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13-1Right-Angle 13-1 Right-AngleTrigonometry Trigonometry Warm Up Lesson Presentation Lesson Quiz Holt Algebra Holt Algebra 22 13-1 Right-Angle Trigonometry Warm Up Given the measure of one of the acute angles in a right triangle, find the measure of the other acute angle. 1. 45° 45° 2. 60° 30° 3. 24° 66° 4. 38° Holt Algebra 2 52° 13-1 Right-Angle Trigonometry Warm Up Continued Find the unknown length for each right triangle with legs a and b and hypotenuse c. 5. b = 12, c =13 6. a = 3, b = 3 Holt Algebra 2 a=5 13-1 Right-Angle Trigonometry Objectives Understand and use trigonometric relationships of acute angles in triangles. Determine side lengths of right triangles by using trigonometric functions. Holt Algebra 2 13-1 Right-Angle Trigonometry Vocabulary trigonometric function sine cosine tangent cosecants secant cotangent Holt Algebra 2 13-1 Right-Angle Trigonometry A trigonometric function is a function whose rule is given by a trigonometric ratio. A trigonometric ratio compares the lengths of two sides of a right triangle. The Greek letter theta θ is traditionally used to represent the measure of an acute angle in a right triangle. The values of trigonometric ratios depend upon θ. Holt Algebra 2 13-1 Right-Angle Trigonometry Holt Algebra 2 13-1 Right-Angle Trigonometry The triangle shown at right is similar to the one in the table because their corresponding angles are congruent. No matter which triangle is used, the value of sin θ is the same. The values of the sine and other trigonometric functions depend only on angle θ and not on the size of the triangle. Holt Algebra 2 13-1 Right-Angle Trigonometry Example 1: Finding Trigonometric Ratios Find the value of the sine, cosine, and tangent functions for θ. sin θ = cos θ = tan θ = Holt Algebra 2 13-1 Right-Angle Trigonometry Check It Out! Example 1 Find the value of the sine, cosine, and tangent functions for θ. sin θ = cos θ = tan θ = Holt Algebra 2 13-1 Right-Angle Trigonometry You will frequently need to determine the value of trigonometric ratios for 30°,60°, and 45° angles as you solve trigonometry problems. Recall from geometry that in a 30°-60°-90° triangle, the ration of the side lengths is 1: 3 :2, and that in a 45°-45°-90° triangle, the ratio of the side lengths is 1:1: 2. Holt Algebra 2 13-1 Right-Angle Trigonometry Holt Algebra 2 13-1 Right-Angle Trigonometry Example 2: Finding Side Lengths of Special Right Triangles Use a trigonometric function to find the value of x. ° The sine function relates the opposite leg and the hypotenuse. Substitute 30° for θ, x for opp, and 74 for hyp. Substitute x = 37 Holt Algebra 2 for sin 30°. Multiply both sides by 74 to solve for x. 13-1 Right-Angle Trigonometry Check It Out! Example 2 Use a trigonometric function to find the value of x. ° The sine function relates the opposite leg and the hypotenuse. Substitute 45 ° for θ, x for opp, and 20 for hyp. Substitute for sin 45°. Multiply both sides by 20 to solve for x. Holt Algebra 2 13-1 Right-Angle Trigonometry Example 3: Sports Application In a waterskiing competition, a jump ramp has the measurements shown. To the nearest foot, what is the height h above water that a skier leaves the ramp? Substitute 15.1° for θ, h for opp., and 19 for hyp. 5≈h Multiply both sides by 19. Use a calculator to simplify. The height above the water is about 5 ft. Holt Algebra 2 13-1 Right-Angle Trigonometry Caution! Make sure that your graphing calculator is set to interpret angle values as degrees. Press . Check that Degree and not Radian is highlighted in the third row. Holt Algebra 2 13-1 Right-Angle Trigonometry Check It Out! Example 3 A skateboard ramp will have a height of 12 in., and the angle between the ramp and the ground will be 17°. To the nearest inch, what will be the length l of the ramp? Substitute 17° for θ, l for hyp., and 12 for opp. Multiply both sides by l and divide by sin 17°. Use a calculator to simplify. l ≈ 41 The length of the ramp is about 41 in. Holt Algebra 2 13-1 Right-Angle Trigonometry When an object is above or below another object, you can find distances indirectly by using the angle of elevation or the angle of depression between the objects. Holt Algebra 2 13-1 Right-Angle Trigonometry Example 4: Geology Application A biologist whose eye level is 6 ft above the ground measures the angle of elevation to the top of a tree to be 38.7°. If the biologist is standing 180 ft from the tree’s base, what is the height of the tree to the nearest foot? Step 1 Draw and label a diagram to represent the information given in the problem. Holt Algebra 2 13-1 Right-Angle Trigonometry Example 4 Continued Step 2 Let x represent the height of the tree compared with the biologist’s eye level. Determine the value of x. Use the tangent function. Substitute 38.7 for θ, x for opp., and 180 for adj. 180(tan 38.7°) = x Multiply both sides by 180. 144 ≈ x Use a calculator to solve for x. Holt Algebra 2 13-1 Right-Angle Trigonometry Example 4 Continued Step 3 Determine the overall height of the tree. x + 6 = 144 + 6 = 150 The height of the tree is about 150 ft. Holt Algebra 2 13-1 Right-Angle Trigonometry Check It Out! Example 4 A surveyor whose eye level is 6 ft above the ground measures the angle of elevation to the top of the highest hill on a roller coaster to be 60.7°. If the surveyor is standing 120 ft from the hill’s base, what is the height of the hill to the nearest foot? Step 1 Draw and label a diagram to represent the information given in the problem. Holt Algebra 2 60.7° 120 ft 13-1 Right-Angle Trigonometry Check It Out! Example 4 Continued Step 2 Let x represent the height of the hill compared with the surveyor’s eye level. Determine the value of x. Use the tangent function. Substitute 60.7 for θ, x for opp., and 120 for adj. 120(tan 60.7°) = x Multiply both sides by 120. 214 ≈ x Use a calculator to solve for x. Holt Algebra 2 13-1 Right-Angle Trigonometry Check It Out! Example 4 Continued Step 3 Determine the overall height of the roller coaster hill. x + 6 = 214 + 6 = 220 The height of the hill is about 220 ft. Holt Algebra 2 13-1 Right-Angle Trigonometry The reciprocals of the sine, cosine, and tangent ratios are also trigonometric ratios. They are trigonometric functions, cosecant, secant, and cotangent. Holt Algebra 2 13-1 Right-Angle Trigonometry Example 5: Finding All Trigonometric Functions Find the values of the six trigonometric functions for θ. Step 1 Find the length of the hypotenuse. a2 + b2 = c2 c2 = 242 + 702 Pythagorean Theorem. Substitute 24 for a and 70 for b. c2 = 5476 Simplify. c = 74 Holt Algebra 2 Solve for c. Eliminate the negative solution. 70 θ 24 13-1 Right-Angle Trigonometry Example 5 Continued Step 2 Find the function values. Holt Algebra 2 13-1 Right-Angle Trigonometry Helpful Hint In each reciprocal pair of trigonometric functions, there is exactly one “co” Holt Algebra 2 13-1 Right-Angle Trigonometry Check It Out! Example 5 Find the values of the six trigonometric functions for θ. Step 1 Find the length of the hypotenuse. a2 + b2 = c2 c2 = 182 + 802 Pythagorean Theorem. Substitute 18 for a and 80 for b. c2 = 6724 Simplify. c = 82 Holt Algebra 2 Solve for c. Eliminate the negative solution. 80 θ 18 13-1 Right-Angle Trigonometry Check It Out! Example 5 Continued Step 2 Find the function values. Holt Algebra 2 13-1 Right-Angle Trigonometry Lesson Quiz: Part I Solve each equation. Check your answer. 1. Find the values of the six trigonometric functions for θ. Holt Algebra 2 13-1 Right-Angle Trigonometry Lesson Quiz: Part II 2. Use a trigonometric function to find the value of x. 3. A helicopter’s altitude is 4500 ft, and a plane’s altitude is 12,000 ft. If the angle of depression from the plane to the helicopter is 27.6°, what is the distance between the two, to the nearest hundred feet? 16,200 ft Holt Algebra 2