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9.4/9.5 Warmup Find the measure of the missing leg in the 𝒚𝟏 right triangle, and then calculate the ratio . 𝒙𝟏 1. 9 2. The two triangles are _____________ so two angles in each triangle are ___________. March 28, 2016 Geometry 9.5 Trigonometric Ratios 1 Geometry 9.4 & 9.5 Trigonometric Ratios [email protected] Essential Question How is a right triangle used to find the sine, cosine, and tangent of an acute angle? March 28, 2016 Geometry 9.5 Trigonometric Ratios 3 Goals Find the sine, cosine, and tangent of an acute angle. Solve problems using trigonometric ratios. March 28, 2016 Geometry 9.5 Trigonometric Ratios 4 Terminology No one, except stuffed-shirt mathematics teachers, uses the word trigonometry. It’s March 28, 2016 Geometry 9.5 Trigonometric Ratios 5 What is trig? Literally, the measure of triangles. An extremely useful, practical and powerful math tool. A branch of math that finds its way into practically everything we do. Usually learned in high school. March 28, 2016 Geometry 9.5 Trigonometric Ratios 6 What you will learn… The basic terms and methods of solving right triangles. How to use a calculator’s trig functions. How to solve problems using trig. March 28, 2016 Geometry 9.5 Trigonometric Ratios 7 Trig Ratios Based on the sides of a right triangle. We will study only three: Sine Cosine Tangent March 28, 2016 Geometry 9.5 Trigonometric Ratios 8 Right Triangle Leg From A, this leg is the Opposite side. A Leg From A, this leg is the Adjacent side. March 28, 2016 Geometry 9.5 Trigonometric Ratios 9 Right Triangle B From B, this leg is the Adjacent side. Leg From A, this leg is the Opposite side. A Leg From A, this leg is the Adjacent side. From B, this leg is the Opposite side. March 28, 2016 Geometry 9.5 Trigonometric Ratios 10 Right Triangle Opposite A March 28, 2016 Adjacent Geometry 9.5 Trigonometric Ratios 11 Trig Ratio Definition: Sine Opposite A Adjacent Opposite Sine of A = Hypotenuse March 28, 2016 Geometry 9.5 Trigonometric Ratios 12 Trig Ratio Definition: Cosine Opposite A Adjacent Adjacent Cosine of A = Hypotenuse March 28, 2016 Geometry 9.5 Trigonometric Ratios 13 Trig Ratio Definition: Tangent Opposite A Adjacent Opposite Tangent of A = Adjacent March 28, 2016 Geometry 9.5 Trigonometric Ratios 14 Abbreviations Opposite A = Hypotenuse Sine sin of A Adjacent cos A Cosine of A = Hypotenuse Opposite A = Adjacent Tangent tan of A March 28, 2016 Geometry 9.5 Trigonometric Ratios 15 Memory Aid Sine is Opposite over Hypotenuse. Cosine is Adjacent over Hypotenuse. Tangent is Opposite over Adjacent. SOH CAH TOA March 28, 2016 Geometry 9.5 Trigonometric Ratios 16 Trig Ratios A March 28, 2016 Geometry 9.5 Trigonometric Ratios 17 Writing Ratios SOH CAH TOA 4? sin B 5? 3 ? cos B 5 ? 4 ? tan B 3 ? March 28, 2016 B 5 3 4 Geometry 9.5 Trigonometric Ratios 3? sin A 5? 4 ? cos A 5? A 3? tan A 4? 18 Writing Ratios SOH CAH TOA 4 sin B 5 3 cos B 5 4 tan B 3 March 28, 2016 B 5 3 4 Geometry 9.5 Trigonometric Ratios 3 sin A 5 4 cos A 5 A 3 tan A 4 19 Example 1 Find sin S, cos S, and tan S. Write each answer as a fraction and as a decimal rounded to four places. sin S = 80 40 = = .9756 82 41 cos S = 18 82 tan S = 80 40 = = 4.444 18 9 = March 28, 2016 9 41 = .2195 Geometry 9.5 Trigonometric Ratios 20 Your Turn Find sin R, cos R, and tan R. Write each answer as a fraction and as a decimal rounded to four places. March 28, 2016 sin R = 18 82 cos R = 80 40 = = .9756 82 41 tan R = 18 9 = = .2250 80 40 Geometry 9.5 Trigonometric Ratios = 9 41 = .2195 21 Calculators Make sure your calculator is in DEGREE mode. Always use four decimal places of accuracy when using trig functions. All demonstrations here are from a TI graphing calculator. March 28, 2016 Geometry 9.5 Trigonometric Ratios 22 Mode Setting Press MODE Use the cursor arrows and move to Degree. Press ENTER. Press 2nd Quit. Press Clear March 28, 2016 Geometry 9.5 Trigonometric Ratios 23 Using Trig Functions To find the sin 78: Press ‘sin’ Enter 78 Press ENTER. Answer is .9781 March 28, 2016 Geometry 9.5 Trigonometric Ratios 24 Find these values: March 28, 2016 sin 15 cos 45 tan 45 cos 80 sin 10 tan 5 cos 60 sin 90 .2588 .7071 1 .1736 .1736 .0875 .5 1 Geometry 9.5 Trigonometric Ratios 25 Solving Triangles Carefully analyze the given information. Decide what you are trying to find. Ask: Which trig function fits this problem? WRITE AN EQUATION. (SOH CAH TOA) Solve. March 28, 2016 Geometry 9.5 Trigonometric Ratios 26 Example 2 Find x. From the 28 angle, x is the ? Opposite side, and 15 is the x 15 Hypotenuse. What trig ratio is this? 28 Sine (SOH CAH TOA) March 28, 2016 Geometry 9.5 Trigonometric Ratios 27 Example 2 Find x. Write the equation and solve. x sin 28 15 15sin 28 x x 15 28 x 7.0 March 28, 2016 Geometry 9.5 Trigonometric Ratios 28 Example 3 Find y. Write the equation and solve. y cos31 56 56cos31 y y 48.0 March 28, 2016 56 31 y Geometry 9.5 Trigonometric Ratios 29 Example 4 Find a. Write the equation and solve. a tan 40 8 8 tan 40 a a 8 a 6.7 March 28, 2016 40 Geometry 9.5 Trigonometric Ratios 30 Fraction Reminder If b ac 8 2 4 March 28, 2016 Then b ca 8 4 2 Geometry 9.5 Trigonometric Ratios 31 Example 5 Find a. Write the equation and solve. 17 tan 40 a a tan 40 17 17 40 17 a tan 40 x 20.3 March 28, 2016 a Geometry 9.5 Trigonometric Ratios 32 Example 6 x y Find x & y. x tan 78 150 150 tan 78 x x 705.7 78 150 cos 78 y 150 y cos 78 y 721.5 150 March 28, 2016 Geometry 9.5 Trigonometric Ratios 33 Angle of Elevation Angle of Elevation Horizontal March 28, 2016 Geometry 9.5 Trigonometric Ratios 34 Angle of Depression Horizontal Angle of Depression March 28, 2016 Geometry 9.5 Trigonometric Ratios 35 Example 7 Standing 30 yards from a tree, the angle of elevation to the top of the tree is 15. How tall is the tree? h tan15 30 h 30 tan15 h 8.0 h 15 30 yd March 28, 2016 Geometry 9.5 Trigonometric Ratios 36 Example 8 Isabella is 30 feet from a fearsome monster. The angle of elevation to the top of the monster’s head is 42. How tall is the monster? 42 March 28, 2016 x ft 30 ft Geometry 9.5 Trigonometric Ratios 37 Solution x tan 42 30 x 30 tan 42 30(.9004) 27 42 March 28, 2016 x ft 30 ft Geometry 9.5 Trigonometric Ratios 38 Solution x tan 42 30 x 30 tan 42 30(.9004) 27 42 March 28, 2016 27 ft 30 ft Geometry 9.5 Trigonometric Ratios 39 Your Turn You are skiing on a mountain. You start at an altitude of 8400 feet and ski down to an altitude of 7200. The angle of depression is 21°. Find the distance x you ski down the mountain to the nearest foot. y You ski about 3349 ft down the mountain. March 28, 2016 Geometry 9.5 Trigonometric Ratios 40 Summary Trig ratios are based on acute angles in right triangles. They are Sine, Cosine, Tangent. SOH CAH TOA Angle of elevation is from the ground up. March 28, 2016 Geometry 9.5 Trigonometric Ratios 41 Homework March 28, 2016 Geometry 9.5 Trigonometric Ratios 42