Transcript Slide 1
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,
measurements
.
the study of trigonometry is the study of triangle
A ratio of the lengths of the sides of a right triangle
sine
, is called a
cosine
, and
trigonometric ratio tangent
. . The three most common trigonometric ratios are
Trigonometric Ratios
Only Apply to Right Triangles
In right triangles :
• • • • “Hyp.”.
A
AC
Hyp.
Opp.
Angle of Perspective
B C
Adj.
The “angle of perspective” determines how to label the sides.
Segment adjacent to (next to) the Angle of Perspective ( ) 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 Adj.
A B
Opp.
Hyp.
C
AC BC AB
Hyp
Opp
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
Adj
A
Hyp
B
Opp
C
AC
Hyp BC
Opp AB
Adj
C then
A
Hyp Opp
B C
Adj
AC
Hyp AB
Opp BC
Adj
The 3 Trigonometric Ratios
•
The 3 ratios are Sine, Cosine and Tangent
Sine Ratio
Opposite Side Hypotenuse Co
sin
e Ratio
Adjacent Side Hypotenuse Tangent Ratio
Opposite Side Adjacent Side
Trigonometric Ratios
To help you remember these trigonometric
S
relationships, you can use
C H
ypotenuse the mnemonic device,
SOH-CAH-TOA
, where the first letter of each word of
T
an A =
O
pposite side
TOA
the trigonometric ratios is represented in the correct order.
A b C a c B
Trigonometric Ratios
C A
side adjacent
B sin
Oh Hell Soh
cos
Another Hour Cah
tan
Of Algebra Toa
SohCahToa
Soh Cah Toa
sin
opposite hypotenuse
cos
adjacent hypotenuse
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 calculator display Sample keystroke sequences 74
sin sin
74
ENTER 74 COS COS 74 ENTER
0.961262695
0.275637355
3.487414444
74 TAN TAN ENTER 74
Rounded Approximation 0.9613
0.2756
3.4874
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.
Example 1: cos 42
x
15
Trig. ratio used to find side adj. to 42
˚
angle.
x
15cos 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
x
9.5
sin 66
x
10.40
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
tan
S
= leg opposite
S
leg adjacent to
S
= 4 3 4 = 3 ≈ 1.7321
tan
R
= leg opposite
R
leg adjacent to
R
= 4 4 3 = 1 3 ≈ 0.5774
Example 2 Use a Calculator for Tangent Approximate tan 47° to four decimal places.
SOLUTION Calculator keystrokes
47
or
47
Display
1.07236871
Rounded value
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 tan S = tan R = 3 4 4 3 = 0.75
; ≈ 1.3333
2.
ANSWER tan S = 5 12 tan R = 12 5 ≈ 0.4167
= 2.4
;
Your Turn:
Use a calculator to approximate the value to four decimal places.
3.
tan 35° ANSWER 0.7002
4.
5.
tan 85° tan 10° ANSWER 11.4301
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° = 3
x
x · tan 22° = 3
x
= 3 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
opposite leg tan 35° = adjacent leg tan 35° = 4
x
x · tan 35° = 4
Method 2
opposite leg tan 55° = adjacent leg tan 55° =
x
4 4 tan 55° =
x
Example 4 Find Leg Length
x
= 4 tan 35°
x
≈ 4 0.7002
x
≈ 5.7
4(1.4281) ≈
x 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 tan 59° =
h
45 45 tan 59° =
h
Write ratio.
Substitute.
Multiply each side by 45 .
45(1.6643) ≈ 74.9 ≈
h h
Use a calculator or table to approximate tan 59° .
Simplify.
Example 5 Estimate Height ANSWER The tree is about 75 feet tall.
Your Turn:
Write two equations you can use to find the value of
x
.
6.
ANSWER and tan 46° =
x
8 7.
8.
ANSWER and tan 53° =
x
4 ANSWER and tan 31° =
x
5
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
Assignment 10.4
• Pg. 560 – 562: #1 – 43 odd
PRACTICE SECTION 10.5 SINE AND COSINE RATIOS
SohCahToa
Soh Cah Toa
sin
opposite hypotenuse
cos
adjacent hypotenuse
tan
opposite adjacent
Example 1 Find Sine and Cosine Ratios Find sin A and cos A .
SOLUTION
sin
A
= = leg opposite
A
hypotenuse 3 5 cos
A
= leg adjacent to
A
hypotenuse = 4 5
Write ratio for sine.
Substitute.
Write ratio for cosine.
Substitute.
Your Turn:
Find sin A and cos A .
1.
ANSWER sin A = 15 17 ; cos A = 8 17 2.
ANSWER sin A = 24 25 ; cos A = 7 25 3.
ANSWER sin A = 4 5 ; cos A = 3 5
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 ≈ 0.3846
cos
A
= leg adjacent to
A
hypotenuse = 12 13 ≈ 0.9231
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 = cos A = 40 41 9 41 ≈ 0.9756
; ≈ 0.2195
5.
ANSWER sin A = cos A = 2 2 2 2 ≈ 0.7071
; ≈ 0.7071
6.
ANSWER sin A = cos A = 39 8 ≈ 0.7806
; 5 8 ≈ 0.625
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
74 74
or
74
Display
0.961261696
0.275637356
Rounded value
0.9613
0.2756
Your Turn:
Use a calculator to approximate the value to four decimal places.
7.
sin 43° ANSWER 0.6820
8.
9.
10.
cos 43° sin 15° cos 15° ANSWER 0.7314
ANSWER 0.2588
ANSWER 0.9659
Your Turn:
Use a calculator to approximate the value to four decimal places.
11.
cos 72° ANSWER 0.3090
12.
13.
14.
sin 72° cos 90° sin 90° ANSWER 0.9511
ANSWER 0 ANSWER 1
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 sin 32° =
a
10 10(sin 32°) =
a
10(0.5299) ≈
a
5.3 ≈
a
ANSWER
cos
A
= cos 32° = 10(cos 32°) = 10
b
10(0.8480) ≈
b
8.5 ≈
b
leg adjacent to
A b
hypotenuse
In the triangle,
BC
is about 5.3
about 8.5
.
and
AC
is
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
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 = 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 of x.” x is read as “the inverse sine
•
On your calculator, this means you will be punching the 2 nd 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
If If sin tan
x x
a b a b
, then
x
, then
x
sin tan 1 If cos
x
a
, then
b x
cos 1 1
a b
.
a b
.
a 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.
) 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.
Example: (sin 1 )sin sin
A
sin sin 23 1 1 14 14 23 14 23
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.
37.50
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:
•
Calculate the length of y.
a (opposite) 26 o 7cm ( adjacent )
•
(opposite) y
The appropriate ratio to use is Tangent, i.e. TOA
•
5m 34 o (hypotenuse)
The appropriate ratio is sine, SOH
• • • • •
Tangent Ratio
Opposite
Tan 26 0 = a/7 a/7 = Tan 26 0 a = 7 x Tan 26 0
Adjacent
a = 7 x 0.4877
a =3.41cm
Side Side
• • • • •
Sine Ratio
Opposite Side Hypotenuse
Sine 34 0 = y/5 y/5 = Sine 34 0 y = 5 x Sine 34 0 y = 5 x 0.559
y =2.80m
Your Turn:
• • • • •
Calculate the length of side x
Hypotenuse 10.6m
67 0 x(Adjacent)
•
Calculate the length of y y (Opposite) Hypotenuse 6.2m
42 0
•
The appropriate ratio to use is Cosine, i.e. CAH
Co
s
Ratio
Adjacent Side
,
Hypotenuse
Cos 67 0 = x/10.6
0.39 = x/10.6
x =10.6 X 0.39
x =4.14m
•
The appropriate ratio is Sine, i.e. SOH
• • • • •
Sine Ratio
Opposite Side Hypotenuse
Sine 42 0 = y/6.2
y/6.2 = Sine 42 0 y = 6.2 X Sine 42 0 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)
•
To find the size of angle y
25cm 50cm Hypotenuse a Adjacent ( Opposite ) 30cm y 0
•
The appropriate ratio to use is Tangent, i.e. TOA
• • • •
Tangent Ratio
Opposite Adjacent Side Side
Tan a = 32/25 Tan a = 1.28
a = Tan a = 52.0
-1 0 1.28
• • • • •
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 = 36.9
0
•
Find angle b
Opposite 6cm Hypotenuse 12cm b
•
The appropriate ratio to use is Sine, i.e. SOH
Sine Ratio
Opposite Side Hypotenuse
• • • •
Sin b = 6/12 Sin b = 0.5
b = Sin -1 0.5
b = 30 0 Your Turn:
•
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 = 50.28
0
Solving Trigonometric Equations
There are only three possibilities for the placement of the variable ‘x”.
A
Opp Hyp
A
x Hyp
A
x
x 12 cm
B
X
25 cm x
C 25
X = 28.6854
12 cm x 25
B
x Sin 25 =
12
0.4226 = x
1 12
x = (12) (0.4226)
C
x = 5.04 cm
12 cm 25
B
Sin 25 =
12
x
C 1
x =
12 0.4226
x = 28.4 cm x
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 Since
tan
A
= 8 10 = 0.8
,
tan –1 0.8 =
m
A
.
Expression
tan –1 0.8
Calculator keystrokes
0.8
or
0.8
Display
38.65980825
ANSWER Because tan –1 0.8 ≈ 38.7° ,
m
A ≈ 38.7° .
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
c
2 = 3 2 + 2 2
c
2 = 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
m
B
.
Since
tan
B
2 = ≈ 0.6667
,
3
m
B
≈ tan –1 0.6667 ≈ 33.7°
.
c.
A
and
B m
A
are complementary, so
≈ 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.
3.
tan A = 2 tan A = 0.4402
ANSWER 63.4° ANSWER 23.8°
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°
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 –1 0.55
.
sin –1 0.55 ≈ 33.36701297
, so
m
A
≈ 33.4°
.
b. Since
cos
A
= 0.48
,
m
A
= cos –1 0.48
.
cos –1 0.48 ≈ 61.31459799
, so
m
A
≈ 61.3°
.
Example 4 Solve a Right Triangle Solve
GHJ
by finding each measure. Round decimals to the nearest tenth.
a.
m
G
b.
m
H
c.
g
SOLUTION a. Since
cos
G
16 = = 0.64
,
25
m
cos –1 0.64 ≈ 50.2081805
, so
m G
= cos
G
–1 0.64
≈ 50.2°
.
.
b.
G
and
H m
H
= 90° –
are complementary.
m
G
≈ 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 16 2 +
g
2 = 25 2 256 +
g
2 = 625
g
2 = 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.
9.
cos A = 0.92
sin A = 0.1149
ANSWER 23.1° ANSWER 6.6° 10.
11.
12.
cos A = 0.5
sin A = 0.25
cos A = 0.45
ANSWER 60° ANSWER 14.5° ANSWER 63.3°
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°
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:
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.
Your turn:
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 ° has a measure of x.
angle
Example 2
Definition of cosine ratio Substitution Simplify.
B.
C.
D.
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.
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
Use a calculator to find y.
Multiply each side by 60.
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.
Step 1
Example 4
Find m
A by using a tangent ratio.
Definition of inverse tangent 29.7448813
≈ m
A Use a calculator.
So, the measure of
A is about 30
.
Example 4
Step 2
Find m
B using complementary angles.
m
A + m
B = 90 Definition of complementary angles 30 + m
B ≈ 90 m
B ≈ 60 m So, the measure of
B is about 60
.
A ≈ 30 Subtract 30 from each side.
Example 4
Step 3
Find AB by using the Pythagorean Theorem.
(AC) 2 + (BC) 2 = (AB) 2 7 2 + 4 2 65 = (AB) = (AB) 2 2 Pythagorean Theorem Substitution 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:
m
A ≈ 30, m
B ≈ 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. 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
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 tan 59 ° ° = = opposite adjacent h 45 45 tan 59 ° = h 45 (1.6643) ≈ h 74.9 ≈ h Write the ratio Substitute values Multiply each side by 45 Use a calculator or table to find tan 59 ° 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.
adj.
tan x ° = 15.7
59 Use correct ratio Substitute 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 point in its descent. Round your answer to the nearest tenth.
tan 19 ° = tan 19 ° = 5 tan 19 ° =
The shuttle’s altitude is about 1.7 miles.
1.7 ≈ h
distance to runway
5 miles opp.
adj.
h 5 h 5 Use correct ratio Substitute values 5 Isolate h by multiplying by 5.
Approximate using 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?
20 75˚ x Using the 75˚ Use
adj hyp
angle as a reference, we know hypotenuse and adjacent side. cos cos 75˚ =
x 20
20 (cos 75˚) = x About 5 ft.
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˚ Use
opp adj
angle as a reference, we know opposite and adjacent side. tan tan 62˚ =
x 18
18 (tan 62˚) = x About 34 m 18 (1.8807) = x x ≈ 33.9
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˚ Use
opp hyp
angle as a reference, we know hypotenuse and opposite side. sin sin 55˚ =
x 85
85 (sin 55˚) = x About 70 m 85 (.8192) = x x ≈ 69.6