section 7-3 the sine and cosine functions objective: to use the definitions of sine and cosine to...
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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/2012/08/trigonometry-evaluating-base-angles.html
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Trig Trick!
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Draw a UCIdentify the quadrants; what is the sign of (x,y) in each quadrant?
Q IQ II
Q III Q IV
(+,+)(-,+)
(-,-) (+,-)
(0,0)
(0,+1)
(-1,0)
(0,-1)
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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:
sin y
rWe define the cosine of θ, denoted cos θ, by:
cos x
r
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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 4th quadrant angle
sin 5
13
Find cos.
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Click here to see graph
Graph of last example.
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Unit Circle
The circle x2 + y2 = 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