Let 𝟑, 𝟒 be a point on the terminal side of 𝜽. Determine the value of the six trigonometric functions for 𝜽. 3 4 5
𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽=
𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽=
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§4.4: Trig Functions of Any Angle

8. §4.4: Trig Functions of Any Angle
Example 2
Let 𝟏𝟎 𝟏𝟎 ,− 𝟑 𝟏𝟎 𝟏𝟎 be a point on the terminal side of 𝜽. Determine the value of the six trigonometric functions for 𝜽.
𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽=
𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽=
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§4.4: Trig Functions of Any Angle

9. §4.4: Trig Functions of Any Angle
Your Turn
Let 𝟏,−𝟏 be a point on the terminal side of 𝜽. Determine the value of the six trigonometric functions for 𝜽.
𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽=
𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽=
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§4.4: Trig Functions of Any Angle

10. §4.4: Trig Functions of Any Angle
Example 3
Let 𝜽 be in Quadrant II. Given 𝐬𝐢𝐧 𝜽= 𝟏 𝟑 , determine the value of the six trigonometric functions for 𝜽. 3 1 x
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§4.4: Trig Functions of Any Angle

11. §4.4: Trig Functions of Any Angle
Example 3
Let 𝜽 be in Quadrant II. Given 𝐬𝐢𝐧 𝜽= 𝟏 𝟑 , determine the value of the six trigonometric functions for 𝜽. 1 3 x
𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽=
𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽=
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§4.4: Trig Functions of Any Angle

12. §4.4: Trig Functions of Any Angle
Example 4
Let 𝐜𝐨𝐭 𝜽<𝟎. Given 𝐜𝐬𝐜 𝜽=𝟒 , determine the value of the six trigonometric functions for 𝜽.
𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽=
𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽=
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§4.4: Trig Functions of Any Angle

13. §4.4: Trig Functions of Any Angle
Example 5
Given 𝒚= 𝟏 𝟑 𝒙 and 𝜽 is in Quadrant III, determine the value of the six trigonometric functions for 𝜽. -3
𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽=
𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽= -1
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§4.4: Trig Functions of Any Angle

14. §4.4: Trig Functions of Any Angle
Your Turn
Let 𝐭𝐚𝐧 𝜽>𝟎. Given 𝐜𝐨𝐬 𝜽=− 𝟐 𝟑 , determine the value of the six trigonometric functions for 𝜽.
𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽=
𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽=
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§4.4: Trig Functions of Any Angle

15. §4.4: Trig Functions of Any Angle
Assignment
Page , odd, odd (omit 29)
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§4.4: Trig Functions of Any Angle