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Trigonometric Functions
Chapter 5 Trigonometric Functions © 2011 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved © 2011 Pearson Education, Inc. All rights reserved 1
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Trigonometric Functions of Any Angle: The Unit Circle
SECTION 5.3 Evaluate trigonometric functions of any angle. Determine the signs of the trigonometric functions in each quadrant. Find a reference angle. Use reference angles to find trigonometric function values. Define the trigonometric functions using the unit circle. 1 2 3 4 5
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TRIGONOMETRIC FUNCTIONS OF ANGLES
© 2011 Pearson Education, Inc. All rights reserved
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DEFINITION OF THE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE θ
Let P(x,y) be any point on the terminal ray of an angle in standard position (other than the origin) and let We define © 2011 Pearson Education, Inc. All rights reserved
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© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 1 Finding Trigonometric Function Values Suppose is an angle whose terminal side contains the point P(–1,3). Find the exact values of the six trigonometric functions of . Solution © 2011 Pearson Education, Inc. All rights reserved
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© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 1 Finding Trigonometric Function Values Solution continued Now, with x = –1, y = 3, and r = , we have © 2011 Pearson Education, Inc. All rights reserved
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TRIGONOMETRIC FUNCTION VALUES OF QUADRANTAL ANGLES
© 2011 Pearson Education, Inc. All rights reserved
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© 2011 Pearson Education, Inc. All rights reserved
COTERMINAL ANGLES Because the value of each trigonometric function of an angle in standard position is completely determined by the position of the terminal side, the following statements are true. Coterminal angles are assigned identical values by the six trigonometric functions. The signs of the values of the trigonometric functions are determined by the quadrant containing the terminal side. For any integer n, θ, and θ + n360° are coterminal angles and θ and θ + 2πn are coterminal angles. © 2011 Pearson Education, Inc. All rights reserved
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TRIGONOMETRIC FUNCTION VALUES OF COTERMINAL ANGLES
in degrees in radians These equations hold for any integer n. © 2011 Pearson Education, Inc. All rights reserved
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SIGNS OF THE TRIGONOMETRIC FUNCTIONS
Suppose the angle is not quadrantal and its terminal side contains the point (x,y). The only value other than x and y used in defining the trigonometric functions, , is always positive. Therefore, the signs of x and y determine the signs of the trigonometric functions. © 2011 Pearson Education, Inc. All rights reserved
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© 2011 Pearson Education, Inc. All rights reserved
Determining the Quadrant in Which an Angle Lies EXAMPLE 4 If tan θ > 0 and cos θ < 0, in which quadrant does θ lie? Solution Because tan θ > 0, θ lies either in quadrant I or in quadrant III. However, cos θ > 0 for θ in quadrant I; so θ must lie in quadrant III. © 2011 Pearson Education, Inc. All rights reserved
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© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 5 Evaluating Trigonometric Functions Solution Since tan θ > 0 and cos θ < 0, θ lies in Quadrant III; both x and y must be negative. © 2011 Pearson Education, Inc. All rights reserved
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© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 5 Evaluating Trigonometric Functions Solution continued , © 2011 Pearson Education, Inc. All rights reserved
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DEFINITION OF A REFERENCE ANGLE
Let be an angle in standard position that is not a quadrantal angle. The reference angle for is the positive acute angle (“theta prime”) formed by the terminal side of and the x-axis. © 2011 Pearson Education, Inc. All rights reserved
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DEFINITION OF A REFERENCE ANGLE
Quadrant I Quadrant II © 2011 Pearson Education, Inc. All rights reserved
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DEFINITION OF A REFERENCE ANGLE
Quadrant III Quadrant IV © 2011 Pearson Education, Inc. All rights reserved
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© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 6 Identifying Reference Angles Find the reference angle for each angle . a. = 250º b. = c. = 5.75 Solution Because 250º lies in quadrant III, = - 180º. So = 250º - 180º = 70º. Because lies in quadrant II, = π - . So = π © 2011 Pearson Education, Inc. All rights reserved
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© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 6 Identifying Reference Angles Solution continued Since no degree symbol appears in θ = 5.75, has radian measure. Now ≈ 4.71 and 2π ≈ So lies in quadrant IV and = 2π - . So = 2π – 5.75 ≈ 6.28 – 5.75 = 0.53. © 2011 Pearson Education, Inc. All rights reserved
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USING REFERENCE ANGLES TO FIND TRIGONOMETRIC FUNCTION VALUES
Step 1 Assuming that > 360º or θ < 0°, find a coterminal angle for with degree measure between 0º and 360º. Otherwise, go to Step 2. Step 2 Find the reference angle for the angle resulting from Step 1. Write the trigonometric function of . © 2011 Pearson Education, Inc. All rights reserved
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USING REFERENCE ANGLES TO FIND TRIGONOMETRIC FUNCTION VALUES
Step 3 Choose the correct sign for the trigonometric function value of θ based on the quadrant in which it lies. Write the given trigonometric function of θ in terms of the same trigonometric function of θ with the appropriate sign. © 2011 Pearson Education, Inc. All rights reserved
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© 2011 Pearson Education, Inc. All rights reserved
Using the Reference Angle to Find Values of Trigonometric Functions EXAMPLE 8 Find the exact value of each expression. Solution Step 1 0º < 330º < 360º; find its reference angle. a. Step º is in Q IV; its reference angle is . © 2011 Pearson Education, Inc. All rights reserved
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© 2011 Pearson Education, Inc. All rights reserved
Using the Reference Angle to Find Values of Trigonometric Functions EXAMPLE 8 Solution continued Step 3 In Q IV, tan θ is negative, so . b. Step 1 is between 0 and 2π coterminal with . © 2011 Pearson Education, Inc. All rights reserved
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is in Q IV; its reference angle is
Using the Reference Angle to Find Values of Trigonometric Functions EXAMPLE 8 Solution continued Step 2 is in Q IV; its reference angle is . Step 3 In Q IV, sec θ > 0; so . © 2011 Pearson Education, Inc. All rights reserved
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TRIGONOMETRIC FUNCTIONS AND THE UNIT CIRCLE
A circle with radius 1 centered at the origin of a rectangular coordinate system is a unit circle. In a unit circle, s = rθ = 1·θ = θ, so the radian measure and the arc length of an arc intercepted by a central angle in a unit circle are numerically identical. The correspondence between real numbers and endpoints of arcs on the unit circle is used to define the trigonometric functions of real numbers, or the circular functions. © 2011 Pearson Education, Inc. All rights reserved
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UNIT CIRCLE DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS
Let t be any real number and let P = (x,y) be the point on the unit circle associated with t. Then © 2011 Pearson Education, Inc. All rights reserved
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