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Lesson 7-3 The Sine and Cosine Functions

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Objective:

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

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Trigonometry

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The sine function is abbreviated sin

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

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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

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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

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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

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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

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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

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

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Unit Circle

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

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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:

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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:

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Unit Circle

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

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

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

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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.

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++ --

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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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+ + - -

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Find:

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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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We summarize these facts by saying that the sine and cosine functions are periodic and that they both have a fundamental period of 360 0 or 2π radians.

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Assignment: Pgs. 272-274 1-41 odd, omit 31

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