Geometry 9.5 Trigonometric Ratios 9.5 Warmup 1. Can these 3 side lengths 2. Find tan B. make a Triangle? If yes, classify the triangle. 7, 9, 12 3. Find x & y. X February 22, 2019 Geometry 9.5 Trigonometric Ratios

9.4 & 9.5 Trigonometric Ratios Geometry 9.4 & 9.5 Trigonometric Ratios mbhaub@mpsaz.org

Geometry 9.5 Trigonometric Ratios Essential Question How is a right triangle used to find the sine, cosine, and tangent of an acute angle? February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Goals Find the sine, cosine, and tangent of an acute angle. Solve problems using trigonometric ratios. February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Right Triangle Hypotenuse Opposite A Adjacent February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Sine and Cosine Ratios February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Trig Ratios A February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Memory Aid Sine is Opposite over Hypotenuse. Cosine is Adjacent over Hypotenuse. Tangent is Opposite over Adjacent. SOH CAH TOA February 22, 2019 Geometry 9.5 Trigonometric Ratios

Writing Ratios SOH CAH TOA 3 4 5 A B February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Example 1 Find sin J, cos J, and tan J. Write each answer as a fraction and as a decimal rounded to four places. 24 40 = 3 5 = .6 sin J = cos J = tan J = 32 40 = 4 5 = .8 24 32 = 3 4 =.75 Geometry 9.5 Trigonometric Ratios February 22, 2019

Geometry 9.5 Trigonometric Ratios Your Turn Find sin K, cos K, and tan K. Write each answer as a fraction and as a decimal rounded to four places. 32 40 = 4 5 = .8 sin K = cos K = tan K = 24 40 = 3 5 = .6 32 24 = 4 3 =1.3333 February 22, 2019 Geometry 9.5 Trigonometric Ratios

Sine and Cosine of Complementary Angles February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Example 2 Find the cosine that is equivalent to the given sine. sin 83° sin 34° sin 57° February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Your Turn Find the sine that is equivalent to the given cosine. cos 26° cos 73° cos 45° February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Using Special Right Triangles to find sine and cosine of a 30, 60 or 45 angle. 2x Use a special right triangle to find the sine and cosine of a 60° angle. x sin 60= 𝑜𝑝𝑝 ℎ𝑦𝑝 sin 60= 3 2 Check: sin 60≈.8660 x 3 cos 60= 𝑎𝑑𝑗 ℎ𝑦𝑝 cos 60= 1 2 Check: sin 60≈.5 February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Using Special Right Triangles to find sine and cosine of a 30, 60 or 45 angle. 2x Use a special right triangle to find the sine and cosine of a 30° angle. x sin 30= 𝑜𝑝𝑝 ℎ𝑦𝑝 sin 30= 1 2 Check: sin 30≈.5 x 3 cos 30= 𝑎𝑑𝑗 ℎ𝑦𝑝 cos 30= 3 2 Check: cos 30≈.8660 February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Using Special Right Triangles to find sine and cosine of a 30, 60 or 45 angle. x 2 Use a special right triangle to find the sine and cosine of a 45° angle. x sin 45= 𝑜𝑝𝑝 ℎ𝑦𝑝 sin 45= 1 2 sin 45= 2 2 Check: sin 45≈.7071 cos 45= 𝑎𝑑𝑗 ℎ𝑦𝑝 cos 45= 1 2 cos 45= 2 2 Check: cos 45≈.7071 x ∙ 2 2 ∙ 2 2 February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Calculators Remember to make sure your calculator is in DEGREE mode. Always use four decimal places of accuracy when using trig functions. February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Find these values: sin 15 cos 45 tan 45 cos 80 sin 10 tan 5 cos 60 .2588 .7071 1 .1736 .0875 .5 February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Solving Triangles To solve a Right Triangle, all we need is ONE acute angle and ONE side. Carefully analyze the given information. Decide what you are trying to find. Ask: How do you use the given information to solve the problem? WRITE AN EQUATION. (SOH CAH TOA) Solve. February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Example 3 Find x. From the 28 angle, x is the ? Opposite side, and 15 is the Hypotenuse. What trig ratio is this? Sine (SOHCAHTOA) 28 x 15 February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Example 3 Find x. Write the equation and solve. 28 x 15 February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Example 4 Find y. Write the equation and solve. 56 31 y February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Example 5 Find x & y. tan 26= 𝑜𝑝𝑝 𝑎𝑑𝑗 tan 26= 𝑥 14 14 tan 26=𝑥 𝑦≈6.8283 cos 26= 𝑎𝑑𝑗 ℎ𝑦𝑝 cos 26= 14 𝑦 y cos 26=14 𝑦= 14 𝑐𝑜𝑠26 𝑦≈15.6 26 y 14 x February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios 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. February 22, 2019 Geometry 9.5 Trigonometric Ratios

Geometry 9.5 Trigonometric Ratios Homework February 22, 2019 Geometry 9.5 Trigonometric Ratios