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Lesson 7-3 The Sine and Cosine Functions. Objective:

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Presentation on theme: "Lesson 7-3 The Sine and Cosine Functions. Objective:"— Presentation transcript:

1 Lesson 7-3 The Sine and Cosine Functions

2 Objective:

3 To use the definitions of sine and cosine to find values of these functions and to solve simple trigonometric equations.

4 Trigonometry

5 The sine function is abbreviated sin

6 Trigonometry The sine function is abbreviated sin The cosine function is abbreviated cos

7 Let P(x,y) be any point on the circle x 2 + y 2 = r 2 and θ be an angle in standard position with terminal ray OP, as shown below. P(x,y) rθ O

8 Let P(x,y) be any point on the circle x 2 + y 2 = r 2 and θ be an angle in standard position with terminal ray OP, as shown below. We define the sin θ by: P(x,y) rθ O

9 Let P(x,y) be any point on the circle x 2 + y 2 = r 2 and θ be an angle in standard position with terminal ray OP, as shown below. We define the sin θ by: P(x,y) rθ O

10 Let P(x,y) be any point on the circle x 2 + y 2 = r 2 and θ be an angle in standard position with terminal ray OP, as shown below. We define the sin θ by: We define the cos θ by: P(x,y) rθ O

11 Let P(x,y) be any point on the circle x 2 + y 2 = r 2 and θ be an angle in standard position with terminal ray OP, as shown below. We define the sin θ by: We define the cos θ by: P(x,y) rθ O

12 If the terminal ray of an angle θ in standard position passes through (-3,2), find sin θ and cos θ.

13

14 Unit Circle

15 The unit circle is a circle with a center at the origin and has a radius of 1.

16 Unit Circle The unit circle is a circle with a center at the origin and has a radius of 1. Therefore its equation is simply:

17 Unit Circle The unit circle is a circle with a center at the origin and has a radius of 1. Therefore its equation is simply:

18 Unit Circle

19 Which now allows us to take our two formulas for sin θ and cos θ and change them to:

20 Unit Circle Which now allows us to take our two formulas for sin θ and cos θ and change them to:

21 Unit Circle Which now allows us to take our two formulas for sin θ and cos θ and change them to:

22 Now angles of rotations can locate you anywhere in the four quadrants. Since sin θ can now be determined strictly by the y-values, that means the sine of an angle will always be positive if your angle of rotation locates you in the 1 st of 2 nd quadrants.

23 ++ --

24 Likewise, cos θ can now be determined by the x-values, so the cosine function will always be positive if the angle of rotation locates you in the 1 st or 4 th quadrants.

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26 Find:

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31 From the previous examples and the definitions of sin θ and cos θ we can see that the sine and cosine functions repeat their values every 360° or 2π radians. Formally, this means that for all θ:

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34 We summarize these facts by saying that the sine and cosine functions are periodic and that they both have a fundamental period of or 2π radians.

35 Assignment: Pgs odd, omit 31


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