Name Class Date. Investigating a Ratio in a Right Triangle

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1 Name lass Date Trigonometric Ratios Going Deeper Essential question: How do you find the tangent, sine, and cosine ratios for acute angles in a right triangle? In this chapter, you will be working etensively with right triangles, so some new vocabulary will be helpful. Given a right triangle,, with a right angle at verte, the leg adjacent to is the leg that forms one side of. The leg opposite is the leg that does not form a side of Video Tutor 1 M9 12.G.SRT.6 EXPLORE Investigating a Ratio in a Right Triangle Use geometry software to draw a horizontal segment. Label one endpoint of the segment. Hypotenuse Leg opposite Select point, go to the Transform menu, and choose Mark enter. Select the segment, go to the Transform menu, and choose Rotate. Enter 30 for the angle of rotation. Label the endpoint of the rotation image. Leg adjacent to D Select point and the original line segment. Use the onstruct menu to construct a perpendicular from to the segment. Plot a point at the point of intersection and label the point. E Use the Measure menu to measure and. Then use the alculate tool to calculate the ratio. F Drag the points and lines to change the size and location of the triangle. Notice what happens to the measurements. G Repeat the above steps using a different angle of rotation. = 1.94 cm = 3.37 cm = 0.58 REFLET 1a. ompare your findings with those of other students. For an acute angle in a right triangle, what can you say about the ratio of the length of the opposite leg to the length of the adjacent leg? Module Lesson 1

2 You may have discovered that in a right triangle the ratio of the length of the leg opposite an acute angle to the length of the leg adjacent to the angle is constant. You can use what you know about similarity to see why this is true. onsider the right triangles and DEF, in which D, as shown. y the Similarity riterion, DEF. This means the lengths of the sides of DEF are each k times the lengths of the corresponding D sides of. EF = k DF k = This shows that the ratio of the length of the leg opposite an acute angle to the length of the leg adjacent to the angle is constant. This ratio is called the tangent of the angle. Thus, the tangent of, written tan, is defined as follows: length of leg opposite tan = length of leg adjacent to = You can find the tangent of an angle using a calculator or by using lengths that are given in a figure, as in the following eample. E F 2 M9 12.G.SRT.6 EXMPLE Finding the Tangent of an ngle Find the tangent of J and K. Write each ratio as a fraction and as a decimal rounded to the nearest hundredth. length of leg opposite J tan J = length of leg adjacent to J = KL JL = 24 = 12 length of leg opposite K tan K = length of leg adjacent to K = JL KL = 10 = 5 = K 10 L J REFLET 2a. What do you notice about the ratios you wrote for tan J and tan K? Do you think this will always be true for the two acute angles in a right triangle? 2b. Why does it not make sense to ask for the value of tan L? Module Lesson 1

3 When you know the length of a leg of a right triangle and the measure of one of the acute angles, you can use the tangent to find the length of the other leg. This is especially useful in real-world problems. 3 M9 12.G.SRT.8 EXMPLE Solving a Real-World Problem long ladder leans against a building and makes an angle of 68 with the ground. The base of the ladder is 6 feet from the building. To the nearest tenth of a foot, how far up the side of the building does the ladder reach? Write a tangent ratio that involves the unknown length,. length of leg opposite tan = length of leg adjacent to = 6 Use the fact that m = 68 to write the equation as tan 68 = ft Solve for. 6 tan 68 = 6 = So, the ladder reaches about Multiply both sides by 6. Use a calculator to find tan 68. Do not round until the final step of the solution. Multiply. Round to the nearest tenth. up the side of the building. REFLET 3a. Why is it best to wait until the final step before rounding? What happens if you round the value of tan 68 to the nearest tenth before multiplying? 3b. student claims that it is possible to solve the problem using the tangent of. Do you agree or disagree? If it is possible, show the solution. If it is not possible, eplain why not. Module Lesson 1

4 trigonometric ratio is a ratio of two sides of a right triangle. You have already seen one trigonometric ratio, the tangent. It is also possible to define two additional trigonometric ratios, the sine and the cosine, that involve the hypotenuse of a right triangle. The sine of, written sin, is defined as follows: length of leg opposite sin = = The cosine of, written cos, is defined as follows: length of leg adjacent to cos = = 4 M9 12.G.SRT.6 EXMPLE Finding the Sine and osine of an ngle Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. R length of leg opposite R sin R = = PQ = 20 RQ length of leg opposite Q sin Q = = RP RQ = 29 P 20 Q length of leg adjacent to R cos R = = length of leg adjacent to Q D cos Q = = REFLET 4a. What do you notice about the sines and cosines you found? Do you think this relationship will be true for any pair of acute angles in a right triangle? Eplain. Module Lesson 1

5 You may have discovered a relationship between the sines and cosines of the acute angles in a right triangle. In particular, if and are the acute angles in a right triangle, then sin = cos and sin = cos. Note that the acute angles in a right triangle are complementary. The above observation leads to a more general fact: the sine of an angle is equal to the cosine of its complement, and the cosine of an angle is equal to the sine of its complement. 5 M9 12.G.SRT.7 EXMPLE Using omplementary ngles Given that sin , write the cosine of a complementary angle. Find the measure of an angle that is complementary to a 57 angle = 90, so = Use the fact that the cosine of an angle is equal to the sine of its complement. cos Given that cos 60 = 0.5, write the sine of a complementary angle. Find the measure y of an angle that is complementary to a 60 angle. y + 60 = 90, so y = D Use the fact that the sine of an angle is equal to the cosine of its complement. sin = 0.5 REFLET 5a. Is it possible to find m J in the figure? Eplain. 5b. What can you conclude about the sine and cosine of 45? Eplain m J L 839 m K 5c. Is it possible for the sine of an angle to equal 1? Why or why not? Module Lesson 1

6 6 M9 12.G.SRT.8 EXMPLE Solving a Real-World Problem loading dock at a factory has a 16-foot ramp in front of it, as shown in the figure. The ramp makes an angle of 8 with the ground. To the nearest tenth of a foot, what is the height of the loading dock? How far does the ramp etend in front of the loading dock? (The figure is not drawn to scale, so you cannot measure it to solve the problem.) 8 16 ft y Loading dock Find the height of the loading dock. length of leg opposite sin = =, so sin 8 = Solve the equation for. Use a calculator to evaluate the epression, then round. So, the height of the loading dock is about. Find the distance y that the ramp etends in front of the loading dock. length of leg adjacent to cos = =, so cos =. Solve the equation for y. Use a calculator to evaluate the epression, then round. y So, the distance the ramp etends in front of the loading dock is about. REFLET 6a. student claimed that she found the height of the loading dock by using the cosine. Eplain her thinking. 6b. Suppose the owner of the factory decides to build a new ramp for the loading dock so that the new ramp makes an angle of 5 with the ground. How far will this ramp etend from the loading dock? Eplain. Module Lesson 1

7 practice Find the tangent of and. Write each ratio as a fraction and as a decimal rounded to the nearest hundredth Find the value of to the nearest tenth. 4. P T S J 210 N M U 60 H 54 G 7. hiker whose eyes are 5.5 feet above ground stands 25 feet from the base of a redwood tree. She looks up at an angle of 71 to see the top of the tree. To the nearest tenth of a foot, what is the height of the tree? 8. Error nalysis To find the distance XY across a large rock formation, a student stands facing one endpoint of the formation, backs away from it at a right angle for 20 meters, and then turns 55 to look at the other endpoint of the formation. The student s calculations are shown. ritique the student s work. X 20 m ft 5.5 ft 55 Y Z tan 55 = 20 XY XY tan 55 = 20 XY = 20 tan m Module Lesson 1

8 Find the given trigonometric ratios. Write each ratio as a fraction and as a decimal rounded to the nearest hundredth. 9. sin R, cos R 10. cos D, cos E 11. sin M, sin N P 30 Q D P M E R F N 12. Given that sin , write the cosine of a complementary angle. 13. Given that cos , write the sine of a complementary angle. Find the value of to the nearest tenth J 16. U K L W 9 40 V 17. You are building a skateboard ramp from a piece of wood that is 3.1 meters long. You want the ramp to make an angle of 25 with the ground. To the nearest tenth of a meter, what is the length of the ramp s base? What is its height? 18. Error nalysis Three students were asked to find the value of in the figure. The equations they used are shown at right. Which students, if any, used a correct equation? Eplain the other students errors and then find the value of. M P R m 15 Lee s equation: sin 57 = 15 Jamila s equation: cos 33 = 15 Tyler s equation: sin 33 = 15 N T S Module Lesson 1

9 Name lass Date dditional Practice 10-1 Use the figure for Eercises 1 6. Write each trigonometric ratio as a simplified fraction and as a decimal rounded to the nearest hundredth. 1. sin 2. cos 3. tan 4. sin 5. cos 6. tan Use special right triangles to write each trigonometric ratio as a simplified fraction. 7. sin cos tan tan cos tan 60 Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. 13. sin cos tan 15 Find each length. Round to the nearest hundredth XZ HI KM ST EF DE Module Lesson 1

10 Problem Solving 1. ramp is used to load a 4-wheeler onto a truck bed that is 3 feet above the ground. The angle that the ramp makes with the ground is 32. What is the horizontal distance covered by the ramp? Round to the nearest hundredth. 2. Find the perimeter of the triangle. Round to the nearest hundredth. 3. right triangle has an angle that measures 55. The leg adjacent to this angle has a length of 43 cm. What is the length of the other leg of the triangle? Round to the nearest tenth. 4. The hypotenuse of a right triangle measures 9 inches, and one of the acute angles measures 36. What is the area of the triangle? Round to the nearest square inch. hoose the best answer foot ladder makes a 62 angle with the ground. To the nearest foot, how far up the house does the ladder reach? 6 ft 7 ft 12 ft D 16 ft 7. What is EF, the measure of the longest side of the sail on the model? Round to the nearest inch. 31 in. 35 in. 40 in. D 60 in. 6. To the nearest inch, what is the length of the springboard shown below? F 24 in. G 36 in. H 38 in. J 127 in. 8. Right triangle is graphed on the coordinate plane and has vertices at ( 1, 3), (0, 5), and (4, 3). What is the measure of to the nearest degree? F 27 G 29 H 32 J 43 Module Lesson 1