Download 13 - Mira Costa High School

13.1 Use Trigonometry with Right Triangles
Goal  Use trigonometric functions to find lengths.
Your Notes
VOCABULARY
Sine, cosine, tangent, cosecant, secant, cotangent
The six trigonometric functions that consist of ratios of a right triangle’s side lengths
RIGHT TRIANGLE DEFINITIONS OF TRIGONOMETRIC FUNCTIONS
Let  be an acute angle of a right triangle. The six trigonometric functions of  are
defined as follows:
hypotenuse
csc  =
opposite
opposite
sin  =
hypotenuse
cos  =
adjacent
hypotenus
sec  =
opposite
adjacent
tan  =
hypotenuse
adjacent
adjacen
cot  = opposit
The abbreviations opp, adj, and hyp are often used to represent the side lengths of the
right triangle. Note that the ratios in the second column are reciprocals of the ratios in the
first column:
csc  = 1
sin 
sec  =
1
cos 
cot  =
1
tan 
Your Notes
Example 1
Evaluate trigonometric functions
Evaluate the six trigonometric functions of the angle 0.
From the Pythagorean theorem, the length of the hypotenuse is
2
289_17_.
82  15 =
opp
hy

8
17
Sin  =
csc 
hy

opp
cos  =
adj
hy
tan  = opp
adj

sec  = hyp
15
17

adj
cot  = adj
8
15
opp
1
8


17
15
15
8
TRIGONOMETRIC VALUES FOR SPECIAL ANGLES
The table below gives the values of the six trigonometric functions for the angles 30°,
45°, and 60°. You can obtain these values from the triangles shown.
0
30
sin 
45
60
cos 
tan 
3
2
3
3
1
2
2
2
1
2
2
1
2
3
2

csc 
30
2
sec 
3
cot 
2 3
3
45
2
60
2 3
3
2
2
3
1
3
3
Your Notes
Example 2
Use a calculator to solve a right triangle
Solve ABC.
Solution
A and B are complemetary angels, so A = 90  _56_ = _34_
opp
sec hyp
56 =
ad
b
c =
sec 56
13
13
tan 56 = ad
tan 56 =
__13(tan 56)__ = b
_13_
_19.27_  b
 1 
 co 56   c


_23.25_  c
So, A = _34_, b  _19.27_, and c  _23.25_.
Checkpoint Complete the following exercises.
1.
Evaluate the six trigonometric functions of the angle 0.
3 13
sin 0 =
csc  =
13
cos 0 =
2 13
13
13
3
sec  =
13
2
tan  =
3
cot  =
2
2
3
2.
Solve ABC.
B = 53, a  13.24, and b  17.57
Your Notes
Example 3
Use an angle of elevation
Building Height You are measuring the height of your school building. You stand 25 feet
from the base of the school. The angle of elevation from a point on the ground to the top
of the school is 62°. Estimate the height of the school to the nearest foot.
Solution
1.
Draw a diagram that represents the situation.
2.
Write and solve an equation to find the height h.
tan _62_ = h
25
_25_(tan 62) = h
_47_  h
The height of the school is about _47_ feet.
Checkpoint Complete the following exercise.
3.
A kite makes an angle of 59° with the ground. If the string on the kite is 40 feet,
how far above the ground is the kite itself? Round to the nearest foot.
About 34 ft
Homework
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