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

Right Triangle Trigonometry SECTION 5.2 Define trigonometric functions of acute angles. Evaluate trigonometric functions of acute angles. Evaluate trigonometric functions for the special angles 30°, 45°, and 60°. Use fundamental identities. Use right triangle trigonometry in applications. 1 2 3 4 5

TRIGONOMETRIC RATIOS AND FUNCTIONS a = length of the side opposite  b = length of the side adjacent to  c = length of the hypotenuse Six ratios can be formed with the lengths of these sides: © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved RIGHT TRIANGLE DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE θ © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved Finding the Values of Trigonometric Functions EXAMPLE 1 Find the exact values for the six trigonometric functions of the angle  in the figure. Solution © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved Finding the Values of Trigonometric Functions EXAMPLE 1 Solution continued Now, with c = 4, a = 3, and b = , we have © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved Finding the Trigonometric Function Values for 45°. EXAMPLE 3 Use the figure to find sin 45°, cos 45°, and tan 45°. Solution © 2011 Pearson Education, Inc. All rights reserved

TRIGONOMETRIC FUNCTION VALUES OF SOME COMMON ANGLES © 2011 Pearson Education, Inc. All rights reserved

COFUNCTION IDENTITIES The value of any trigonometric function of an angle  is equal to the cofunction of the complement of . This is true whether  is measured in degrees or in radians.  in degrees If  is measured in radians, replace 90º with © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved Finding Trigonometric Function Values of a Complementary Angle EXAMPLE 5 a. Given that cot 68° ≈ 0.4040, find tan 22°. b. Given that cos 72° ≈ 0.3090, find sin 18°. Solution a. tan 22° = tan (90° – 68°) = cot 68° ≈ 0.4040 b. sin 18° = sin (90° – 72°) = cos 72° ≈ 0.3090 © 2011 Pearson Education, Inc. All rights reserved

RECIPROCAL AND QUOTIENT IDENTITIES Reciprocal Identities Quotient Identities © 2011 Pearson Education, Inc. All rights reserved

PYTHAGOREAN IDENTITIES The cofunction, reciprocal, quotient, and Pythagorean identities are called the Fundamental identities. © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved APPLICATIONS Angles that are measured between a line of sight and a horizontal line occur in many applications and are called angles of elevation or angles of depression. If the line of sight is above the horizontal line, the angle between these two lines is called the angle of elevation. If the line of sight is below the horizontal line, the angle between the two lines is called the angle of depression. © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 8 Measuring the Height of Mount Kilimanjaro A surveyor wants to measure the height of Mount Kilimanjaro by using the known height of a nearby mountain. The nearby location is at an altitude of 8720 feet, the distance between that location and Mount Kilimanjaro’s peak is 4.9941 miles, and the angle of elevation from the lower location is 23.75º. See the figure on the next slide. Use this information to find the approximate height of Mount Kilimanjaro (in feet). © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 8 Measuring the Height of Mount Kilimanjaro © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 8 Measuring the Height of Mount Kilimanjaro Solution The sum of the side length h and the location height of 8720 feet gives the approximate height of Mount Kilimanjaro. Let h be measured in miles. Use the definition of sin , for  = 23.75º. h = (4.9941) sin θ = (4.9941) sin 23.75° h ≈ 2.0114 © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 8 Measuring the Height of Mount Kilimanjaro Solution continued Because 1 mile = 5280 feet, 2.0114 miles = (2.0114)(5280) ≈ 10,620 feet. Thus, the height of Mount Kilimanjaro ≈ 10,620 + 8720 = 19,340 feet. © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 9 Finding the Width of a River To find the width of a river, a surveyor sights straight across the river from a point A on her side to a point B on the opposite side. See the figure on the next slide. She then walks 200 feet upstream to a point C. The angle  that the line of sight from point C to point B makes with the river bank is 58º. How wide is the river? © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 9 Finding the Width of a River © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 9 Finding the Width of a River Solution The points A, B, and C are the vertices of a right triangle with acute angle 58º. Let w be the width of the river. The river is about 320 feet wide at the point A. © 2011 Pearson Education, Inc. All rights reserved