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7.4 Trigonometry Find trigonometric ratios using right triangles Solve problems using trigonometric ratios The word trigonometry originates from two Greek terms, trigon, which means triangle, and metron, which means measure. Thus, the study of trigonometry is the study of triangle measurements. A ratio of the lengths of the sides of a right triangle is called a trigonometric ratio. The three most common trigonometric ratios are sine, cosine, and tangent. For right ∆ABC… sin A = opposite side = a hypotenuse c A c b cos A = adjacent side = b hypotenuse c tan A = opposite side = a adjacent side b C a B To help you remember A these trigonometric Sin Arelationships, = Opposite side youSOH can Hypotenuse use the mnemonic Cos A = Adjacent side CAH device, SOH-CAH-TOA, b Hypotenuse where the first letter of Tan A = Opposite side each word of theTOA Adjacent side trigonometric ratios is C represented in the correct order. c a B Find sin L, cos L, tan L, sin N, cos N, and tan N. Express each ratio as a fraction and as a decimal. Answer: Find sin A, cos A, tan A, sin B, cos B, and tan B. Express each ratio as a fraction and as a decimal. Answer: Use a calculator to find tan thousandth. KEYSTROKES: TAN Answer: 56 to the nearest ten ENTER 1.482560969 Use a calculator to find cos thousandth. KEYSTROKES: COS Answer: 90 to the nearest ten ENTER 0 a. Use a calculator to find sin 48° to the nearest ten thousandth. Answer: b. Use a calculator to find cos 85° to the nearest ten thousandth. Answer: You can use a calculator or a trigonometric table to find the missing measures of a right triangle if you are given the measures of two sides of the triangle or one side and one acute angle. EXERCISING A fitness trainer sets the incline on a treadmill to The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor? Let y be the height of the treadmill from the floor in inches. The length of the treadmill is 5 feet, or 60 inches. Multiply each side by 60. Use a calculator to find y. KEYSTROKES: 60 SIN 7 ENTER 7.312160604 Answer: The treadmill is about 7.3 inches high. CONSTRUCTION The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about how high does the ramp rise off the ground to the nearest inch? Answer: about 15 in. COORDINATE GEOMETRY Find mX in right XYZ for X(–2, 8), Y(–6, 4), and Z(–3, 1). Explore You know the coordinates of the vertices of a right triangle and that is the right angle. You need to find the measures of one of the angles. Plan Use the Distance Formula to find the measure of each side. Then use one of the trigonometric ratios to write an equation. Use the inverse to find Solve or or or Use the cosine ratio. Simplify. Solve for x. Use a calculator to find KEYSTROKES: 2ND 4 ENTER Examine Use the sine ratio to check the answer. Simplify. 5 ) KEYSTROKES: 2ND ENTER Answer: The measure of is about 36.9. 3 5 ) COORDINATE GEOMETRY Answer: about 56.3 Pre-AP Geometry: Pg. 368 #18 – 48, 50, 51, 56 – 58 Geometry: Pg. 368 #18 – 48