College Algebra & Trigonometry, 2018a

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Chapter 9 Graphing the Trigonometric Functions 9.1 Trigonometric Functions of Non-Acute Angles In Chapter 1, we learned about the trigonometric functions of positive acute angles that occur within right triangles. If we wish to extend the definition of the trigonometric functions, then we need to define how to determine the values for the sine and cosine of other angles. To do this, consider a right triangle drawn on the coordinate axes. The positive acute angle θ will be the angle created between the x-axis and the hypotenuse of the triangle. The lengths of the two legs of the triangle will be the x and y coordinates of a point in the first quadrant. (x, y) 1 y θ x 383
384 CHAPTER 9. GRAPHING THE TRIGONOMETRIC FUNCTIONS (x, y) =(cosθ, sin θ) 1 y opposite θ x adjacent In the picture above we see a triangle in the first quadrant with a hypotenuse of 1. In this situation, the value of sin θ = opp hyp = y 1 = y, which is just the y-coordinate of the point at the top of the triangle. Correspondingly, the value of cos θ = adj hyp = x 1 = x, or the value of the x-coordinate of the same point. This allows us to find the sine or cosine for what are known as the quadrantal angles - the angles that are multiples of 90 ◦ . If we look at the unit circle (the circle with a radius of 1), then we can see the values of the sine and cosine for these angles. (cos 90 ◦ , sin 90 ◦ ) (0, 1) (cos 180 ◦ , sin 180 ◦ ) (−1, 0) (cos 0 ◦ , sin 0 ◦ ) (1, 0) (0, −1) (cos 270 ◦ , sin 270 ◦ )
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College Algebra & Trigonometry
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c○2018 Richard W. Beveridge This
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4 CONTENTS 2.4 Solution of Rational
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6 CONTENTS 8.2 The Trigonometric Ra
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8 CONTENTS
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88 CHAPTER 2. POLYNOMIAL AND RATION
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100 CHAPTER 2. POLYNOMIAL AND RATIO
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192 CHAPTER 4. FUNCTIONS 4.2 Domain
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194 CHAPTER 4. FUNCTIONS Exercises
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258 CHAPTER 5. CONIC SECTIONS - CIR
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288 CHAPTER 6. SEQUENCES AND SERIES
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College Algebra & Trigonometry, 2018a

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