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Solving a Right Triangle
A trigonometric ratio is a ratio of the lengths of two sides of 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 parts.
You can solve a right triangle if you know either of the following:
 Two side lengths
 One side length and one acute angle measure
As you learned in Lesson 9.5, you can use the side lengths of a right triangle
to find trig ratios for the acute angles of the triangle.
As you will see in this lesson, once you know the sine, the cosine, or the
tangent of an acute angle, you can use a calculator to find the measure of
the angle.
Solving a Right Triangle
A trigonometric ratio is a ratio of the lengths of two sides of 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 parts.
You can solve a right triangle if you know either of the following:
 Two side lengths
 One side length and one acute angle measure
In general, for an acute angle A:
if sin A = x, then sine -1 x = mA.
if cos A = y, then cos -1 y = mA.
if tan A = z, then tan -1 z = mA.
Solving a Right Triangle
SOLVE the right triangle. Round decimals to the nearest tenth.
R
c
2
SOLUTION: Step 1 find the missing side.
T
S
3
Begin by using the Pythagorean Theorem to find the length of the
hypotenuse.
(hypotenuse)2 = (leg)2 + (leg)2
c2 = 32 + 22
c2 = 9 + 4
c2 = 13
c = 13  3.6
R
c
2
hyp.
opp.
T
3 adj.
S
Solving a Right Triangle
SOLVE the right triangle. Round decimals to the nearest tenth.
R
SOLUTION: Step 2
find the first
missing .
2
T
Then use a calculator to find the measure of S:
 For the TI-30xa classroom calculator:
2 
3
(
(
this just
appears
(
2 
R
c
3
(
TAN
S
3
TAN
2nd
 For the TI-30xs recommended calculator:
2nd
c  3.6
Enter
2
opp.
T
mS  33.7
hyp.
3 adj.
S
Solving a Right Triangle
SOLVE the right triangle. Round decimals to the nearest tenth.
R
c
SOLUTION: Step 3 find the last
missing .
2
T
3
S
Finally, because R and S are complements, you can write:
mR = 90 - mS  90 - 33.7 = 56.3
 (conclusion)
 The side lengths of the triangle are 2, 3, and √13, or about 3.6.
 The angle measurements are 90, about 33.7, and about 56.3.
Solving a Right Triangle
Solve the right triangle. Round decimals to the nearest tenth.
SOLUTION: Step 1 Use trigonometric ratios to find the values of g and h.
G
sin H =
opp.
hyp.
opp.
=
sin 25º =
hyp.
0.4226º =
h
13
13(0.4226º)  h
5.5  h
cos H =
h
13
cos 25º =
adj.
hyp.
adj.
=
hyp.
0.9063º =
13(0.9063º)
11.8  g
g
13
g
13
g
13
h
25º
J
g
H
Solving a Right Triangle
Solve the right triangle. Round decimals to the nearest tenth.
SOLUTION: Step 2 Use the Triangle Sum Theorem to find mG.
G
13
Because H and G are complements, you can write:
mG = 90 - mH = 90 - 25 = 65
h
25º
J
g
 (conclusion)
 The side lengths of the triangle are about 5.5, about 11.8, and 13.
 The angle measurements are 90, 65, and 25.
H
Trigonometric Ratios for 45º
Find the sine, the cosine, and the tangent of 45º.
SOLUTION
Because all such triangles are similar, you can make calculations simple by
choosing 1 as the length of each leg.
From the 45º-45º-90º Triangle Theorem, it follows that the length of the
hypotenuse is 2 .
sin 45º =
opp. =
hyp.
1
=
2
2
2
 0.7071
2
1
cos 45º =
adj. =
hyp.
1
=
2
2
2
 0.7071
45º
1
tan 45º =
1
opp.
=
1
adj.
=1
hyp.
Finding Trigonometric Ratios
The sine or cosine of an acute angle is always less than 1.
The reason is that these trigonometric ratios involve the ratio of a leg of a right
triangle to the hypotenuse.
The length of a leg of a right triangle is always less than the length of its
hypotenuse, so the ratio of these lengths is always less than one.
Because the tangent of an acute angle involves the ratio of one leg to another leg,
the tangent of an angle can be less than 1, equal to 1, or greater than 1.
Finding Trigonometric Ratios
The sine or cosine of an acute angle is always less than 1.
The reason is that these trigonometric ratios involve the ratio of a leg of a right
triangle to the hypotenuse.
The length of a leg of a right triangle is always less than the length of its
hypotenuse, so the ratio of these lengths is always less than one.
Because the tangent of an acute angle involves the ratio of one leg to another leg,
the tangent of an angle can be less than 1, equal to 1, or greater than 1.
TRIGONOMETRIC IDENTITIES
A trigonometric identity is an equation involving trigonometric ratios that is
true for all acute angles. The following are two examples of identities:
B
(sin A) 2 + (cos A) 2 = 1
tan A =
sin A
cos A
c
a
C
A
b
Using Trigonometric Ratios in Real Life
Suppose you stand and look up at a point in the distance, such as the top of
the tree. The angle that your line of sight makes with a line drawn horizontally
is called the angle of elevation.
Indirect Measurement
FORESTRY You are measuring the height of a Sitka spruce tree in Alaska.
You stand 45 feet from the base of a tree. You measure the angle of elevation
from a point on the ground to 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.
tan 59° =
opposite
adjacent
Write ratio.
tan 59° =
opposite
h
adjacent
45
Substitute.
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.
Estimating a Distance
ESCALATORS The escalator at the Wilshire/Vermont Metro Rail Station in
Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels
on the escalator stairs, you can write a trigonometric ratio that involves the
hypotenuse and the known leg length of 76 feet.
sin 30° =
opposite
hypotenuse
short leg = 76
Write ratio for sine of 30°.
Whoa! This is a 30-60-90 triangle!!!
d
hypotenuse = 76(2)
What is d?
76 ft
30°
d = 152
Simplify.
A person travels 152 feet on the escalator stairs.
Estimating a Distance
ESCALATORS The escalator at the Wilshire/Vermont Metro Rail Station in
Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels
on the escalator stairs, you can write a trigonometric ratio that involves the
hypotenuse and the known leg length of 76 feet.
sin 30° =
opposite
hypotenuse
Write ratio for sine of 30°.
sin 30° =
76opposite
hypotenuse
d
Substitute.
d sin 30° = 76
76
d=
sin 30°
76
d=
0.5
d = 152
Multiply each side by d.
d
76 ft
Divide each side by sin 30°.
30°
Substitute 0.5 for sin 30°.
Simplify.
A person travels 152 feet on the escalator stairs.