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10-3 Trigonometry Do Now Lesson Presentation Exit Ticket 10-3 Trigonometry DO NOW #14 Write each fraction as a decimal rounded to the nearest hundredth. 1. 0.67 2. 0.29 Solve each equation. 3. 4. x = 7.25 x = 7.99 10-3 Trigonometry Connect to Mathematical Ideas (1)(F) By the end of today’s lesson, SWBAT Find the sine, cosine, and tangent of an acute angle. Use trigonometric ratios to find side lengths in right triangles and to solve real-world problems. 10-3 Trigonometry Vocabulary trigonometric ratio sine cosine tangent 10-3 Trigonometry By the AA Similarity Postulate, a right triangle with a given acute angle is similar to every other right triangle with that same acute angle measure. So ∆ABC ~ ∆DEF ~ ∆XYZ, 𝐵𝐶 𝐸𝐹 𝑌𝑍 and 𝐴𝐶 = 𝐷𝐹 = 𝑋𝑍 . These are trigonometric ratios. A trigonometric ratio is a ratio of two sides of a right triangle. 10-3 Trigonometry Writing Math In trigonometry, the letter of the vertex of the angle is often used to represent the measure of that angle. For example, the sine of A is written as sin A. 10-3 Trigonometry 10-3 Trigonometry Example 1: Writing Trigonometric Ratios What are the sine, cosine, and tangent ratios for T ? opposite 8 𝐬𝐢𝐧 𝑻 = = hypotenuse 17 adjacent 15 𝐜𝐨𝐬 𝑻 = = hypotenuse 17 opposite 𝐭𝐚𝐧 𝑻 = adjacent 8 = 15 10-3 Trigonometry Example 2: Using a Trigonometric Ratios to Find Distance Landmarks In 1990, the Leaning Tower of Pisa was closed to the public due to safety concerns. The tower reopened in 2001 after a 10-year project to reduce its tilt from vertical. Engineers’ efforts were successful and resulted in a tilt of 5o, reduced from 5.5o. Suppose someone drops an object from the tower at a height of 150 ft. How far from the base of the tower will the object land? Round to the nearest foot. The given side is adjacent to the given angle. The side you want to find its opposite the given angle. 𝑥 = 150 𝑥 = 150(𝐭𝐚𝐧 5o ) 𝐭𝐚𝐧 5o Use the tangent ratio. Multiply each side by 150. 𝑥 ≈ 13.12329953 Use a calculator. 13 ft 10-3 Trigonometry Caution! Do not round until the final step of your answer. Use the values of the trigonometric ratios provided by your calculator. 10-3 Trigonometry Example 3: Using Inverses What is the mX in the nearest degree? You know the lengths of the hypotenuse and the side opposite X. 𝑼𝒔𝒆 𝒕𝒉𝒆 𝒔𝒊𝒏𝒆 𝒓𝒂𝒕𝒊𝒐. 6 𝐬𝐢𝐧 𝑿 = 10 6 −1 𝑿 = sin 10 𝒎𝑿 ≈ 37o You know the lengths of the hypotenuse and the side adjacent to X. 𝑼𝒔𝒆 𝒕𝒉𝒆 𝒄𝒐𝒔𝒊𝒏𝒆 𝒓𝒂𝒕𝒊𝒐. 15 Write the ratio. 𝐜𝐨𝐬 𝑿 = 20 15 −1 𝑿 = cos Use the inverse. 20 𝒎𝑿 ≈ 41o Use a calculator. 10-3 Trigonometry Example 4: Using the Tangent Ratio An urban planner needs to know the measure of the angle formed by a street and the edge of a bike path. What is the measure of DEG to the nearest degree? ∆DEG is a right triangle. You know the lengths of the side adjacent to and the side opposite DEG. 𝑼𝒔𝒆 𝒕𝒉𝒆 𝒕𝒂𝒏𝒈𝒆𝒏𝒕 𝒓𝒂𝒕𝒊𝒐. 150 Write the ratio. 𝐭𝐚𝐧 𝐷𝐸𝐺 = 115 150 −1 DEG = tan Use the inverse. 115 𝒎𝑫𝑬𝑮 ≈ 53o Use a calculator. 10-3 Trigonometry Got It ? Solve With Your Partner Problem 1 Writing Trigonometric Ratios What are the sine, cosine, and tangent ratios for G ? opposite 15 𝐬𝐢𝐧 𝑮 = = hypotenuse 17 adjacent 8 𝐜𝐨𝐬 𝑮 = = hypotenuse 17 opposite 𝐭𝐚𝐧 𝑮 = adjacent 15 = 8 10-3 Trigonometry Got It ? Solve With Your Partner Problem 2 Using Trigonometric Ratio to Find Distance Find the value of w to the nearest tenth. B. A. 13.8 C. 1.9 3.8 10-3 Trigonometry Got It ? Solve With Your Partner Problem 3 Using Trigonometric Ratio to Find Distance A section of Filbert Street in San Francisco rises at an angle of about 17o. If you walk 150 ft up this section, what is your vertical rise? Round to the nearest foot. 44 ft 10-3 Trigonometry Closure: Communicate Mathematical Ideas (1)(G) How could you determine the value of sin 35o without using a calculator ? Draw a right triangle with one acute of 35o. Measure the length of the opposite side and the hypotenuse. Then find the ratio of the length of the opposite side to the length of the hypotenuse. 10-3 Trigonometry Exit Ticket: Apply Mathematics (1)(A) Use a special right triangle to write each trigonometric ratio as a fraction. 1. sin 60° 2. cos 45° Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 3. tan 84° 4. cos 13° Find each length. Round to the nearest tenth. 5. CB 6. AC