1. Right Triangle Trigonometry
Section 10.4 Tangent Ratio
Section 10.5 Sine and Cosine Ratios
Section 10.6 Solving Right Triangles

2. Goals Find trigonometric ratios using right triangles.
Use trigonometric ratios to find angle measures in right triangles.

3. 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

4. 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.

5. 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.

6. 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.

7. Only Apply to Right Triangles
Trigonometric Ratios
Only Apply to Right Triangles

8. In right triangles :
The segment across from the right angle ( ) is labeled the hypotenuse “Hyp.”.
The “angle of perspective” determines how to label the sides.
Segment opposite from the Angle of Perspective( ) is labeled “Opp.”
Segment adjacent to (next to) the Angle of Perspective ( ) is labeled “Adj.”.
* The angle of Perspective is never the right angle.
Hyp.
Opp.
Angle of Perspective
Adj.

9. Labeling sides depends on the Angle of PerspectiveIf
is the Angle of Perspective then ……
Angle of Perspective
Hyp.
Adj.
Opp.
*”Opp.” means segment opposite from Angle of Perspective
“Adj.” means segment adjacent from Angle of Perspective

10. If the Angle of Perspective is
then
then
Hyp
Adj
Hyp
Opp
Opp
Adj

11. The 3 Trigonometric Ratios
The 3 ratios are Sine, Cosine and Tangent

12. Trigonometric Ratios
To help you remember these trigonometric relationships, you can use the mnemonic device, SOH-CAH-TOA, where the first letter of each word of the trigonometric ratios is represented in the correct order. A
Sin A = Opposite side        SOH             Hypotenuse
Cos A = Adjacent side         CAH            Hypotenuse
Tan A = Opposite side    TOA                   Adjacent side c b a C B

13. Trigonometric Ratios
hypotenuse
side opposite 
side adjacent 

14. SohCahToa

15. The Amazing Legend of…
Chief SohCahToa

16. 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….

17. 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).

18. 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.

19. Sample keystrokes sin sin ENTER COS COS ENTER TAN TAN ENTER 74
Sample keystroke sequences
Sample calculator display
Rounded
Approximation 74
0.9613
0.2756
3.4874
sin
sin
ENTER 74
COS
COS
ENTER 74
TAN
TAN
ENTER

20. 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

21. 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.
Example 1:
Trig. ratio used to find side adj. to 42˚ angle.
Multiply both sides by 15 to solve for x.
Use calculator to find the length of the adj. side.

22. Using a Calculator – Trigonometric Ratios
Example 2:
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.

23. 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

24. Practice Section 10.4 Tangent Ratio

25. 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
tan S =
leg adjacent to S = ≈ 4 3
tan R =
leg opposite R
leg adjacent to R = ≈ 4 3 1 25

26. Approximate tan 47° to four decimal places.
Example 2
Use a Calculator for Tangent
Approximate tan 47° to four decimal places.
SOLUTION
Calculator keystrokes 47 or
Display
Rounded value
1.0724 26

27. 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
tan S = 3 4
= 0.75;
tan R = ≈ 2.
ANSWER
tan S = 5 12
≈ ;
tan R =
= 2.4

28. Your Turn:
Use a calculator to approximate the value to four decimal places. 3.
tan 35°
ANSWER
0.7002 4.
tan 85°
ANSWER 5.
tan 10°
ANSWER
0.1763

29. Example 3
Find Leg Length
Use a tangent ratio to find the value of x. Round your answer to the nearest tenth.
SOLUTION
tan 22° =
opposite leg
adjacent leg
Write the tangent ratio.
tan 22° = 3 x
Substitute.
x · tan 22° = 3
Multiply each side by x.
x = 3
tan 22°
Divide each side by tan 22°.
x ≈ 3
0.4040
Use a calculator or table to approximate tan 22°.
x ≈ 7.4
Simplify. 29

30. First, find the measure of the other acute angle: 90° – 35° = 55°.
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
tan 35° =
opposite leg
adjacent leg
Method 2
tan 55° =
opposite leg
adjacent leg
tan 35° = 4 x
tan 55° = x 4
x · tan 35° = 4
4 tan 55° = x 30

31. The two methods yield the same answer: x ≈ 5.7.
Example 4
Find Leg Length
x = 4
tan 35°
4(1.4281) ≈ x
x ≈ 4
0.7002
x ≈ 5.7
x ≈ 5.7
ANSWER
The two methods yield the same answer: x ≈ 5.7. 31

32. 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
tan 59° =
opposite leg
adjacent leg
Write ratio.
tan 59° = h 45
Substitute.
45 tan 59° = h
Multiply each side by 45.
45(1.6643) ≈ h
Use a calculator or table to approximate tan 59°.
74.9 ≈ h
Simplify. 32

33. The tree is about 75 feet tall.
Example 5
Estimate Height
ANSWER
The tree is about 75 feet tall. 33

34. Your Turn:
Write two equations you can use to find the value of x. 6.
ANSWER
tan 44° = 8 x
and
tan 46° = 7.
tan 37° = 4 x
and
tan 53° =
ANSWER 8.
tan 59° = 5 x
and
tan 31° =
ANSWER

35. Your turn:
Find the value of x. Round your answer to the nearest tenth. 9.
ANSWER
10.4
10.
ANSWER
12.6
11.
ANSWER
34.6

36. Assignment 10.4
Pg. 560 – 562: #1 – 43 odd

37. Practice Section 10.5 Sine and cosine Ratios

38. SohCahToa

39. Example 1 Find sin A and cos A. SOLUTION leg opposite A sin A =
Find Sine and Cosine Ratios
Find sin A and cos A.
SOLUTION
sin A =
leg opposite A
hypotenuse
Write ratio for sine. 3 5 =
Substitute.
cos A =
leg adjacent to A
hypotenuse
Write ratio for cosine. 4 5 =
Substitute. 39

40. Your Turn:
Find sin A and cos A. 1.
ANSWER
sin A = 15 17 ;
cos A = 8 2.
ANSWER
sin A = 24 25 ;
cos A = 7 3.
ANSWER ;
sin A = 4 5
cos A = 3

41. 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
sin A =
leg opposite A
hypotenuse 5 13 = ≈ =
cos A =
leg adjacent to A
hypotenuse 12 13 ≈ 41

42. Your Turn:
Find sin A and cos A. Write your answers as fractions and as decimals rounded to four decimal places. 4.
ANSWER ;
sin A = 40 41 ≈
cos A = 9 ≈ 5. ;
ANSWER
sin A = ≈ 2
cos A = 6.
ANSWER
cos A = 5 8
≈ 0.625 ≈
sin A = 39 ;

43. Calculator keystrokes
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 74 or
Display
Rounded value
0.9613 74 or
0.2756 43

44. 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

45. 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
14.
sin 90°
ANSWER 1

46. In the triangle, BC is about 5.3 and AC is about 8.5.
Example 4
Find Leg Lengths
Find the lengths of the legs of the triangle. Round your answers to the nearest tenth.
SOLUTION
sin A =
leg opposite A
hypotenuse
cos A =
leg adjacent to A = a 10
sin 32° b
cos 32°
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. 46

47. Your Turn:
Find the lengths of the legs of the triangle. Round your answers to the nearest tenth.
15.
ANSWER
a ≈ 3.9; b ≈ 5.8
16.
ANSWER
a ≈ 10.9; b ≈ 5.1
17.
ANSWER
a ≈ 3.4; b ≈ 3.7

48. Assignment 10.5
Pg. 566 – 568: #1 – 31 odd, 37 – 47 odd

49. 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.

50. 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 = mA
If cos A = y, then cos-1 y = mA
If tan A = z, then tan-1 z = mA
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.

51. Using a Calculator – Inverse Trigonometric Ratios
“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”.

52. Example Given the trig. Ratio, solve for the angle.
KEYSTROKES: [COS]
2nd ( ÷ )
ENTER
Answer: So, the measure of P is approximately 46.8°.

53. Using a Calculator – Inverse Trigonometric Ratios
Given a trigonometric ratio in a right triangle, use inverse trig. ratios to solve for an acute angle.
Example:
Trig. ratio for ∠A.
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.

54. 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˚

55. 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:
The appropriate ratio to use is Tangent, i.e. TOA
Tan 260 = a/7
a/7 = Tan 260
a = 7 x Tan 260
a = 7 x
a =3.41cm
Calculate the length of y.
The appropriate ratio is sine, SOH
Sine 340 = y/5
y/5 = Sine 340
y = 5 x Sine 340
y = 5 x 0.559
y =2.80m 5m
(hypotenuse) a y
(opposite)
(opposite)
34o
26o
7cm
(adjacent)

56. Your Turn: (Opposite) Hypotenuse y 10.6m Hypotenuse 670 6.2m 420 x
Calculate the length of y
The appropriate ratio is Sine, i.e. SOH
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
Calculate the length of side x
The appropriate ratio to use is Cosine, i.e. CAH
Cos 670 = x/10.6
0.39 = x/10.6
x =10.6 X 0.39
x =4.14m
(Opposite)
Hypotenuse y
10.6m
Hypotenuse
670
6.2m
420 x
(Adjacent)

57. To use Inverse Trigonometric Ratios to find an interior angle, given two sides of a RAT
To find the size of angle y
The appropriate ratio is sin , i.e. SOH
Sine y = 30/50
Sine y = 0.6
y = Sine
y = 36.90
To find angle a
The appropriate ratio to use is Tangent, i.e. TOA
Tan a = 32/25
Tan a = 1.28
a = Tan
a = 52.00
(Opposite)
32cm
50cm
25cm
(Opposite)
Hypotenuse a
Adjacent
30cm y0

58. Your Turn: Find angle b The appropriate ratio to use is Sine, i.e. SOH
Find angle y
The appropriate ratio is Cosine, i.e. CAH
Cos y = 12.4/19.7
Cos y = 0.639
y = Cos
y =
Find angle b
The appropriate ratio to use is Sine, i.e. SOH
Sin b = 6/12
Sin b = 0.5
b = Sin-1 0.5
b = 300
(Adjacent)
Hypotenuse
12.4m
Opposite y
12cm
6cm
Hypotenuse b
19.7m

59. Solving Trigonometric Equations
There are only three possibilities for the placement of the variable ‘x”.
Sin =
Sin =
Sin =
Sin =
Sin =
Sin = =
Sin = 0.48 =
X = Sin (0.48)
x =
x = (12) (0.4226)
x = 5.04 cm
X =
x = 28.4 cm

60. Practice Section 10.6 Solving Right Triangles

61. Calculator keystrokes
Example 1
Use Inverse Tangent
Use a calculator to approximate the measure of A to the nearest tenth of a degree.
SOLUTION
tan A = 8 10
= 0.8, tan–1 0.8 = mA.
Since
Expression
tan–1 0.8
Calculator keystrokes
0.8 or
Display
ANSWER
Because tan–1 0.8 ≈ 38.7°, mA ≈ 38.7°. 61

62. Find each measure to the nearest tenth.
Example 2
Solve a Right Triangle
Find each measure to the nearest tenth. a. c b.
mB c.
mA
SOLUTION a.
Use the Pythagorean Theorem to find c.
(hypotenuse)2 = (leg)2 + (leg)2
Pythagorean Theorem
c2 =
Substitute.
c2 = 13
Simplify.
Find the positive square root.
c = 13
c ≈ 3.6
Use a calculator to approximate. 62

63. Use a calculator to find mB.
Example 2
Solve a Right Triangle b.
Use a calculator to find mB.
Since tan B = ≈ , mB ≈ tan– ≈ 33.7°. 2 3 c.
A and B are complementary, so mA ≈ 90° – 33.7° = 56.3°. 63

64. 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 =
ANSWER
23.8°

65. Your Turn:
Find the measure of A to the nearest tenth of a degree. 4.
ANSWER
29.1° 5.
ANSWER
40.4° 6.
ANSWER
58.0°

66. 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, mA = sin–
sin– ≈ , so mA ≈ 33.4°. b.
Since cos A = 0.48, mA = cos–
cos– ≈ , so mA ≈ 61.3°. 66

67. G and H are complementary. mH = 90° – mG ≈ 90° – 50.2° = 39.8°
Example 4
Solve a Right Triangle
Solve GHJ by finding each measure. Round decimals to the nearest tenth. a.
mG c. g b.
mH
SOLUTION
cos– ≈ , so mG ≈ 50.2°. a.
Since cos G = = 0.64, mG = cos– 16 25 b.
G and H are complementary.
mH = 90° – mG ≈ 90° – 50.2° = 39.8° 67

68. Use the Pythagorean Theorem to find g.
Example 4
Solve a Right Triangle c.
Use the Pythagorean Theorem to find g.
(leg)2 + (leg)2 = (hypotenuse)2
Pythagorean Theorem
162 + g2 = 252
Substitute.
256 + g2 = 625
Simplify.
g2 = 369
Subtract 256 from each side.
Find the positive square root.
g =
369
g ≈ 19.2
Use a calculator to approximate. 68

69. 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 =
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°

70. Your Turn:
Solve the right triangle. Round decimals to the nearest tenth.
13.
ANSWER
x = 5; mA ≈ 36.9°; mB ≈ 53.1°
14.
ANSWER
y ≈ 4.5; mD ≈ 41.8°; mE ≈ 48.2°
15.
ANSWER
z ≈ 4.9; mG ≈ 44.4°; mH ≈ 45.6°

71. Assignment 10.6
Pg. 572 – 575: #1 – 45 odd

72. Review Practice

73. Example 1a
A. Express sin L as a fraction and as a decimal to the nearest hundredth.
Answer:

74. Example 1b
B. Express cos L as a fraction and as a decimal to the nearest hundredth.
Answer:

75. Example 1c
C. Express tan L as a fraction and as a decimal to the nearest hundredth.
Answer:

76. Example 1d
D. Express sin N as a fraction and as a decimal to the nearest hundredth.
Answer:

77. Example 1e
E. Express cos N as a fraction and as a decimal to the nearest hundredth.
Answer:

78. Example 1f
F. Express tan N as a fraction and as a decimal to the nearest hundredth.
Answer:

79. Your Turn:
A. Find sin A. A. B. C. D.

80. Your Turn:
B. Find cos A. A. B. C. D.

81. Your Turn:
C. Find tan A. A. B. C. D.

82. Your Turn:
D. Find sin B. A. B. C. D.

83. Your Turn:
E. Find cos B. A. B. C. D.

84. Your turn:
F. Find tan B. A. B. C. D.

85. 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.

86. Example 2
Definition of cosine ratio
Substitution
Simplify.

87. 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.

88. 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.

89. Example 3 Multiply each side by 60. Use a calculator to find y.
Answer: The treadmill is about 7.3 inches high.

90. 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.

91. Example 4
Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree.

92. Example 4 Step 1 Find mA by using a tangent ratio.
Definition of inverse tangent
≈ mA Use a calculator.
So, the measure of A is about 30.

93. Example 4 Step 2 Find mB using complementary angles.
mA + mB = 90 Definition of complementary angles
30 + mB ≈ 90 mA ≈ 30
mB ≈ 60 Subtract 30 from each side.
So, the measure of B is about 60.

94. Example 4 Step 3 Find AB by using the Pythagorean Theorem.
(AC)2 + (BC)2 = (AB)2 Pythagorean Theorem
= (AB)2 Substitution
65 = (AB)2 Simplify.
Take the positive square root of each side.
8.06 ≈ AB Use a calculator.

95. Example 4 So, the measure of AB is about 8.06.
Answer: mA ≈ 30, mB ≈ 60, AB ≈ 8.06

96. 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°

97. Your Turn:
Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.
A. mA = 36°, mB = 54°, AB = 13.6
B. mA = 54°, mB = 36°, AB = 13.6
C. mA = 36°, mB = 54°, AB = 16.3
D. mA = 54°, mB = 36°, AB = 16.3

98. Application Problems

99. 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.

100. Solution The tree is about 75 feet tall. opposite tan 59° = adjacent
Write the ratio
tan 59° = h 45
Substitute values
Multiply each side by 45
45 tan 59° = h
Use a calculator or table to find tan 59°
45 (1.6643) ≈ h
Simplify
74.9 ≈ h
The tree is about 75 feet tall.

101. 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.

102. Solution:
Glide  = x°
15.7 miles
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.
59 miles
opp.
tan x° =
Use correct ratio
adj.
15.7
Substitute values
tan x° = 59
tan 15.7/59 ≈ 14.9
 When the space shuttle’s altitude is about 15.7 miles, the glide angle is about 14.9°.

103. B. Solution
Glide  = 19° h
When the space shuttle is 5 miles from the runway, its glide angle is about 19°. Find the shuttle’s altitude at this point in its descent. Round your answer to the nearest tenth.
5 miles
opp.
tan 19° =
Use correct ratio
adj. h
Substitute values
tan 19° = 5 h 5
Isolate h by multiplying by 5.
5 tan 19° = 5
 The shuttle’s altitude is about 1.7 miles.
1.7 ≈ h
Approximate using calculator

104. Your Turn: 20 building ladder x Use cos
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? 20
building
ladder
75˚ x
Using the 75˚ angle as a reference, we know hypotenuse and adjacent side.
Use
cos
cos 75˚ =
About 5 ft.
20 (cos 75˚) = x
20 (.2588) = x
x ≈ 5.2

105. Your Turn:
When the sun is 62˚ above the horizon, a building casts a shadow 18m long. How tall is the building? x
62˚ 18
shadow
Using the 62˚ angle as a reference, we know opposite and adjacent side.
Use
tan
tan 62˚ =
About 34 m
18 (tan 62˚) = x
18 (1.8807) = x
x ≈ 33.9

106. Your Turn: kite 85 x string Use sin
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
string
55˚
Using the 55˚ angle as a reference, we know hypotenuse and opposite side.
Use
sin
sin 55˚ =
85 (sin 55˚) = x
About 70 m
85 (.8192) = x
x ≈ 69.6