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Welcome to Precalculus! Get notes out. Get homework out Write “I Can” statements I can: - Evaluate trigonometric functions of acute angles, and use a calculator to evaluate trigonometric functions. Use the fundamental trigonometric identities. Use trigonometric functions to model and solve real-life problems. 1 Focus and Review Attendance Questions from yesterday’s work 2 4.3 Right Triangle Trigonometry Copyright © Cengage Learning. All rights reserved. Objectives Evaluate trigonometric functions of acute angles, and use a calculator to evaluate trigonometric functions. Use the fundamental trigonometric identities. Use trigonometric functions to model and solve real-life problems. 4 The Six Trigonometric Functions 5 The Six Trigonometric Functions This section introduces the trigonometric functions from a right triangle perspective. Consider a right triangle with one acute angle labeled , as shown below. Relative to the angle , the three sides of the triangle are the hypotenuse, the opposite side (the side opposite the angle ), and the adjacent side (the side adjacent to the angle ). 6 The Six Trigonometric Functions Using the lengths of these three sides, you can form six ratios that define the six trigonometric functions of the acute angle . sine cosecant cosine secant tangent cotangent 7 The Six Trigonometric Functions In the following definitions, it is important to see that 0 < < 90 ( lies in the first quadrant) and that for such angles the value of each trigonometric function is positive. 8 Memory Aid 9 The Six Trigonometric Functions In the following definitions, it is important to see that 0 < < 90 ( lies in the first quadrant) and that for such angles the value of each trigonometric function is positive. 10 Example 1 – Evaluating Trigonometric Functions Use the triangle in Figure 4.20 to find the values of the six trigonometric functions of . Solution: By the Pythagorean Theorem, (hyp)2 = (opp)2 + (adj)2, it follows that Figure 4.20 11 Example 1 – Solution cont’d So, the six trigonometric functions of are 12 Example 1 – Solution cont’d 13 Practice Problems 7, 8, 13, 18 14 The 45-45-90 Triangle and the 30-60-90 Triangle 15 The Six Trigonometric Functions In Example 1, you were given the lengths of two sides of the right triangle, but not the angle . Often, you will be asked to find the trigonometric functions of a given acute angle . To do this, construct a right triangle having as one of its angles. 16 The 45-45-90 Triangle 17 The 45-45-90 Triangle 18 The 45-45-90 Triangle 19 The 45-45-90 Triangle 20 The 45-45-90 Triangle 21 The 45-45-90 Triangle 22 The 45-45-90 Triangle 23 The 30-60-90 Triangle 24 The 30-60-90 Triangle 25 The 30-60-90 Triangle 2 2 1 1 26 The 30-60-90 Triangle 2 1 27 The 30-60-90 Triangle 28 The Six Trigonometric Functions 29 The Six Trigonometric Functions 30 The Six Trigonometric Functions 31 Applications Involving Right Triangles 32 Applications Involving Right Triangles Many applications of trigonometry involve a process called solving right triangles. In this type of application, you are usually given one side of a right triangle and one of the acute angles and are asked to find one of the other sides, or you are given two sides and are asked to find one of the acute angles. In Example 8, the angle you are given is the angle of elevation, which represents the angle from the horizontal upward to an object. 33 Applications Involving Right Triangles Find the measure of each side indicated. Round to the nearest tenth. A B A 13 C B C 34 Applications Involving Right Triangles Find the measure of each angle indicated. Round to the nearest tenth. 3 200 yd 400 yd 3 35 Applications Involving Right Triangles In the following example, the angle you are given is the angle of elevation, which represents the angle from the horizontal upward to an object. In other applications you may be given the angle of depression, which represents the angle from the horizontal downward to an object. (See next slide.) 36 Applications Involving Right Triangles 37 Example 8 – Using Trigonometry to Solve a Right Triangle A surveyor is standing 115 feet from the base of the Washington Monument, as shown in figure below. The surveyor measures the angle of elevation to the top of the monument as 78.3. How tall is the Washington Monument? 38 Example 8 – Solution From the figure, you can see that where x = 115 and y is the height of the monument. So, the height of the Washington Monument is y = x tan 78.3 115(4.82882) 555 feet. 39 Applications Involving Right Triangles 40 Cofunctions of Complimentary Angles In the box, note that sin 30 = = cos 60. This occurs because 30 and 60 are complementary angles. In general, it can be shown from the right triangle definitions that cofunctions of complementary angles are equal. That is, if is an acute angle, then the following relationships are true. sin(90 – ) = cos cos(90 – ) = sin tan(90 – ) = cot cot(90 – ) = tan sec(90 – ) = csc csc(90 – ) = sec 41 Cofunctions of Complimentary Angles 42 Cofunctions of Complimentary Angles 43 Cofunctions of Complimentary Angles 44 Cofunctions of Complimentary Angles Equivalent statements can be derived relating cosine, cotangent, and cosecant to sine, tangent, and secant, respectively. Therefore, the following cofunction relationships hold: sin(90 – ) = cos cos(90 – ) = sin tan(90 – ) = cot cot(90 – ) = tan sec(90 – ) = csc csc(90 – ) = sec 45 Trigonometric Identities 46 Trigonometric Identities 47 Cofunctions of Complimentary Angles 48 Trigonometric Identities In trigonometry, a great deal of time is spent studying relationships between trigonometric functions (identities). 49 Important Note You need to memorize all of the Fundamental Trigonometric Identities found on page 282 verbatim. Stating these identities will be included on all of the quizzes and tests for the remainder of Chapter 4. 50 Example 5 – Applying Trigonometric Identities Let be an acute angle such that sin = 0.6. Find the values of (a) cos and (b) tan using trigonometric identities. Solution: a. To find the value of cos , use the Pythagorean identity sin2 + cos2 = 1. So, you have (0.6)2 + cos2 = 1 Substitute 0.6 for sin . cos2 = 1 – (0.6)2 Subtract (0.6)2 from each side. cos2 = 0.64 Simplify. 51 Example 5 – Solution cont’d cos = Extract positive square root. cos = 0.8. Simplify. b. Now, knowing the sine and cosine of , you can find the tangent of to be = 0.75. Use the definitions of cos and tan , and the triangle shown in Figure 4.23, to check these results. Figure 4.23 52 Example 6 – Applying Trigonometric Identities 53 Independent Practice Section 4.3 (page 286) # 1 – 4 (vocabulary), 5, 6, 10, 13, 14, 15, 16, 17, 19, 31 – 34, 41, 42, 45, 46, 47, 49, 51, 53, 55, 57, 59, 61, 63, 66, 69, 71, 73 Extra Credit: # 76 (5 points on next quiz) IMPORTANT NOTE: Problem 65 has been changed to Problem 66. 54