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Section 7-3 The Sine and Cosine Functions Objective: To use the definitions of sine and cosine to find values of these functions and to solve simple trigonometric equations.

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Review topics Draw a circle: Label each quadrants as 1-4 Note the positive or negative x and y values in each quadrant

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Trig or Trick! Draw an outline of your non-dominate hand on your paper. Spread your fingers so that your thumb and pinky make approximately 90 degrees.

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http://mathrescue.blogspot.com/201 2/08/trigonometry-evaluating-base- angles.html

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Trig Trick!

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Draw a UC Identify the quadrants; what is the sign of (x,y) in each quadrant?

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Review of geometry concepts Given a right triangle, do you remember the definition of sin and cos ?

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An angle in the standard position

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We can study the same angle… It is in the standard position. Its vertex is on point (0,0). It is on the coordinate system.

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Now what if the same triangle was within a circle on

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Sine and Cosine Functions –We define the sine of θ, denoted sin θ, by: We define the cosine of θ, denoted cos θ, by:

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What if r=1 If r=1 what does sin θ and cos θ equal to? sin θ=y cos θ=x

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The Unit Circle The circle of radius 1 is called the unit circle.

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We can determine the value of many angles on the unit circle. Find A.) sin 90° B.) sin 450° C.) cos (-π)

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Example 1 If the terminal ray of an angle θ in standard position passes through (-3, 2), find sin θ and cos θ. Solution: On a grid, locate (-3,2) Use this point to draw a right triangle, where one side is on the x-axis, and the hypotenuse is line segment between (-3,2) and (0,0).

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Click here to see graph Graph for last example

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Example 2 Given θ is a 4 th quadrant angle Find cos .

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Click here to see graph Graph of last example.

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Unit Circle The circle x 2 + y 2 = 1 has radius 1 and is therefore called the unit circle. This circle is the easiest one with which to work because, as the diagram shows, sin θ and cos θ are simply the y- and x-coordinates of the point where the terminal ray of θ intersects the circle. Sin θ = y / r = y / 1 = y Cos θ = x / r = x / 1 = x When a circle is used to define the trigonometric functions, they are sometimes called circular functions.

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Example 4 Solve sin θ = 1 for θ in degrees.

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Repeating Sin and Cos Values Sin (θ + 360°) = sin θ Cos (θ + 360°) = cos θ Sin (θ + 2π) = sin θ Cos (θ + 2π) = cos θ Sin (θ + 360°n) = sin θ Cos (θ + 360°n) = cos θ Sin (θ + 2πn) = sin θ Cos (θ + 2πn) = cos θ

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Homework sec 7.3 written exercises Day 1: Problems # 1-16 All Day 2 #17-28, 33-42 ALL

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