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NAME ______________________________________________ DATE 8-4 ____________ PERIOD _____ Lesson Reading Guide Trigonometry Get Ready for the Lesson Read the introduction to Lesson 8-4 in your textbook. • Why is it important to determine the relative positions accurately in navigation? (Give two possible reasons.) • What does calibrated mean? Read the Lesson 1. Refer to the figure. Write a ratio using the side lengths in the figure to represent each of the following trigonometric ratios. A. sin N M N B. cos N P C. tan N D. tan M E. sin M F. cos M a. sin 20 i. the degree measure of an acute angle whose cosine is 0.8 b. cos 20 ii. the ratio of the length of the leg adjacent to the 20° angle to the length of hypotenuse in a 20°-70°-90° triangle c. sin⫺1 0.8 d. tan⫺1 0.8 e. tan 20 f. cos⫺1 0.8 iii.the degree measure of an acute angle in a right triangle for which the ratio of the length of the opposite leg to the length of the adjacent leg is 0.8 iv. the ratio of the length of the leg opposite the 20° angle to the length of the leg adjacent to it in a 20°-70°-90° triangle v. the ratio of the length of the leg opposite the 20° angle to the length of hypotenuse in a 20°-70°-90° triangle vi. the degree measure of an acute angle in a right triangle for which the ratio of the length of the opposite leg to the length of the hypotenuse is 0.8 Remember What You Learned 3. How can the co in cosine help you to remember the relationship between the sines and cosines of the two acute angles of a right triangle? Chapter 8 27 Glencoe Geometry Lesson 8-4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 2. Assume that you enter each of the expressions in the list on the left into your calculator. Match each of these expressions with a description from the list on the right to tell what you are finding when you enter this expression. NAME ______________________________________________ DATE 8-4 ____________ PERIOD _____ Study Guide and Intervention Trigonometry Trigonometric Ratios The ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The three most common ratios are sine, cosine, and tangent, which are abbreviated sin, cos, and tan, respectively. leg opposite ⬔R hypotenuse r ⫽ ᎏᎏ t sin R ⫽ ᎏᎏ leg adjacent to ⬔R cos R ⫽ ᎏᎏᎏ hypotenuse s t S r t T s leg opposite ⬔R tan R ⫽ ᎏᎏᎏ leg adjacent to ⬔R r s ⫽ ᎏᎏ ⫽ ᎏᎏ Example Find sin A, cos A, and tan A. Express each ratio as a decimal to the nearest thousandth. B 13 5 C opposite leg hypotenuse sin A ⫽ ᎏᎏ adjacent leg hypotenuse cos A ⫽ ᎏᎏ opposite leg adjacent leg ⫽ ᎏᎏ ⫽ ᎏᎏ 12 13 ⫽ ᎏᎏ ⬇ 0.923 ⬇ 0.417 ⫽ ᎏᎏ ⫽ ᎏᎏ 5 13 ⬇ 0.385 A 12 tan A ⫽ ᎏᎏ AC AB BC AB ⫽ ᎏᎏ R BC AC 5 12 Exercises 1. sin A B 30 20 2. tan B C 3. cos A 4. cos B 5. sin D 6. tan E 7. cos E 8. cos D Chapter 8 E 34 28 16 A D 16 12 F Glencoe Geometry Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Find the indicated trigonometric ratio as a fraction and as a decimal. If necessary, round to the nearest ten-thousandth. NAME ______________________________________________ DATE 8-4 Study Guide and Intervention ____________ PERIOD _____ (continued) Trigonometry Use Trigonometric Ratios In a right triangle, if you know the measures of two sides or if you know the measures of one side and an acute angle, then you can use trigonometric ratios to find the measures of the missing sides or angles of the triangle. Example Find x, y, and z. Round each measure to the nearest whole number. A z B a. Find x. y 18 C c. Find z. b. Find y. x ⫹ 58 ⫽ 90 x ⫽ 32 x⬚ 58⬚ y 18 y tan 58° ⫽ ᎏᎏ 18 18 z 18 cos 58° ⫽ ᎏᎏ z cos A ⫽ ᎏᎏ tan A ⫽ ᎏᎏ y ⫽ 18 tan 58° y ⬇ 29 z cos 58° ⫽ 18 18 cos 58° z ⫽ ᎏᎏ z ⬇ 34 Exercises 1. 2. x 3. x⬚ 16 12 28⬚ 32 x⬚ 4. 12 1 x⬚ 5 Lesson 8-4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Find x. Round to the nearest tenth. 4 5. 40⬚ 16 Chapter 8 6. 64⬚ 15 x 29 x Glencoe Geometry