The Trigonometric Functions

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© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 1 of 39 Chapter 8 The Trigonometric Functions.

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Presentation transcript:

The Trigonometric Functions Chapter 8 The Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter Outline Radian Measure of Angles The Sine and the Cosine Differentiation and Integration of sin t and cos t The Tangent and Other Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 8.2 The Sine and the Cosine Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section Outline Sine and Cosine Sine and Cosine in a Right Triangle Sine and Cosine in a Unit Circle Properties of Sine and Cosine Calculating Sine and Cosine Using Sine and Cosine Determining an Angle t The Graphs of Sine and Cosine Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Sine & Cosine Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Sine & Cosine in a Right Triangle Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Sine & Cosine in a Unit Circle Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Properties of Sine & Cosine Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Calculating Sine & Cosine EXAMPLE Give the values of sin t and cos t, where t is the radian measure of the angle shown. SOLUTION Since we wish to know the sine and cosine of the angle that measures t radians, and because we know the length of the side opposite the angle as well as the hypotenuse, we can immediately determine sin t. Since sin2t + cos2t = 1, we have Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Calculating Sine & Cosine CONTINUED Replace sin2t with (1/4)2. Simplify. Subtract. Take the square root of both sides. Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Using Sine & Cosine EXAMPLE If t = 0.4 and a = 10, find c. SOLUTION Since cos(0.4) = 10/c, we get Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Determining an Angle t EXAMPLE Find t such that –π/2 ≤ t ≤ π/2 and t satisfies the stated condition. SOLUTION One of our properties of sine is sin(-t) = -sin(t). And since -sin(3π/8) = sin(-3π/8) and –π/2 ≤ -3π/8 ≤ π/2, we have t = -3π/8. Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. The Graph of Sin(t) Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. The Graphs of Sine & Cosine Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Incorporating Technology Copyright © 2014, 2010, 2007 Pearson Education, Inc.