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43 selfcheck
1.
© Ohio Resource Center for Mathematics, Science, and Reading, 2008 Page 1 of 4 SelfCheck 23 Laws of Sines and Cosines 1. In ∆XYZ, angle X measures 48.5°, side x measures 8.1 feet, and angle Z measures 25°. Apply the law of sines to find the length of side z. Include a sketch with your answer. 2. In ∆ABC, angle C measures 125°, side b measures 5.3 cm, and side c measures 8.9 cm. Find all missing parts of the triangle. This is a two‐step question that requires the use of the inverse sine. (See guy‐wire problem in tutorial #22 for calculator directions for finding an inverse trig function.) Include a sketch with your answer. 3.
Given ∆KLM with angle K equal to 141.43°, side l measuring 6.5 cm, and side m measuring 7.9 cm. Find the length of side k. Include a sketch with your answer. 4. In ∆ABC, side a measures 8.2 cm, side b is 3.7 cm, and side c measures 10.8 cm. Find the measure of angle C. Don’t forget that the largest angle of a triangle is opposite the longest side and the smallest angle is opposite the shortest side. (See guy‐wire problem in tutorial #22 for calculator directions for finding an inverse function.) Include a sketch with your answer. 5. Using information and sketch from problem 4, find the measures of angles A and B. 6. Using the figure below, compare the cost of constructing a road on a rocky surface between points X and Y at the rate of $30/ft with the cost of building the road on a soil surface with an indirect route from point X to point Z to point Y for $20/ft. Is the shorter route is the most economical? X 120° x = 560 ft y = 430 ft z YZ
2.
© Ohio Resource Center for Mathematics, Science, and Reading, 2008 Page 2 of 4 Solutions to SelfCheck 23 Laws of Sines and Cosines 1. In ∆XYZ, angle X measures 48.5°, side x measures 8.1 feet, and angle Z measures 25°. Apply the law o fi
the of side z. of sines t nd length sin 48.5° 8.1 sin 25° · si 8 .1 · sin 25° n 4 .5° 8 8.1 · sin 25° sin 48.5° 4.570641261 4.5 2. In ∆ABC, angle C measures 125°, side b measures 5.3 cm, and side c measures 8.9 cm. Find all missing parts of the triangle. This is a two‐step question that requires the use of the inverse sine. (See guy‐wire problem in tutorial #22 for calculator directions for finding an inverse trig function.) sin 125° 8.9 5.3 sin 5.3 · sin 125° 8.9 sin 78 . 48 096444 sin sin .4878096444 29.19671722 or m 29.2° Since the sum of angles A, B, and C is 180°, angle A is approximately 25.8°. Apply the law of sine d ti .s a secon me sin 25.8° 8.9 sin 125° 8.9 sin 25.8° sin 125° 4.728739691 or measure of side is approximately 4.73 inches z x = 8.1 ft y X Y Z 48.5° 25° C B A c = 8.9 b = 5.3 cm 125°
3.
© Ohio Resource Center for Mathematics, Science, and Reading, 2008 Page 3 of 4 Solutions to SelfCheck 23 Laws of Sines and Cosines 3. Given ∆KLM with angle K equal to 141.43°, side l measuring 6.5 cm, and side m measuring 7.9 cm. n
the law of cosines. Fi d the length of side k. Apply 2 cos 2 6.5 7.9 cos 141.43° 6.5 7.9 184.8956898 13.59763545 or measure of side is approximately 13.6 feet 4. In ∆ABC, side a measures 8.2 cm, side b is 3.7 cm, and side c measures 10.8 cm. Find the measure of angle C. Don’t forget that the largest angle of a triangle is opposite the longest side and the smallest angle is opposite the shortest side. (See guy‐wire problem in tutorial #22 for calcula inverse trig function.) tor directions for finding an 2 cos . os 10.8 8.2 3.7 2 8.2 · 3 7 · c 69 60.68 cos 116.64 67.24 13. 35.71 60.68 cos .5884970336 cos .5884970336 126.0504255 U m 126.1° K L M m = 7.9 cm k l = 6.5 cm 141.43° B A C b = 3.7 cm c = 10.8 cm a = 8.2 cm
4.
© Ohio Resource Center for Mathematics, Science, and Reading, 2008 Page 4 of 4 Solutions to SelfCheck 23 Laws of Sines and Cosines 5. Using information, solution, and sketch from problem 4, find the measures of angles A and B. When given three sides of a triangle, first use the law of cosines to find the measure of the largest angle (problem 4). Next, use the law of sines to find the measure of a second angle. Finally, knowing that the sum of angles A, B, and C is 180°, calculate the measure of the third angle. sin sin 126.1° sin
. 80799 10.8 3878 · 8.2 sin . 07481 6 sin . 1347 347 sin .61 m 37.8° m 180° 126.1° 37.8° or m 16.1° 6. Using the figure below, compare the cost of constructing a road on a rocky surface between points X and Y at the rate of $30/ft with the cost of building the road on a soil surface with an indirect route from point X to point Z to point Y for $20/ft. Is the shorter route is the most cost effe ivct e? cos 430 2 560 430 cos 120° 560 739300 859.8255673 or measure of side is approximately 859.83ft Approximate cost for direct stony surface route ($30/ft) is $25,794 and cost for the indirect soil surface route ($20/ft) is $19,860. Therefore, the longer, indirect route is most cost effective. B A C b = 3.7 cm c = 10.8 cm a = 8.2 cm X 120° x = 560 ft y = 430 ft z YZ
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