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1 TRIG-Fall 2008-Morrison Trigonometry, 7th edition, Lial/Hornsby/Schneider, Addison Wesley, 2001 Chapter 2: Acute Angles and Right Triangles Section 2.1 I. Trigonometric Functions of Acute Angles Right-Triangle Based Definitions of Trigonometric Functions For any acute angle A in standard position, Example 1 II. sin A = y opp = r hyp csc A = hyp r = opp y cos A = x adj = r hyp sec A = r hyp = x adj tan A = y opp = x adj cot A = adj x = opp y Find the values of sin A, cos A, and tan A in the right triangle shown. Cofunction Identities For any acute angle A, sin A = cos (90° - A) cos A = sin (90° - A) tan A = cot (90° - A) csc A = sec (90° - A) sec A = csc (90° - A) cot A = tan (90° - A) Example 2 Write each function in terms of its cofunction. a) cos 38° b) sec 78° 2 Example 3 III. Find one solution for the equation: cot (4θ + 8°) = tan (2θ + 4°) Assume all angles are acute angles. Special Triangles Example 4 Give the exact value. a) cos 30° b) cot 45° 3 IV. Function Values of Special Angles Example 5 Find the exact value of each part labeled with a variable. 4 Section 2.2 I. Trigonometric Functions of Non-Acute Angles Reference Angles A reference angle for an angle θ is the positive acute angle made by the terminal side of angle θ and the x-axis. Caution: The reference angle is always found with reference to the x-axis, never the y-axis. Example 1 II. Find the reference angle for each angle. a) 218° b) 1387° Finding Trigonometric Function Values for any Nonquadrantal Angle θ 1. If θ > 360°, or if θ < 0°, then find a coterminal angle by adding or subtracting 360° as many times as needed to get an angle greater than 0° but less than 360°. 2. Find the reference angle θ′. 3. Find the trigonometric function values for reference angle θ′. 4. Determine the correct signs for the values found in Step 3. This gives the values of the trigonometric functions for angle θ. Example 2 Find exact values of the six trigonometric functions for 210°. 5 Example 3 III. Find exact values of the six trigonometric functions for -1020°. Finding the Angle Given the Trigonometric Function Value Example 4 Find all values of θ in the interval [0°, 360°) that has the given function value. tan 3 6 Section 2.3 I. II. Finding Trigonometric Function Values Using a Calculator Function Values Using a Calculator A. When evaluating trigonometric functions of angles given in degrees, remember that the calculator must be set in degree mode. B. Remember that most calculator values of trigonometric functions are approximations. Example 1 Use a calculator to give a decimal approximation for each value. Give as many digits as your calculator displays. a) sin 38° 24′ b) cot 68.4832° Example 2 Use a calculator to evaluate the expression: cos 75° cos 14° - sin 75° sin 14° Finding Angle Measures Using a Calculator A. Graphing calculators have three inverse trigonometric functions. B. If sin θ = x, then θ = sin -1 x for θ in the interval [0°, 90°]. Example 3 Find a value of θ in the interval [0°, 90°] that satisfies each statement. a) sin θ = .8535508 b) sec θ = 2.486879 7 Section 2.4 I. Significant Digits for Angles A. A significant digit is a digit obtained by actual measurement. B. When performing calculations, your answer is no more accurate than the least accurate number in your calculation. C. Start by determining the least number of significant digits in the given numbers and round your final answer to the same number of significant digits as this number. Number of Significant Digits 2 3 4 5 II. Solving Right Triangles Angle Measure to Nearest: Degree Ten minutes, or nearest tenth of a degree Minute, or nearest hundredth of a degree Tenth of a minute, or nearest thousandth of a degree Example 52˚ 52˚ 30′ = 52.5˚ 52˚ 45′ = 52.75˚ 52˚ 40.5′ = 52.675˚ Solving Triangles A. To solve a triangle means to find the measures of all the angles and sides of the triangle. B. Denote the angles of a triangle by capital letters. Then use the corresponding lower case letters to denote the respective opposite sides. C. If the letters A, B, and C, are used to denote the angles of a right triangle, then it is usually assumed that C is the right angle. Example 1 Solve the following right triangle. Example 2 Solve right triangle ABC if b = 219 m and c = 647 m and C = 90˚. (When two sides are given, give angles in degrees and minutes.) 8 III. Solving Applied Trigonometry Problems A. Draw a sketch, and label it with the given information. Label the quantity to be found with a variable. B. Use the sketch to write an equation relating the given quantities to the variable. C. Solve the equation, and check that your answer makes sense. Example 3 IV. Find the altitude of an isosceles triangle having base 184.2 cm if the angle opposite the base is 68˚ 44′. Angles of Elevation and Depression When a horizontal line of sight is used as a reference line, the angle measured above the line of sight is called an angle of elevation, while the angle measured below the line of sight is called an angle of depression. Example 4 An airplane is flying 10,500 feet above the level ground. The angle of depression from the plane to the base of a tree is 13˚ 50′. How far horizontally must the plane fly to be directly over the tree? 9 Section 2.5 I. Further Applications of Right Triangles Bearing A. When a single angle is given as a bearing, it is understood that the bearing is measured in a clockwise direction from the north. bearing of 32° B. bearing of 229° The second method for expressing bearing starts with a north-south line and uses an acute angle to show the direction, either east or west, from this line. Example 1 A ship travels 50 km on a bearing of 27°, then travels on a bearing of 117° for 140 km. Find the distance traveled from the starting point to the ending point. 10 Example 2 II. The bearing from Atlanta to Macon is S 27° E, and the bearing from Macon to Augusta is N 63° E. An automobile traveling at 60 mph needs 1 ¼ hour to go from Atlanta to Macon and 1 ¾ hour to go from Macon to Augusta. Find the distance from Atlanta to Augusta. Problems Involving Angles of Elevation/Depression Example 3 Sean wants to know the height of a Ferris wheel. From a given point on the ground, he finds the angle of elevation to the top of the Ferris wheel is 42.3°. He then moves back 75 ft. From the second point, the angle of elevation to the top of the Ferris wheel is 25.4°. Find the height of the Ferris wheel.