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1. L E S S O N Trigonometric Ratios 12.1 Trigonometry is the study of the relationships between the sides and the angles of triangles. In this lesson you will discover some of these relationships for right triangles.Research is what I amdoing when I don’t know Astronomywhat I’m doing.WERNHER VON BRAUN Trigonometry has origins in astronomy. The Greek astronomer Claudius Ptolemy (100–170 C.E.) used tables of chord ratios in his book known as Almagest. These chord ratios and their related angles were used to describe the motion of planets in what were thought to be circular orbits. This woodcut shows Ptolemy using astronomy tools. When studying right triangles, early mathematicians discovered that whenever the ratio of the shorter leg’s length to the longer leg’s length was close to a specific fraction, the angle opposite the shorter leg was close to a specific measure. They found this (and its converse) to be true for all similar right triangles. For example, in every right triangle in which the ratio of the shorter leg’s length to the longer leg’s length is , the angle opposite the shorter leg is almost exactly 31°. What early mathematicians discovered is supported by what you know about similar triangles. If two right triangles each have an acute angle of the same measure, then the triangles are similar by the AA Similarity Conjecture. And if the triangles are similar, then corresponding sides are proportional. For example, in the similar right triangles shown below, these proportions are true: This side is called the opposite leg because it is across from the 20° angle. This side is called the adjacent leg because it is next to the 20° angle. The ratio of the length of the opposite leg to the length of the adjacent leg in a right triangle came to be called the tangent of the angle.640 CHAPTER 12 Trigonometry © 2008 Key Curriculum Press
In Chapter 11, you used mirrors and shadows to measure heights indirectly. Trigonometry gives you another indirect measuring method. EXAMPLE A At a distance of 36 meters from a tree, the angle from the ground to the top of the tree is 31°. Find the height of the tree. Solution As you saw in the right triangles on page 640, the ratio of the length of the side opposite a 31° angle divided by the length of the side adjacent to a 31° angle is approximately , or 0.6. You can set up a proportion using this tangent ratio. tan 31° The definition of tangent. 0.6 The tangent of 31° is approximately 0.6. 0.6 Substitute 36 for HA. HT 36 · 0.6 Multiply both sides by 36 and reduce the left side. HT 22 Multiply. The height of the tree is approximately 22 meters. In order to solve problems like Example A, early mathematicians made tables that related ratios of side lengths to angle measures. They named six possible ratios. You will work with three. Sine, abbreviated sin, is the ratio of the length of the opposite leg to the length of the hypotenuse. Cosine, abbreviated cos, is the ratio of the length of the adjacent leg to the length of the hypotenuse. Tangent, abbreviated tan, is the ratio of the length of the opposite leg to the length of the adjacent leg.© 2008 Key Curriculum Press LESSON 12.1 Trigonometric Ratios 641
Trigonometric Ratios For any acute angle A in a right triangle: In ABC above: sine of A sin A cosine of A cos A tangent of A tan A Trigonometric Tables In this investigation you will make a small table of trigonometric ratios for angles a protractor measuring 20° and 70°. a ruler a calculator Step 1 Use your protractor to make a large right triangle ABC with m A = 20°, m B = 90°, and m C = 70°. Step 2 Measure AB, AC, and BC to the nearest millimeter. Step 3 Use your side lengths and the definitions of sine, cosine, and tangent to complete a table like this. Round your calculations to the nearest thousandth. Step 4 Share your results with your group. Calculate the average of each ratio within your group. Create a new table with your group’s average values. Step 5 Discuss your results. What observations can you make about the trigonometric ratios you found? What is the relationship between the values for 20° and the values for 70°? Explain why you think these relationships exist. Go to www.keymath.com/DG to find complete tables of trigonometric ratios.642 CHAPTER 12 Trigonometry © 2008 Key Curriculum Press
Today, trigonometric tables have been replaced by calculators that have sin, cos, and tan keys. Step 6 Experiment with your calculator to determine how to find the sine, cosine, and tangent values of angles. Step 7 Use your calculator to find sin 20°, cos 20°, tan 20°, sin 70°, cos 70°, and tan 70°. Check your group’s table. How do the trigonometric ratios found by measuring sides compare with the trigonometric ratios you found on the calculator? Using a table of trigonometric ratios, or using a calculator, you can find the approximate lengths of the sides of a right triangle given the measures of any acute angle and any side. EXAMPLE B Find the length of the hypotenuse of a right triangle if an acute angle measures 20° and the leg opposite the angle measures 410 feet. Solution Sketch a diagram. The trigonometric ratio that relates the lengths of the opposite leg and the hypotenuse is the sine ratio. sin 20° Substitute 20° for the measure of A and substitute 410 for the length of the opposite side. The length of the hypotenuse is unknown, so use x. x · sin 20° = 410 Multiply both sides by x and reduce the right side. x Divide both sides by sin 20°and reduce the left side. From your table in the investigation, or from a calculator, you know that sin 20° is approximately 0.342. x Sin 20° is approximately 0.342. x 1199 Divide. The length of the hypotenuse is approximately 1199 feet.© 2008 Key Curriculum Press LESSON 12.1 Trigonometric Ratios 643
With the help of a calculator, it is also possible to determine the size of either acute angle in a right triangle if you know the length of any two sides of that triangle. For instance, if you know the ratio of the legs in a right triangle, you can find the measure of one acute angle by using the inverse tangent, or tan−1, function. Let’s look at an example. The inverse tangent of x is defined as the measure of the acute angle whose tangent is x. The tangent function and inverse tangent function undo each other. That is, tan −1(tan A) = A and tan(tan −1 x) = x. EXAMPLE C A right triangle has legs of length 8 inches and 15 inches. Find the measure of the angle opposite the 8-inch leg. Solution Sketch a diagram. In this sketch the angle opposite the 8-inch leg is A . The trigonometric ratio that relates the lengths of the opposite leg and the adjacent leg is the tangent ratio. tan A Substitute 8 for the length of the opposite leg and substitute 15 for the length of the adjacent leg. tan−1 (tan A) Take the inverse tangent of both sides. A The tangent function and inverse tangent functionTo find the angle that has a undo each other.specific tangent value, use A 28 Use your calculator to evaluatethe tan−1 feature on yourcalculator. The measure of the angle opposite the 8-inch leg is approximately 28°. You can also use inverse sine, or sin−1, and inverse cosine, or cos −1 , to find angle measures. EXERCISES You will need For Exercises 1–3, use a calculator to find each trigonometric ratio accurate to the nearest ten thousandth. 1. sin 37° 2. cos 29° 3. tan 8° For Exercises 4–6, solve for x. Express each answer accurate to the nearest hundredth of a unit. 4. sin 40° 5. cos 52° 6. tan 29° 644 CHAPTER 12 Trigonometry © 2008 Key Curriculum Press
For Exercises 7–9, find each trigonometric ratio. 7. sin A 8. sin 9. sin A sin B cos A cos cos A cos B tan A tan tan A tan B For Exercises 10–13, find the measure of each angle accurate to the nearest degree. 10. sin A = 0.5 11. cos B = 0.6 12. tan C = 0.5773 13. tan x = For Exercises 14–20, find the values of a–g accurate to the nearest whole unit. 14. 15. 16. 17. 18. 19. 20. 21. Find the perimeter of 22. Find x. this quadrilateral.© 2008 Key Curriculum Press LESSON 12.1 Trigonometric Ratios 645
Review For Exercises 23 and 24, solve for x. 23. 24. 25. Application Which is the better buy—a pizza with a 16-inch diameter for $12.50 or a pizza with a 20-inch diameter for $20.00? 26. Application Which is the better buy—ice cream in a cylindrical container with a base diameter of 6 inches and a height of 8 inches for $3.98 or ice cream in a box (square prism) with a base edge of 6 inches and a height of 8 inches for $4.98? 27. A diameter of a circle is cut at right angles by a chord into a 12 cm segment and a 4 cm segment. How long is the chord? 28. Find the volume and surface area of this sphere. 3-by-3 Inductive Reasoning Puzzle II Sketch the figure missing in the lower right corner of this pattern.646 CHAPTER 12 Trigonometry © 2008 Key Curriculum Press
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