Historical Note Trigonometry Ratios via Similarity

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1 Section 12-6 Trigonometry Ratios via Similarity Trigonometry Ratios via Similarity h ft of elevation Figure Measurements of buildings, structures, and some other objects are frequently required in many occupations. For eample, in a tourist brochure advertising alifornia, one region boasts a sequoia tree of great height, such as the one pictured in Figure From the previous section, we know how to find the height of the tree by using shadows and similar triangles. Trigonometry might also be used to find the required height of the tree. Trigonometry developed from a need to compute distances and angle measures, especially in map making, surveying, and range finding for artillery use. Today, trigonometry is an indispensable tool in many applied problems in both science and technology. The word trigonometry is derived from the Greek words trigonom, which means triangle, and metron, which means measurement. In this section, we study the basics of right triangle trigonometry, which has applications to measuring distances and angles. The definition of trigonometric functions in a right triangle is based on properties of similar triangles. Earlier in this chapter, we saw that corresponding sides of similar triangles are proportional. onsequently, in two similar triangles the ratio of one side to another in one triangle will be the same as the ratio of the corresponding sides in the second triangle. onsider the two similar triangles in Figure oth are right triangles, and each has an angle with measure cm 30 cm 30 F 90 cm D Figure E We could find using the following proportion: 90 = which implies that = 45 cm. If we use this property, it follows that in any right triangle the ratio between the side opposite the angle with measure 30 and the side opposite the right angle, the hypotenuse, will be the same as that ratio in the smaller triangle in Figure 12-84, that is, 15 30, or 1 2. Historical Note Hipparchus of Nicea (ca E) is sometimes called the father of trigonometry because he is one of the first to try and organize a set of values associating arcs and chords of a circle. This work helped those who came later in the development of modern notions of trigonometry.

2 2 onstructions, ongruence, and Similarity Suppose we use a protractor to construct a right triangle with an angle measuring 37 and measure both the hypotenuse and the side opposite this angle, as shown in Figure The ratio between the side opposite the angle with measure 37 and the hypotenuse is 100, or 0.6. Since measurements are approimate, the ratio also is only an approimation. This ratio is the sine of 37, or sin 37. Thus, in a right triangle having an angle with measure 37, sin 37 = Length of side opposite the 37 angle Length of hypotenuse Hypotenuse a = 60 cm c = 100 cm 37 = m (j 1) b Side adjacent to j 1 Side opposite to j 1 Figure Other ratios such as the cosine (cos) and tangent (tan) between lengths of sides in a right triangle are also useful and are defined in reference to Figure as follows: Hypotenuse c a 1 b Side opposite to j 1 Side adjacent to j 1 Figure sin1 12 = a c cos1 12 = b c tan1 12 = a b Length of opposite side = Length of hypotenuse Length of adjacent side = Length of hypotenuse Length of opposite side = Length of adjacent side REMRK Technically, we should write sin( m 1), but this is typically shortened to sin( 1).

3 Section 12-6 Trigonometry Ratios via Similarity 3 1. The ratios found corresponded to the sines, cosines, and tangents. 2. It is not necessary to find the measures of the sides of a triangle to determine the sine, cosine, and tangent ratios. TEHNOLOGY ORNER Using The Geometer s Sketchpad, draw a right triangle with right angle at. Measure each side of the triangle and find the measures of and. Use The Geometer s Sketchpad to find measures, calculate ratios, and build a table with headings similar to those in Table Table 12-2 Meas. Opp. Side Meas. dj. Side Meas. Hyp. Meas. Opp. Hypot. Sine of 1. What is true about the ratios you found using measurements and the sines, cosines, and tangents found automatically? 2. Eplain whether you think it is necessary to find the measures of the sides of a triangle to determine the sine, cosine, and tangent ratios. dj. Hypot. osine of Opp. dj. Tangent of Using Trigonometric Ratios on a alculator Tables of trigonometric ratios were essential in all parts of the world until calculators became readily available. Trigonometry in right triangles as we use it today is primarily done using a scientific calculator, graphing calculator, or computer. For eample, in a given right triangle with an acute angle of 40, we can use the SIN, OS, and TN buttons to find the respective values of 0.643, 0.766, and assigned to a 40 angle. Each of the trigonometry buttons is a function button that requires an angle measure input either in degrees or in radians. ecause we use degrees in this tet, make sure your calculator is set to use degrees and not radians. One method for telling whether your calculator is in the degree mode is to find the sine of 30. If your answer is 0.5, or.5, then the calculator is in the proper mode. If not, consult a manual to see what you must do to put your calculator in degree mode. REMRK heck your calculator to determine if you should use 3 0 SIN or SIN 3 0. SIN 3 0 ) or Eample Use a calculator to solve for in Figure cm (a) 10 cm 35 (b) cm (c) cm (d) 70 Figure 12-87

4 4 onstructions, ongruence, and Similarity Solution a. = sin = 10 sin 35 To find the answer on your calculator, you may have to press the following keys: b. c. d. 1 SIN The resulting display is , or approimately 5.7 cm. = tan = 55 tan 40 L , or approimately 46.2 cm 30 = sin = sin 20 L , or approimately 87.7 cm = cos = 3.2 cos 70 L 1.094, or approimately 1.1 cm 0 * 3 5 ) * nswers can be found in nswers at the back of the tet. NOW TRY THIS In Eample 12-25, use Figure 12-87(a) to find using the cosine function instead of the sine function. Sometimes the lengths of the sides of a right triangle are given and we need to find the measures of the angles of the triangle. If, for eample, a triangle has sides measuring 3 m, 4 m, and 5 m, as shown in Figure 12-88, we can find the measure of angle by first finding sin1 2. We have sin1 2 = 4 The SIN key on your calculator (or possibly the INV key used with the SIN key) can be used to find the measure of the angle as follows: SIN -1 4, 5 ) = 5 m 4 m 3 m Figure The angle has measure approimately 53.1.

5 Section 12-6 Trigonometry Ratios via Similarity 5 Eample If is an angle measure, solve for in each part of Figure 12-89: 72 cm 63 cm 4 m (a) 55 cm (b) 41 cm (c) 3 m Figure Solution a. b. cos = L , or about 40.2 sin = L , or approimately 40.6 c. tan = 3 4 L , or approimately 36.9 Finding Measurements Using Trigonometric Ratios We can use an angle of depression (see Figure 12-90) to find measures, as demonstrated in Eample Eample From the top of Mount Sentinel, the measure of the angle of depression of the administration building is 18, as illustrated in Figure If Mount Sentinel is 1575 ft high, how far through the air is the top of the mountain from the base of the building? 18 of depression 1575 ft d of elevation Mount Sentinel 18 Figure Solution In Figure 12-90, the measure of the angle of elevation must equal the measure of the angle of depression (why?). Using trigonometric ratios, we have the following: sin 18 = 1575 d Thus, the air distance is approimately 5097 ft. d = 1575 sin 18 L 5097

6 6 onstructions, ongruence, and Similarity Eample To measure the height of a flagpole on top of a building, a surveyor measures the distance to the building and the two angles from point, as shown in Figure Find the height D of the flagpole. y D m Figure Solution We need to find D in Figure ecause D is not a length of a side of a right triangle, but y and whose difference is D are, we may find and y first. The solution then is D = y -. In triangle D, we have the following: In triangle, we have the following: tan 33 = tan = = 12 tan 33 y = 12 tan Therefore, D = 12 tan tan 33. Using a calculator, we obtain D L Thus, the height of the pole atop the building is about 5.1 m. 12 y 12 ssessment For each of the following figures, solve for and epress each length to the nearest hundredth: about m 20 m a. 41 b m about m 2. The angle of elevation of a 15-ft ladder is 70. Find out how far the base of the ladder is from the wall. about 5.5 ft 3. Vectors are used in science to depict forces with both magnitudes and directions. If a force of 14 lb is directed at an angle measuring 38 from the horizontal, as depicted in the following figure, determine the vertical and horizontal components as pictured: horizontal: about 11.0 lb; vertical: about 8.6 lb 14 lb Vertical component 38 Horizontal component

7 Section 12-6 Trigonometry Ratios via Similarity 7 4. How many feet of cable will it take to anchor a guy wire to a 30 ft vertical pole if the angle of elevation is 28? ssume the ground is level. about 63.9 ft 5. s a plane takes off, it flies 1500 ft along a straight path and rises at an angle measuring 22. a. What is the vertical rise when it has flown 1500 ft? * b. fter it has flown 1500 ft, how far has it moved horizontally? about ' 6. a. omplete the following chart: Decimals are rounded to four places: 1 Measure 2 Sine 3 osine 4 Square of Sine 5 Square of osine 6 Sum of olumns 4 and Find the height of a tree whose horizontal shadow is 120 ft when the angle of elevation of the Sun from the tip of the shadow measures 68. about 297' 8. In navigational terms, a bearing is defined as the number of degrees a direction is from due north clockwise. If a plane flying 120 mph has flown 1 hr at a bearing of 32, how far has it flown in a horizontal direction (east)? In a vertical direction (north)? * 9. Find in the following: about m 50 m 42 m 10. In problem 6, you likely conjectured the following for any angle : sin 2 + cos 2 = 1, where sin 2 means 1sin 21sin 2. If sin = a in a right triangle where a c and b are legs and the hypotenuse is c, write an equation equivalent to the trigonometric equation in terms of a, b, and c using no fractions in the final result. a 2 + b 2 = c a c b. Make a conjecture about the sum of the squares of the sines and cosines of various angles. The sum of the squares of the sine and cosine is 1 for any angle. b ssessment For each of the following figures, solve for and epress each length to the nearest hundredth: m about 9.53 m a. b. 9 m about m 2. diagonal is drawn in a 12 in. square floor tile. Find the sine, cosine, and tangent of the angle formed by the diagonal and a side. * 3. Determine the height of a tree if it casts a shadow 7 m long on level ground when the angle of elevation of the Sun is 50. about 8.3 m 4. To find the length of a lake, a person set stakes at point and made the following measurements: 170 ft 138 ft a. What is the measure of angle? about 35.7 b. What is the length of the lake? about 99.3 ft 5. gutter cleaner wants to reach a gutter 40 ft above the ground. Find the length of the shortest ladder that can be used if the steepest angle at which it can be leaned against the house has measure ft

8 8 onstructions, ongruence, and Similarity 6. jet plane cruising at 450 mph climbs at an angle measuring 13. Determine how much altitude the jet gains in 2 min. about 8.4 mi 7. highway that has a 6% grade rises 6 ft vertically for every 100 ft horizontally. Which trigonometric ratio is being used in reporting the 6% grade? Eplain why. * 8. The slope of a line in coordinate geometry is described as the rise divided by the run. Which trigonometric ratio does the slope represent? Eplain your answer. * 9. Find in each of the following: 41 cm 41 cm 10 m cm 2 m a. b. about 42.9 about 1.7 m 2 m 40 Mathematical onnections 12-6 ommunication 1. What does your calculator show for the tangent of 90? Why? nswers depend on the calculator. Some will show an error. The tangent of 90 does not eist because division by 0 is undefined. Open-Ended 2. Investigate the origin of the word sine. Did it originally have anything to do with sine as we know it today? nswers may vary, but the answer is no. ooperative Learning 3. a. onstruct a semicircle of any radius. Mark the center. b. Use a protractor to mark angle measures in 10 increments around the semicircle. nswers may vary. c. onstruct a table of values for the sine of each marked angle. nswers may vary. * d. ompare measures of classmates. How do the tables compare? The values should be comparable. Questions from the lassroom 4. student noticed that sin 10 = cos 80, sin 35 = cos 55, and sin 5 = cos 85. The student is wondering if these eamples generalize, and why they are true. How do you respond? * 5. student noticed that tan a = sin a and would like to cos a know if this is always true and if so, why. How do you respond? lways true; tan a = a b = a>c b>c = sin a cos a.