14-3 Right Triangle Trig

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Transcript 14-3 Right Triangle Trig 

14-3 Right Triangle Trig
Hubarth
Algebra II
The trigonometric ratios for a right triangle:
π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’
π‘Ž
sin 𝐴 =
=
β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ 𝑐
π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘
𝑏
cos 𝐴 =
=
β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ 𝑐
tan 𝐴 =
π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ π‘Ž
=
π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝑏
1
β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ 𝑐
csc 𝐴 =
=
=
sin 𝐴
π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’
π‘Ž
1
β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ 𝑐
sec 𝐴 =
=
=
cos 𝐴
π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘
𝑏
cot 𝐴 =
1
π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝑏
=
=
tan 𝐴 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ π‘Ž
A
b
C
c
a
B
Ex. 1 Real-World Connection
A tourist visiting Washington D.C. is seated on
the grass at point A and is looking up at the top of the Washington
Monument. The angle of her line of sight with the ground is 27°.
Given that sin 27° 0.45, cos 27° 0.89, and tan 27° 0.51, find her
approximate distance AC from the base of the monument.
tan 27° =
height of monument
distance from monument
555
tan 27° = AC
555
0.51 = AC
AC =
Definition of tan
Substitute.
Use a calculator in degree mode.
555
β‰ˆ1088
0.51
The distance of the visitor from the monument is approximately 1088 feet.
Ex. 2 Using a Right Triangle to Find Ratios
In
7
PQR,
R is a right angle and cos P = 25 .
Find sin P, tan P, and cos Q in fraction and in decimal form.
Step 1: Draw a diagram.
Step 2: Use the Pythagorean
Theorem to find p.
r2 = p2 + q2
252 = p2 + 72
625 = p2 + 49
576 = p2
24 = p
Step 3: Calculate the ratios.
sin P =
π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’
𝑝
=
β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ π‘Ÿ
π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’
𝑝
tan P = π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ = π‘ž =
cos Q =
π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘
β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’
=
=
25
7
24
7
= 3.43
𝑝
π‘Ÿ
= 25 = 0.96
= 0.96
24
Ex. 3 Real-World Connection
A man 6 feet tall is standing 50 feet from a tree. When he looks at the
top of the tree, the angle of elevation is 42°. Find the height of the tree
to the nearest foot.
In the right triangle, the length of the leg
adjacent to the 42° angle is 50 ft.
The length of the leg opposite the 42° angle is unknown.
You need to find the length of the leg opposite the 42° angle.
Use the tangent ratio.
x
tan 42° = 50
x = 50 tan 42°
45
Definition of tan
Solve for x.
Use a calculator in degree mode.
The height of the tree is approximately 45 + 6 or 51 ft.
Ex. 4 Finding Angle Measures
In KMN, N is a right angle, m = 7, and n = 25. Find m
nearest tenth of a degree.
Step 1: Draw a diagram.
K to the
Step 2: Use a cosine ratio.
cos K
7
=
25
= 0.28
m
K
= cos–1 0.28
73.74° Use a calculator.
Since
K is acute, the other solutions of cos–1 0.28 do not apply.
To the nearest tenth of a degree, m
K is 73.7°.
Ex. 5 Real-World Connection
A straight road that goes up a hill is 800 feet higher at the top than at
the bottom. The horizontal distance covered is 6515 feet. To the
nearest degree, what angle does the road make with level ground?
Let
= the measure of the angle of the slope of the road.
You know the length of the leg opposite of the angle you need to find.
You know the length of the hypotenuse. So, use the tangent ratio.
tan
=
800
6515
= tan–1
7.0
800
6515
Use the inverse of the tangent function.
Use a calculator.
The angle the road makes with level ground is approximately 7°.
Practice
1. Find the length of AB, in the figure.
1222.5
3
2. In βˆ†π·πΈπΉ, < 𝐷 is a right angle and tan 𝐸 = 4 . Draw the diagram and find sin E and sec F in
E
fraction and in decimal form.
3
sin
𝐸
=
= .6
5
5
4
D
3
F
3. Use a trigonometric ratio to find π‘š < 𝐴 in each triangle.
C
B
23.58
5
10
4
C
A
B
9
5
sec 𝐹 = = 1.67
3
56.25
A