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Right Triangle Trigonometry Section 10.4 Tangent Ratio Section 10.5 Sine and Cosine Ratios Section 10.6 Solving Right Triangles Goals • Find trigonometric ratios using right triangles. • Use trigonometric ratios to find angle measures in right triangles. Key Vocabulary • • • • • • • • Trigonometry Trigonometric ratio Sine (sin) Cosine (cos) Tangent (tan) Inverse sine Inverse cosine Inverse tangent • Opposite leg • Adjacent leg • Solve a right triangle History Right triangle trigonometry is the study of the relationship between the sides and angles of right triangles. These relationships can be used to make indirect measurements like those using similar triangles. History Early mathematicians discovered trig by measuring the ratios of the sides of different right triangles. They noticed that when the ratio of the shorter leg to the longer leg was close to a specific number, then the angle opposite the shorter leg was close to a specific number. Trigonometric Ratios • The word trigonometry originates from two Greek terms, trigon, which means triangle, and metron, which means measure. Thus, the study of trigonometry is the study of triangle measurements. • A ratio of the lengths of the sides of a right triangle is called a trigonometric ratio. The three most common trigonometric ratios are sine, cosine, and tangent. Trigonometric Ratios Only Apply to Right Triangles In right triangles : • The segment across from the right angle ( AC ) is labeled the hypotenuse “Hyp.”. A Hyp. Opp. Angle of Perspective C B Adj. • The “angle of perspective” determines how to label the sides. • Segment opposite from the Angle of Perspective( AB ) is labeled “Opp.” • Segment adjacent to (next to) the Angle of Perspective ( BC ) is labeled “Adj.”. * The angle of Perspective is never the right angle. Labeling sides depends on the Angle of Perspective If A is the Angle of Perspective then …… Angle of Perspective A AC Hyp Hyp. Adj. B Opp. C BC Opp AB Adj *”Opp.” means segment opposite from Angle of Perspective “Adj.” means segment adjacent from Angle of Perspective If the Angle of Perspective is A then A Hyp Adj C then A Opp B Opp Hyp C B Adj C AC Hyp AC Hyp BC Opp AB Opp AB Adj BC Adj The 3 Trigonometric Ratios • The 3 ratios are Sine, Cosine and Tangent Opposite Side Sine Ratio Hypotenuse Adjacent Side Co sin e Ratio Hypotenuse Opposite Side Tangent Ratio Adjacent Side Trigonometric Ratios To help you remember these trigonometric Sin A = Opposite sideyou can SOH relationships, use Hypotenuse the mnemonic device, Cos A = Adjacent side CAH SOH-CAH-TOA, where the Hypotenuse first letter of each word of Tan A = Opposite side TOA the trigonometric ratios is Adjacent side represented in the correct order. A c b C a B Trigonometric Ratios A side adjacent B side opposite C sin Oh Hell Soh cos Another Cah Hour tan Of Toa Algebra SohCahToa Soh sin opposite hypotenuse Cah cos adjacent hypotenuse Toa tan opposite adjacent The Amazing Legend of… Chief SohCahToa Chief SohCahToa • Once upon a time there was a wise old Native American Chief named Chief SohCahToa. • He was named that due to an chance encounter with his coffee table in the middle of the night. • He woke up hungry, got up and headed to the kitchen to get a snack. • He did not turn on the light and in the darkness, stubbed his big toe on his coffee table…. Solving a right triangle • Every right triangle has one right angle, two acute angles, one hypotenuse and two legs. To solve a right triangle, means to determine the measures of all six (6) parts. You can solve a right triangle if one of the following two situations exist: – One side length and one acute angle measure (use trigonometric ratios to find other side). – Two side lengths (use inverse trigonometric ratios to find an angle). Using a Calculator – Trigonometric Ratios • Use a calculator to approximate the sine, cosine, and the tangent of 74. • Make sure that your calculator is in degree mode. • The table shows some sample keystroke sequences accepted by most calculators. Sample keystrokes Sample keystroke sequences 74 sin sin 74 Sample calculator display Rounded Approximation 0.961262695 0.9613 0.275637355 0.2756 3.487414444 3.4874 ENTER 74 COS 74 COS ENTER 74 TAN 74 TAN ENTER Using a Calculator – Trigonometric Ratios • Using a calculator to find the following trigonometric ratios. • Sin 37˚ • Sin 63˚ • Cos 82˚ • Cos 16˚ • Tan 29˚ • Tan 55˚ • • • • • • .6018 .8910 .1392 .9613 .5543 1.4281 Using a Calculator – Trigonometric Ratios • Given one acute angle and one side of a right triangle, the trigonometric ratios can be used to find another side of the triangle. Trig. ratio used to find • Example 1: cos 42 x 15 side adj. to 42˚ angle. x15cos 42 Multiply both sides by 15 to solve for x. x 11.15 Use calculator to find the length of the adj. side. Using a Calculator – Trigonometric Ratios • Example 2: sin 66 9.5 x x sin66 9.5 9.5 x sin 66 x10.40 Trig. ratio used to find hyp. of a right triangle. Multiply both sides by x. Divide both sides by sin66˚ to solve for x. Use calculator to find the length of the hyp. Using a Calculator – Trigonometric Ratios • Practice finding a side of a right triangle. Solve for x. • Sin 32 = x/8 • Sin 54 = 21/x • Cos 81 = x/8.8 • Tan 60 = 25/x • • • • 4.24 25.96 1.38 14.43 PRACTICE SECTION 10.4 TANGENT RATIO Example 1 Find Tangent Ratio Find tan S and tan R as fractions in simplified form and as decimals rounded to four decimal places. SOLUTION leg opposite S 4 3 tan S = = = 3 ≈ 1.7321 4 leg adjacent to S leg opposite R 4 = 1 = tan R = ≈ 0.5774 leg adjacent to R 4 3 3 Example 2 Use a Calculator for Tangent Approximate tan 47° to four decimal places. SOLUTION Calculator keystrokes 47 or 47 Display Rounded value 1.07236871 1.0724 Your Turn: Find tan S and tan R as fractions in simplified form and as decimals. Round to four decimal places if necessary. 1. ANSWER 2. ANSWER 3 tan S = = 0.75; 4 4 tan R = ≈ 1.3333 3 5 ≈ 0.4167; 12 12 tan R = = 2.4 5 tan S = Your Turn: Use a calculator to approximate the value to four decimal places. 3. tan 35° ANSWER 0.7002 4. tan 85° ANSWER 11.4301 5. tan 10° ANSWER 0.1763 Example 3 Find Leg Length Use a tangent ratio to find the value of x. Round your answer to the nearest tenth. SOLUTION opposite leg tan 22° = adjacent leg tan 22° = x3 x · tan 22° = 3 3 x= tan 22° x≈ 3 0.4040 x ≈ 7.4 Write the tangent ratio. Substitute. Multiply each side by x. Divide each side by tan 22°. Use a calculator or table to approximate tan 22°. Simplify. Example 4 Find Leg Length Use two different tangent ratios to find the value of x to the nearest tenth. SOLUTION First, find the measure of the other acute angle: 90° – 35° = 55°. Method 1 Method 2 opposite leg tan 35° = adjacent leg opposite leg tan 55° = adjacent leg tan 35° = x4 x · tan 35° = 4 tan 55° = x 4 4 tan 55° = x Example 4 Find Leg Length x= 4 tan 35° x≈ 4 0.7002 4(1.4281) ≈ x x ≈ 5.7 x ≈ 5.7 ANSWER The two methods yield the same answer: x ≈ 5.7. Example 5 Estimate Height You stand 45 feet from the base of a tree and look up at the top of the tree as shown in the diagram. Use a tangent ratio to estimate the height of the tree to the nearest foot. SOLUTION opposite leg tan 59° = adjacent leg h tan 59° = 45 45 tan 59° = h 45(1.6643) ≈ h 74.9 ≈ h Write ratio. Substitute. Multiply each side by 45. Use a calculator or table to approximate tan 59°. Simplify. Example 5 ANSWER Estimate Height The tree is about 75 feet tall. Your Turn: Write two equations you can use to find the value of x. 6. ANSWER x tan 44° = 8x and tan 46° = 8 7. ANSWER x tan 37° = 4x and tan 53° = 4 8. ANSWER x tan 59° = 5x and tan 31° = 5 Your turn: Find the value of x. Round your answer to the nearest tenth. 9. 10. 11. ANSWER 10.4 ANSWER 12.6 ANSWER 34.6 Assignment 10.4 • Pg. 560 – 562: #1 – 43 odd PRACTICE SECTION 10.5 SINE AND COSINE RATIOS SohCahToa Soh sin opposite hypotenuse Cah cos adjacent hypotenuse Toa tan opposite adjacent Example 1 Find Sine and Cosine Ratios Find sin A and cos A. SOLUTION leg opposite A sin A = hypotenuse 3 = 5 leg adjacent to A cos A = hypotenuse 4 = 5 Write ratio for sine. Substitute. Write ratio for cosine. Substitute. Your Turn: Find sin A and cos A. 1. 2. ANSWER sin A = 15 8 ; cos A = 17 17 ANSWER sin A = 24 7 ; cos A = 25 25 ANSWER 4 3 sin A = ; cos A = 5 5 3. Example 2 Find Sine and Cosine Ratios Find sin A and cos A. Write your answers as fractions and as decimals rounded to four decimal places. SOLUTION leg opposite A 5 ≈ 0.3846 = sin A = 13 hypotenuse leg adjacent to A 12 = ≈ 0.9231 cos A = 13 hypotenuse Your Turn: Find sin A and cos A. Write your answers as fractions and as decimals rounded to four decimal places. 4. 5. 6. ANSWER ANSWER ANSWER 40 sin A = ≈ 0.9756; 41 9 cos A = ≈ 0.2195 41 2 sin A = 2 ≈ 0.7071; 2 cos A = 2 ≈ 0.7071 39 ≈ 0.7806; 8 5 cos A = ≈ 0.625 8 sin A = Example 3 Use a Calculator for Sine and Cosine Use a calculator to approximate sin 74° and cos 74°. Round your answers to four decimal places. SOLUTION Calculator keystrokes Display Rounded value 74 or 74 0.961261696 0.9613 74 or 74 0.275637356 0.2756 Your Turn: Use a calculator to approximate the value to four decimal places. 7. sin 43° ANSWER 0.6820 8. cos 43° ANSWER 0.7314 9. sin 15° ANSWER 0.2588 10. cos 15° ANSWER 0.9659 Your Turn: Use a calculator to approximate the value to four decimal places. 11. cos 72° ANSWER 0.3090 12. sin 72° ANSWER 0.9511 13. cos 90° ANSWER 0 14. sin 90° ANSWER 1 Example 4 Find Leg Lengths Find the lengths of the legs of the triangle. Round your answers to the nearest tenth. SOLUTION leg opposite A sin A = hypotenuse sin 32° = a 10 leg adjacent to A cos A = hypotenuse cos 32° = b 10 10(sin 32°) = a 10(cos 32°) = b 10(0.5299) ≈ a 10(0.8480) ≈ b 5.3 ≈ a 8.5 ≈ b ANSWER In the triangle, BC is about 5.3 and AC is about 8.5. Your Turn: Find the lengths of the legs of the triangle. Round your answers to the nearest tenth. 15. 16. ANSWER a ≈ 3.9; b ≈ 5.8 ANSWER a ≈ 10.9; b ≈ 5.1 ANSWER a ≈ 3.4; b ≈ 3.7 17. Assignment 10.5 • Pg. 566 – 568: #1 – 31 odd, 37 – 47 odd Inverse Trigonometry • As we learned earlier, you can use the side lengths of a right triangle to find trigonometric ratios for the acute angles of the triangle. Once you know the sine, cosine, or tangent (trig. ratio) of an acute angle, you can use a calculator to find the measure of the angle. • To find an angle measurement in a right triangle given any two sides, use the inverse of the trig. ratio. Using a Calculator – Inverse Trigonometric Ratios • Given two sides of a right triangle, the inverse trigonometric ratios can be used to find the measure of an acute angle of the triangle. • In general, for an acute angle A: – If sin A = x, then sin-1 x = mA – If cos A = y, then cos-1 y = mA – If tan A = z, then tan-1 z = mA The expression sin-1 x is read as “the inverse sine of x.” • On your calculator, this means you will be punching the 2nd function button usually in yellow prior to doing the calculation. This is to find the degree of the angle. Using a Calculator – Inverse Trigonometric Ratios a 1 a If sin x , then x sin . b b a 1 a If cos x , then x cos . b b a 1 a If tan x , then x tan . b b • “sin-1 x” is read “the angle whose sine is x” or “inverse sine of x”. • arcsin x is the same thing as sin-1 x. • “inverse sin” is the inverse operation of “sin”. Example Given the trig. Ratio, solve for the angle. KEYSTROKES: 2nd [COS] ( 13 ÷ 19 ) ENTER 46.82644889 Answer: So, the measure of P is approximately 46.8°. Using a Calculator – Inverse Trigonometric Ratios • Given a trigonometric ratio in a right triangle, use inverse trig. ratios to solve for an acute angle. 14 Trig. ratio for ∠A. sin A • Example: 23 14 1 1 (sin )sin A sin 23 14 1 Asin 23 mA 37.50 To solve for ∠A, take the sin-1 of both sides of the equation. Inverse operations, sin and sin-1, cancel out. Use calculator to solve for ∠A. Using a Calculator – Inverse Trigonometric Ratios • Practice finding an acute angle of a right triangle. Solve for the indicated angle. • Sin B = 3.5/8 • Cos D = 12/14 • Tan A = 17/12 • ∠B = 25.94˚ • ∠D = 31.0˚ • ∠A = 54.78˚ To use Trigonometric Ratios to find lengths, given an interior angle and one side of a RAT • Finding the length of an unknown side of a right angled triangle: • (opposite) (opposite) 7cm (adjacent) • • 34o The appropriate ratio is sine, SOH The appropriate ratio to use is Tangent, i.e. TOA Tangent Ratio • • • • • 5m (hypotenuse) y a 26o Calculate the length of y. Opposite Side Adjacent Side Tan 260 = a/7 a/7 = Tan 260 a = 7 x Tan 260 a = 7 x 0.4877 a =3.41cm Sine Ratio • • • • • Opposite Side Hypotenuse Sine 340 = y/5 y/5 = Sine 340 y = 5 x Sine 340 y = 5 x 0.559 y =2.80m Your Turn: • Calculate the length of side x • y (Opposite) Hypotenuse 10.6m 670 x(Adjacent) Hypotenuse 6.2m • The appropriate ratio to use is Cosine, i.e. CAH Co s Ratio • • • • 420 • The appropriate ratio is Sine, i.e. SOH Adjacent Side , Hypotenuse Cos 670 = x/10.6 0.39 = x/10.6 x =10.6 X 0.39 x =4.14m Calculate the length of y Sine Ratio • • • • • Opposite Side Hypotenuse Sine 420 = y/6.2 y/6.2 = Sine 420 y = 6.2 X Sine 420 y = 6.2 X 0.669 y =4.19m To use Inverse Trigonometric Ratios to find an interior angle, given two sides of a RAT • To find angle a 32cm (Opposite) 25cm a Adjacent • The appropriate ratio to use is Tangent, i.e. TOA Tangent Ratio • • • • Opposite Side Adjacent Side Tan a = 32/25 Tan a = 1.28 a = Tan -1 1.28 a = 52.00 • To find the size of angle y (Opposite) 30cm 50cm Hypotenuse y0 • The appropriate ratio is sin , i.e. SOH Sine Ratio Opposite Side Hypotenuse • Sine y = 30/50 • Sine y = 0.6 • y = Sine-1 0.6 • y = 36.90 Your Turn: • Find angle b Hypotenuse Opposite 12cm 6cm b • The appropriate ratio to use is Sine, i.e. SOH Opposite Side Sine Ratio Hypotenuse • • • • Sin b = 6/12 Sin b = 0.5 b = Sin-1 0.5 b = 300 • Find angle y 12.4m(Adjacent) y Hypotenuse 19.7m • The appropriate ratio is Cosine, i.e. CAH Co s Ratio Adjacent Side , Hypotenuse • Cos y = 12.4/19.7 • Cos y = 0.639 • y = Cos-1 0.639 • y = 50.280 Solving Trigonometric Equations There are only three possibilities for the placement of the variable ‘x”. Opp Hyp Sin X = Sin = A A 12 cm 25 cm x B x C Sin X =12 25 Sin X = 0.48 X = Sin1 (0.48) X = 28.6854 B x Hyp x Sin 25 = 12 0.4226 = x 12 1 x = (12) (0.4226) x = 5.04 cm = Opp x A x 12 cm 12 cm 25 Sin 25 B C Sin 25 = 0.4226 = 1 x = 12 0.4226 x = 28.4 cm 12 x 12 x C PRACTICE SECTION 10.6 SOLVING RIGHT TRIANGLES Example 1 Use Inverse Tangent Use a calculator to approximate the measure of A to the nearest tenth of a degree. SOLUTION 8 = 0.8, tan–1 0.8 = mA. 10 Since tan A = Expression tan–1 0.8 Calculator keystrokes 0.8 or Display 38.65980825 0.8 ANSWER Because tan–1 0.8 ≈ 38.7°, mA ≈ 38.7°. Example 2 Solve a Right Triangle Find each measure to the nearest tenth. a. c b. mB c. mA SOLUTION a. Use the Pythagorean Theorem to find c. (hypotenuse)2 = (leg)2 + (leg)2 c 2 = 32 + 22 c2 = 13 c = 13 c ≈ 3.6 Pythagorean Theorem Substitute. Simplify. Find the positive square root. Use a calculator to approximate. Example 2 Solve a Right Triangle b. Use a calculator to find mB. Since tan B = 2 ≈ 0.6667, mB ≈ tan–1 0.6667 ≈ 33.7°. 3 c. A and B are complementary, so mA ≈ 90° – 33.7° = 56.3°. Your Turn: A is an acute angle. Use a calculator to approximate the measure of A to the nearest tenth of a degree. 1. tan A = 3.5 ANSWER 74.1° 2. tan A = 2 ANSWER 63.4° 3. tan A = 0.4402 ANSWER 23.8° Your Turn: Find the measure of A to the nearest tenth of a degree. 4. ANSWER 29.1° ANSWER 40.4° ANSWER 58.0° 5. 6. Example 3 Find the Measures of Acute Angles A is an acute angle. Use a calculator to approximate the measure of A to the nearest tenth of a degree. a. sin A = 0.55 b. cos A = 0.48 SOLUTION a. Since sin A = 0.55, mA = sin–1 0.55. sin–1 0.55 ≈ 33.36701297, so mA ≈ 33.4°. b. Since cos A = 0.48, mA = cos–1 0.48. cos–1 0.48 ≈ 61.31459799, so mA ≈ 61.3°. Example 4 Solve a Right Triangle Solve GHJ by finding each measure. Round decimals to the nearest tenth. a. mG b. mH c. g SOLUTION 16 = 0.64, mG = cos–1 0.64. 25 cos–1 0.64 ≈ 50.2081805, so mG ≈ 50.2°. a. Since cos G = b. G and H are complementary. mH = 90° – mG ≈ 90° – 50.2° = 39.8° Example 4 Solve a Right Triangle c. Use the Pythagorean Theorem to find g. (leg)2 + (leg)2 = (hypotenuse)2 162 + g2 = 252 256 + g2 = 625 g2 = 369 g = 369 g ≈ 19.2 Pythagorean Theorem Substitute. Simplify. Subtract 256 from each side. Find the positive square root. Use a calculator to approximate. Your Turn: A is an acute angle. Use a calculator to approximate the measure of A to the nearest tenth of a degree. 7. sin A = 0.5 ANSWER 30° 8. cos A = 0.92 ANSWER 23.1° 9. sin A = 0.1149 ANSWER 6.6° 10. cos A = 0.5 ANSWER 60° 11. sin A = 0.25 ANSWER 14.5° 12. cos A = 0.45 ANSWER 63.3° Your Turn: Solve the right triangle. Round decimals to the nearest tenth. 13. ANSWER x = 5; mA ≈ 36.9°; mB ≈ 53.1° ANSWER y ≈ 4.5; mD ≈ 41.8°; mE ≈ 48.2° ANSWER z ≈ 4.9; mG ≈ 44.4°; mH ≈ 45.6° 14. 15. Assignment 10.6 • Pg. 572 – 575: #1 – 45 odd REVIEW PRACTICE Example 1a A. Express sin L as a fraction and as a decimal to the nearest hundredth. Answer: Example 1b B. Express cos L as a fraction and as a decimal to the nearest hundredth. Answer: Example 1c C. Express tan L as a fraction and as a decimal to the nearest hundredth. Answer: Example 1d D. Express sin N as a fraction and as a decimal to the nearest hundredth. Answer: Example 1e E. Express cos N as a fraction and as a decimal to the nearest hundredth. Answer: Example 1f F. Express tan N as a fraction and as a decimal to the nearest hundredth. Answer: Your Turn: A. Find sin A. A. B. C. D. Your Turn: B. Find cos A. A. B. C. D. Your Turn: C. Find tan A. A. B. C. D. Your Turn: D. Find sin B. A. B. C. D. Your Turn: E. Find cos B. A. B. C. D. Your turn: F. Find tan B. A. B. C. D. Example 2 Use a special right triangle to express the cosine of 60° as a fraction and as a decimal to the nearest hundredth. Draw and label the side lengths of a 30°-60°-90° right triangle, with x as the length of the shorter leg and 2x as the length of the hypotenuse. The side adjacent to the 60° angle has a measure of x. Example 2 Definition of cosine ratio Substitution Simplify. Your Turn: Use a special right triangle to express the tangent of 60° as a fraction and as a decimal to the nearest hundredth. A. B. C. D. Example 3 A fitness trainer sets the incline on a treadmill to 7°. The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor? Let y be the height of the treadmill from the floor in inches. The length of the treadmill is 5 feet, or 60 inches. Example 3 Multiply each side by 60. Use a calculator to find y. Answer: The treadmill is about 7.3 inches high. Example 3 The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about 4.8°, about how high does the ramp rise off the ground to the nearest inch? A. 1 in. B. 11 in. C. 16 in. D. 15 in. Example 4 Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree. Example 4 Step 1 Find mA by using a tangent ratio. Definition of inverse tangent 29.7448813 ≈ mA Use a calculator. So, the measure of A is about 30. Example 4 Step 2 Find mB using complementary angles. mA + mB = 90 30 + mB ≈ 90 mB ≈ 60 Definition of complementary angles mA ≈ 30 Subtract 30 from each side. So, the measure of B is about 60. Example 4 Step 3 Find AB by using the Pythagorean Theorem. (AC)2 + (BC)2 = (AB)2 Pythagorean Theorem 72 + 42 = (AB)2 Substitution 65 = (AB)2 Simplify. Take the positive square root of each side. 8.06 ≈ AB Use a calculator. Example 4 So, the measure of AB is about 8.06. Answer: mA ≈ 30, mB ≈ 60, AB ≈ 8.06 Your Turn: Use a calculator to find the measure of D to the nearest tenth. A. 44.1° B. 48.3° C. 55.4° D. 57.2° Your Turn: Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. A. mA = 36°, mB = 54°, AB = 13.6 B. mA = 54°, mB = 36°, AB = 13.6 C. mA = 36°, mB = 54°, AB = 16.3 D. mA = 54°, mB = 36°, AB = 16.3 APPLICATION PROBLEMS Example 5 • You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of the tree. You measure the angle of elevation from a point on the ground to the top of the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet. Solution tan 59° = opposite adjacent h tan 59° = 45 Write the ratio Substitute values 45 tan 59° = h Multiply each side by 45 45 (1.6643) ≈ h Use a calculator or table to find tan 59° 74.9 ≈ h Simplify The tree is about 75 feet tall. Example 6 • Space Shuttle: During its approach to Earth, the space shuttle’s glide angle changes. • A. When the shuttle’s altitude is about 15.7 miles, its horizontal distance to the runway is about 59 miles. What is its glide angle? Round your answer to the nearest tenth. Solution: • You know opposite and adjacent sides. If you take the opposite and divide it by the adjacent sides, then take the inverse tangent of the ratio, this will yield you the slide angle. Glide = x° altitude 15.7 miles distance to runway 59 miles tan x° = opp. Use correct adj. ratio tan x° = 15.7 Substitute 59 values tan 15.7/59 ≈ 14.9 When the space shuttle’s altitude is about 15.7 miles, the glide angle is about 14.9°. B. Solution Glide = 19° altitude h • When the space shuttle is 5 miles from the runway, its glide angle is about 19°. Find the shuttle’s altitude at this tan 19° = point in its descent. Round your answer to tan 19° = the nearest tenth. distance to runway 5 miles opp. Use correct adj. ratio h Substitute 5 values 5 Isolate h by 5 tan 19° = h 5 multiplying by 5. The shuttle’s altitude is 1.7 ≈ h Approximate using about 1.7 miles. calculator Your Turn: A ladder that is 20 ft is leaning against the side of a building. If the angle formed between the ladder and ground is 75˚, how far is the bottom of the ladder from the base of the building? 75˚ building 20 x Using the 75˚ angle as a reference, we know hypotenuse and adjacent side. adj Use hyp cos cos 75˚ = x 20 About 5 ft. 20 (cos 75˚) = x 20 (.2588) = x x ≈ 5.2 Your Turn: When the sun is 62˚ above the horizon, a building casts a shadow 18m long. How tall is the building? x 18 62˚ shadow Using the 62˚ angle as a reference, we know opposite and adjacent side. opp Use adj tan tan 62˚ = x 18 18 (tan 62˚) = x 18 (1.8807) = x x ≈ 33.9 About 34 m Your Turn: A kite is flying at an angle of elevation of about 55˚. Ignoring the sag in the string, find the height of the kite if 85m of string have been let out. kite 85 x 55˚ Using the 55˚ angle as a reference, we know hypotenuse and opposite side. Use opp hyp sin sin 55˚ = x 85 About 70 m 85 (sin 55˚) = x 85 (.8192) = x x ≈ 69.6