You can solve a right triangle if you know either of the following: Two side lengths One side length and one acute angle measure

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1 Solving a Right Triangle A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs. To solve a right triangle means to determine the measures of all six parts. You can solve a right triangle if you know either of the following: Two side lengths One side length and one acute angle measure As you learned in Lesson 9.5, you can use the side lengths of a right triangle to find trig ratios for the acute angles of the triangle. As you will see in this lesson, once you know the sine, the cosine, or the tangent of an acute angle, you can use a calculator to find the measure of the angle.

2 Solving a Right Triangle A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs. To solve a right triangle means to determine the measures of all six parts. You can solve a right triangle if you know either of the following: Two side lengths One side length and one acute angle measure In general, for an acute angle A: if sin A = x, then sine -1 x = m A. if cos A = y, then cos -1 y = m A. if tan A = z, then tan -1 z = m A.

3 Solving a Right Triangle SOLVE the right triangle. Round decimals to the nearest tenth. R 2 c SOLUTION: Step 1 find the missing side. T 3 S Begin by using the Pythagorean Theorem to find the length of the hypotenuse. (hypotenuse) 2 = (leg) 2 + (leg) 2 c 2 = c 2 = c 2 = 13 c = R 2 opp. T 3 c hyp. adj. S

4 Solving a Right Triangle SOLVE the right triangle. Round decimals to the nearest tenth. R SOLUTION: Step 2 find the first missing. Then use a calculator to find the measure of S: For the TI-30xa classroom calculator: 2 T 3 c 3.6 S 2 3 ( 2nd ( TAN For the TI-30xs recommended calculator: 2nd TAN this just appears ( 2 3 m S 33.7 ( Enter R 2 opp. T 3 c hyp. adj. S

5 Solving a Right Triangle SOLVE the right triangle. Round decimals to the nearest tenth. R SOLUTION: Step 3 find the last 2 c missing. T 3 S Finally, because R and S are complements, you can write: m R = 90 - m S = 56.3 (conclusion) The side lengths of the triangle are 2, 3, and 13, or about 3.6. The angle measurements are 90, about 33.7, and about 56.3.

6 Solving a Right Triangle Solve the right triangle. Round decimals to the nearest tenth. SOLUTION: Step 1 Use trigonometric ratios to find the values of g and h. sin H = sin 25º = º opp. hyp. opp. hyp. = 13(0.4226º) = h 13 h h 13 cos H = cos 25º º = = 13(0.9063º) adj. hyp. adj. hyp. g 13 g = g 13 h G J g 13 25º H 5.5 h 11.8 g

7 Solving a Right Triangle Solve the right triangle. Round decimals to the nearest tenth. SOLUTION: Step 2 Use the Triangle Sum Theorem to find m G. G Because H and G are complements, you can write: m G = 90 - m H = = 65 h J 13 g 25º H (conclusion) The side lengths of the triangle are about 5.5, about 11.8, and 13. The angle measurements are 90, 65, and 25.

8 Trigonometric Ratios for 45º 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. From the 45º-45º-90º Triangle Theorem, it follows that the length of the hypotenuse is 2. opp. sin 45º = = 1 2 = hyp. 2 2 cos 45º adj. = = 1 2 = hyp. 2 2 opp. tan 45º = = 1 = 1 adj hyp. 45º

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 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 tan A = sin A cos A A c b B C a

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 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. tan 59 = opposite adjacent tan 59 = opposite h 45 adjacent 45 tan 59 = h 45(1.6643) h 74.9 h Write ratio. Substitute. Multiply each side by 45. Use a calculator or table to find tan 59. Simplify. The tree is about 75 feet tall.

13 Estimating a Distance 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. sin 30 = opposite hypotenuse Write ratio for sine of 30. short leg = 76 Whoa! This is a triangle!!! hypotenuse = 76(2) What is d? d 76 ft 30 d = 152 Simplify. A person travels 152 feet on the escalator stairs.

14 Estimating a Distance 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. sin 30 = opposite hypotenuse Write ratio for sine of 30. sin 30 = 76 opposite hypotenuse d Substitute. d sin 30 = d = sin 30 d = Multiply each side by d. Divide each side by sin 30. Substitute 0.5 for sin d 76 ft d = 152 Simplify. A person travels 152 feet on the escalator stairs.