2.5 Using the Sine and Cosine Ratios to Calculate Lengths
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1 2.5 Using the ine and Cosine atios to Calculate Lengths FOCU Use the sine and cosine ratios to determine lengths. To use the sine or cosine ratio to find the length of a leg, we need to know: the measure of an acute angle, and the length of the Example 1 Using the ine or Cosine atio to Find the Length of a Leg Find the length of to the nearest tenth of a metre m olution The measure of is known. is the side adjacent to. is the. o, use the cosine ratio. cos side adjacent to adjacent to m cos ubstitute: 28 and 9.6 cos 28 Multiply both sides by cos cos 28 Use a calculator is about 8.5 m long Copyright 2011 Pearson Canada Inc. 97
2 Check 1. Find the length of each indicated side to the nearest tenth of a centimetre. a) AC B cm The measure of B is known. AC is the. BC is the. o, use the ratio. side B A C B BC AC AC is about long. b) DE E F The measure of is known. DE is the. DF is the. o, use the ratio. D cm DE is about long Copyright 2011 Pearson Canada Inc.
3 To use the sine or cosine ratio to find the length of the, we need to know: the measure of an acute angle, and the length of one leg Example 2 Using the ine or Cosine atio to Find the Length of the Hypotenuse Find the length of the to the nearest tenth of a centimetre. N 9.5 cm The is the side opposite the right angle. M 52 P olution The measure of M is known. NP is the side opposite M. MN is the. o, use the sine ratio to write an equation. sin M NP sin M ubstitute: M 52 and NP 9.5 MN 9.5 sin 52 MN Multiply both sides by MN. MN sin Divide both sides by sin 52. MN sin 52 sin 52 side opposite M 9.5 sin 52 M 52 N opposite M 9.5 cm P MN 9.5 sin 52 MN MN is about 12.1 cm long. Use a calculator Copyright 2011 Pearson Canada Inc. 99
4 Check 1. Find the length of each to the nearest tenth of a centimetre. a) K 17.4 cm side sin J M sin J 39 J The measure of J is known. The side opposite J is: The is: Use the sine ratio. sin sin JK JK JK is about long. b) cm is the side. is the. o, use the ratio. is about long Copyright 2011 Pearson Canada Inc.
5 Example 3 Using ine or Cosine to olve a Problem A surveyor makes the measurements shown in the diagram to find the distance between two observation towers on opposite sides of a river. How far apart are the towers? Give the answer to the nearest metre. iver Tower m Tower 2 olution The distance between the towers is the, AC. A Tower 1 B 63 m 73 C Tower 2 The measure of C is known. BC is the side adjacent to C. AC is the. o, use the cosine ratio. side adjacent to C cos C BC cos C ubstitute: C 73 and BC 63 AC 63 cos 73 AC Multiply both sides by AC. AC cos Divide both sides by cos AC cos 73 Use a calculator. AC The distance between the towers is about 215 m Copyright 2011 Pearson Canada Inc. 101
6 Check 1. am and ofia are building a wooden ramp for skateboarding. The height of the ramp is 0.75 m. The ramp makes an angle of 8 with the ground. What length of plywood do am and ofia need for the top of the ramp? Give your answer to the nearest tenth of a metre. E D 8 We want to find the length of DE. The measure of D is known. The side opposite D is: The is: o, use the sine ratio. F 0.75 m sin D opposite sin D ubstitute: sin sin Multiply both sides by. Divide both sides by. DE am and ofia need about of plywood. Practice 1. Which ratio would you use to find each length? a) XY b) T X T Z 29 Y 17.6 cm YZ is the side. XY is the. o, use the ratio. 53 U 15.9 cm T is the side. U is the. o, use the ratio Copyright 2011 Pearson Canada Inc.
7 2. Find the length of each indicated side to the nearest tenth of a centimetre. a) VW U V cm W The measure of is known. The side opposite is. The is. o, use the ratio. side sin sin sin sin VW VW is about long. b) 10.5 cm 36 is the. is the. o, use the ratio. is about long Copyright 2011 Pearson Canada Inc. 103
8 3. Find the length of side PM to the nearest tenth of a metre. P M The measure of is known. N 16.0 m 19 PM is the side. MN is the. o, use the ratio. PM is about long. 4. Find the length of each to the nearest tenth of a centimetre. a) E b) X 29 W cm Y G 8.4 cm F The side opposite is: The is: o, use the sine ratio. WY is the side. WX is the. o, use the ratio. sin opposite sin sin WX WX is about long. sin EF EF EF is about long Copyright 2011 Pearson Canada Inc.
9 5. A straight slide in a playground makes an angle of 28 with the ground. The slide covers a horizontal distance of 4.5 m. How long is the slide? Give your answer to the nearest tenth of a metre. The measure of is known. The side adjacent to is: The is: o, use the ratio m The slide is about long. 6. A 15-m support cable joins the top of a telephone pole to a point on the ground. The cable makes an angle of 32 with the ground. Find the height of the pole to the nearest tenth of a metre. C E 15.0 m 32 D The height of the pole is about Copyright 2011 Pearson Canada Inc. 105
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