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A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.

Finding Trigonometric Ratios. A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles.

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A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.

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  1. Finding Trigonometric Ratios A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan, respectively.

  2. Finding Trigonometric Ratios Let be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows. a c side opposite A hypotenuse = ABC b c side adjacent A hypotenuse = side opposite A side adjacent to A a b = A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. TRIGONOMETRIC RATIOS B sinA= hypotenuse side opposite A c a cosA= A C b tanA= side adjacent to A The value of the trigonometric ratio depends only on the measure of the acute angle, not on the particular right triangle that is used to compute the value.

  3. Finding Trigonometric Ratios B 8.5 4 C A 7.5 B 17 8 A C 15 opposite hypotenuse sinA = 7.5 8.5 4 8.5 4 7.5 15 17 8 17 8 15  0.4706  0.4706 adjacent hypotenuse cosA =  0.8824  0.8824 opposite adjacent tanA =  0.5333  0.5333 Compare the sine, the cosine, and the tangent ratios for A in each triangle below. SOLUTION By the SSS Similarity Theorem, the triangles are similar. Their corresponding sides are in proportion, which implies that the trigonometric ratios for A in each triangle are the same. Trigonometric ratios are frequently expressed as decimal approximations.

  4. Finding Trigonometric Ratios R 13 5 T S 12 opp. adj. opp. hyp. adj. hyp. = = R 5 12 12 13 5 13 13 5 = = S T 12 = = Find the sine, the cosine, and the tangent of the indicated angle. S SOLUTION The length of the hypotenuse is 13. For S, the length of the opposite side is 5, and the length of the adjacent side is 12.  0.3846 sinS hyp.  0.9231 cosS opp. adj.  0.4167 tanS

  5. Finding Trigonometric Ratios R 13 5 T S 12 opp. hyp. opp. adj. adj. hyp. = = R 12 5 5 13 12 13 13 5 = = S T 12 = = Find the sine, the cosine, and the tangent of the indicated angle. R SOLUTION The length of the hypotenuse is 13. For R, the length of the opposite side is 12, and the length of the adjacent side is 5. sinR  0.9231 hyp. cosR  0.3846 adj. opp. tanR = 2.4

  6. Trigonometric Ratios for 45º From the 45º-45º-90º Triangle Theorem, it follows that the length of the hypotenuse is 2 . = = = 1 2 2 2 = = opp. adj. opp. hyp. adj. hyp. 2 1 2 2 2 = = 45º 1 1 = Find the sine, the cosine, and the tangent of 45º. SOLUTION Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. sin45º  0.7071 hyp. 1 cos45º  0.7071 1 tan45º = 1

  7. Trigonometric Ratios for 30º From the 30º-60º-90º Triangle Theorem, it follows that the length of the longer leg is 3 and the length of the hypotenuse is 2. 1 2 = = = = opp. adj. opp. hyp. adj. hyp. 3 2 = 30º 1 3 3 3 = = 3 Find the sine, the cosine, and the tangent of 30º. SOLUTION To make the calculations simple, you can choose 1 as the length of the shorter leg. = 0.5 sin30º 2  0.8660 cos30º 1  0.5774 tan30º

  8. Using a Calculator 74 or 74 74 or 74 74 or 74 cos sin sin tan tan cos 0.961261695 0.275637355 3.487414444 Enter Enter Enter You can use a calculator to approximate the sine, the cosine, and the tangent of 74º. Make sure your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators. 0.9613 0.2756 3.4874

  9. Finding Trigonometric Ratios The sine or cosine of an acute angle is always less than 1. The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse. The length of a leg of a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one. Because the tangent of an acute angle involves the ratio of one leg to another leg, the tangent of an angle can be less than 1, equal to 1, or greater than 1.

  10. Finding Trigonometric Ratios B c a C A b sin A cos A tan A = The sine or cosine of an acute angle is always less than 1. The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse. The length of a leg of a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one. Because the tangent of an acute angle involves the ratio of one leg to another leg, the tangent of an angle can be less than 1, equal to 1, or greater than 1. TRIGONOMETRIC IDENTITIES A trigonometric identity is an equation involving trigonometric ratios that is true for all acute angles. The following are two examples of identities: (sin A)2+ (cos A)2 = 1

  11. Using Trigonometric Ratios in Real Life Suppose you stand and look up at a point in the distance, such as the top of the tree. The angle that your line of sight makes with a line drawn horizontally is called the angle of elevation.

  12. Indirect Measurement opposite adjacent opposite adjacent tan59° = tan59° = h 45 The tree is about 75 feet tall. FORESTRY You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of a tree. You measure the angle of elevation from a point on the ground to the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet. Write ratio. Substitute. 45 tan 59° = h Multiply each side by 45. 45(1.6643) h Use a calculator or table to find tan 59°. 74.9 h Simplify.

  13. Estimating a Distance opposite hypotenuse sin30° = opposite hypotenuse sin30° = 76 d d 76 ft 30° 76 sin 30° d = 76 0.5 d = A person travels 152 feet on the escalator stairs. ESCALATORS The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg length of 76 feet. Write ratio for sine of 30°. Substitute. d sin 30° = 76 Multiply each side by d. Divide each side by sin 30°. Substitute 0.5 for sin 30°. d = 152 Simplify.

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