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Chapter 5 Trigonometric Functions 5.2 Right Triangle Trigonometry Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 Objectives: Use right triangles to evaluate trigonometric functions. Find function values for 30 ,45 , and 60 . 6 4 3 Recognize and use fundamental identities. Use equal cofunctions of complements. Evaluate trigonometric functions with a calculator. Use right triangle trigonometry to solve applied problems. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 The Six Trigonometric Functions The six trigonometric functions are: Function Abbreviation sine sin cosine cos tangent tan cosecant csc secant sec cotangent cot Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Right Triangle Definitions of Trigonometric Functions In general, the trigonometric functions of depend only on the size of angle and not on the size of the triangle. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Right Triangle Definitions of Trigonometric Functions (continued) In general, the trigonometric functions of depend only on the size of angle and not on the size of the triangle. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Evaluating Trigonometric Functions Find the value of the six trigonometric functions in the figure. We begin by finding c. a 2 b2 c 2 c 2 32 42 9 16 25 c 25 5 3 sin 5 4 cos 5 3 tan 4 5 csc 3 5 sec 4 4 cot 3 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Function Values for Some Special Angles A right triangle with a 45°, or radian, angle is 4 isosceles – that is, it has two sides of equal length. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Function Values for Some Special Angles (continued) A right triangle that has a 30°, or 6 has a 60°, or radian, angle also radian angle. In a 30-60-90 triangle, the 3 measure of the side opposite the 30° angle is one-half the measure of the hypotenuse. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Evaluating Trigonometric Functions of 45° Use the figure to find csc 45°, sec 45°, and cot 45°. length of hypotenuse 2 csc45 2 length of side opposite 45 1 length of hypotenuse 2 sec45 2 length of side adjacent to 45 1 length of side adjacent to 45 cot 45 length of side opposite 45 1 1 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Evaluating Trigonometric Functions of 30° and 60° Use the figure to find tan 60° and tan 30°. If a radical appears in a denominator, rationalize the denominator. 3 length of side opposite 60 3 tan 60 length of side adjacent to 60 1 length of side opposite 30 tan 30 length of side adjacent to 30 1 1 3 3 3 3 3 3 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Trigonometric Functions of Special Angles Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Fundamental Identities Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Using Quotient and Reciprocal Identities 5 2 Given sin and cos find the value of each of 3 3 the four remaining trigonometric functions. 2 2 3 2 2 5 2 5 sin 3 tan 5 5 5 5 5 3 5 cos 3 1 3 1 csc 2 2 sin 3 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Using Quotient and Reciprocal Identities (continued) 5 2 Given sin and cos find the value of each of 3 3 the four remaining trigonometric functions. 1 3 1 3 5 3 5 sec cos 5 5 5 5 5 3 1 5 1 5 cot tan 2 5 2 5 2 5 5 5 5 5 5 2 5 25 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 The Pythagorean Identities Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Using a Pythagorean Identity 1 Given that sin and is an acute angle, find the 2 value of cos using a trigonometric identity. 1 2 2 2 cos 1 sin cos 1 4 2 1 cos 2 1 3 2 cos 2 4 3 3 1 2 cos cos 1 4 2 4 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Trigonometric Functions and Complements Two positive angles are complements if their sum is 90° or . Any pair of trigonometric functions f and g for 2 which f ( ) g (90 ) and g ( ) f (90 ) are called cofunctions. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Cofunction Identities Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Using Cofunction Identities Find a cofunction with the same value as the given expression: a. sin 46 cos(90 46) cos 44 6 5 tan b. cot tan tan 12 2 12 12 12 12 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Using a Calculator to Evaluate Trigonometric Functions To evaluate trigonometric functions, we will use the keys on a calculator that are marked SIN, COS, and TAN. Be sure to set the mode to degrees or radians, depending on the function that you are evaluating. You may consult the manual for your calculator for specific directions for evaluating trigonometric functions. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: Evaluating Trigonometric Functions with a Calculator Use a calculator to find the value to four decimal places: a. sin 72.8° (hint: Be sure to set the calculator to degree mode) sin 72.8 0.9553 b. csc 1.5 (hint: Be sure to set the calculator to radian mode) csc1.5 1.0025 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Applications: Angle of Elevation and Angle of Depression An angle formed by a horizontal line and the line of sight to an object that is above the horizontal line is called the angle of elevation. The angle formed by the horizontal line and the line of sight to an object that is below the horizontal line is called the angle of depression. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Problem Solving Using an Angle of Elevation The irregular blue shape in the figure represents a lake. The distance across the lake, a, is unknown. To find this distance, a surveyor took the measurements shown in the figure. What is the distance across the lake? a tan 24 a 750 tan 24 750 a 333.9 The distance across the lake is approximately 333.9 yards. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23