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Review – Right Triangle
Trigonometry
Objectives
• Find trigonometric ratios using right triangles.
• Use trigonometric ratios to find angle
measures in right triangles.
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, 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.
Trigonometric Ratios
Only Apply to Right Triangles
In right triangles :
• The segment across from the right angle ( AC ) is labeled the hypotenuse
“Hyp.”.
A
Hyp.
Opp.
Angle of
Perspective
C
B
Adj.
• The “angle of perspective” determines how to label the sides.
• Segment opposite from the Angle of Perspective( AB ) is labeled “Opp.”
• Segment adjacent to (next to) the Angle of Perspective ( BC ) 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
A
AC  Hyp
Hyp.
Adj.
B
Opp.
C
BC  Opp
AB  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
A
Hyp
Adj
C
then
A
Opp
B
Opp
Hyp
C
B
Adj
C
AC  Hyp
AC  Hyp
BC  Opp
AB  Opp
AB  Adj
BC  Adj
The 3 Trigonometric Ratios
•
The 3 ratios are Sine, Cosine and Tangent
Opposite Side
Sine Ratio 
Hypotenuse
Adjacent Side
Co sin e Ratio 
Hypotenuse
Opposite Side
Tangent Ratio 
Adjacent Side
Trigonometric Ratios
To help you remember these
trigonometric
Sin A
= Opposite sideyou can
SOH
relationships,
use
Hypotenuse
the mnemonic device,
Cos A = Adjacent side
CAH
SOH-CAH-TOA,
where
the
Hypotenuse
first letter of each word of
Tan A = Opposite side
TOA
the trigonometric
ratios is
Adjacent side
represented in the correct
order.
A
c
b
C
a
B
Trigonometric Ratios

A
side adjacent 
B
side opposite 
C
sin  Oh
Hell
Soh
cos  Another Cah
Hour
tan 
Of
Toa
Algebra
SohCahToa
Soh
sin  opposite
hypotenuse
Cah
cos  adjacent
hypotenuse
Toa
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 keystroke
sequences
74 sin
sin 74
Sample calculator display
Rounded
Approximation
0.961262695
0.9613
0.275637355
0.2756
3.487414444
3.4874
ENTER
74
COS
74
COS
ENTER
74
TAN
74
TAN
ENTER
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.
Trig. ratio used to find
• Example 1: cos 42 x
15
side adj. to 42˚ angle.
x15cos 42
Multiple 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
9.5
x
sin 66
x10.40
Trig. ratio used to find
hyp. of a right triangle.
Multiple both sides by x.
Multiple 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
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 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.
Using a Calculator – Inverse
Trigonometric Ratios
a
1  a 
If sin x  , then x  sin  .
b
b
a
1  a 
If cos x  , then x  cos  .
b
b
a
1  a 
If tan x  , then x  tan  .
b
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.
KEYSTROKES: 2nd [COS] ( 13 ÷ 19 )
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.
14
Trig. ratio for ∠A.
sin

A

• Example:
23
 14 

1

1
(sin )sin A  sin  
 23 
 14 

1
Asin  
 23 
mA  37.50
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.
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:
•
(opposite)
(opposite)
7cm (adjacent)
•
•
34o
The appropriate ratio is sine, SOH
The appropriate ratio to use is Tangent, i.e.
TOA
Tangent Ratio 
•
•
•
•
•
5m (hypotenuse)
y
a
26o
Calculate the length of y.
Opposite Side
Adjacent Side
Tan 260 = a/7
a/7 = Tan 260
a = 7 x Tan 260
a = 7 x 0.4877
a =3.41cm
Sine Ratio 
•
•
•
•
•
Opposite Side
Hypotenuse
Sine 340 = y/5
y/5 = Sine 340
y = 5 x Sine 340
y = 5 x 0.559
y =2.80m
Your Turn:
• Calculate the length of side x
•
y (Opposite)
Hypotenuse
10.6m
670
x(Adjacent)
Hypotenuse
6.2m
• The appropriate ratio to use is
Cosine, i.e. CAH
Co s Ratio 
•
•
•
•
420
• The appropriate ratio is Sine,
i.e. SOH
Adjacent Side
,
Hypotenuse
Cos 670 = x/10.6
0.39 = x/10.6
x =10.6 X 0.39
x =4.14m
Calculate the length of y
Sine Ratio 
•
•
•
•
•
Opposite Side
Hypotenuse
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
To use Inverse Trigonometric Ratios to find an
interior angle, given two sides of a RAT
• To find angle a
32cm (Opposite)
25cm
a Adjacent
• The appropriate ratio to use
is Tangent, i.e. TOA
Tangent Ratio 
•
•
•
•
Opposite Side
Adjacent Side
Tan a = 32/25
Tan a = 1.28
a = Tan -1 1.28
a = 52.00
• To find the size of angle y
(Opposite)
30cm
50cm
Hypotenuse
y0
• 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 = 36.90
Your Turn:
• Find angle b
Hypotenuse
Opposite
12cm
6cm
b
• The appropriate ratio to
use is Sine, i.e. SOH
Opposite Side
Sine Ratio 
Hypotenuse
•
•
•
•
Sin b = 6/12
Sin b = 0.5
b = Sin-1 0.5
b = 300
• 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 = 50.280
Solving Trigonometric Equations
There are only three possibilities for the placement of the variable ‘x”.
Opp
Hyp
Sin X =
Sin  =
A
A
12 cm
25 cm
x
B
x
C
Sin X =12
25
Sin X
= 0.48
X = Sin1 (0.48)
X = 28.6854
B
x
Hyp
x
Sin 25
=
12
0.4226 = x
12
1
x = (12) (0.4226)
x = 5.04 cm
= Opp
x
A
x
12 cm
12 cm
25
Sin 
25
B
C
Sin 25
=
0.4226
=
1
x =
12
0.4226
x = 28.4 cm
12
x
12
x
C
EXAMPLES
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:
Your Turn:
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.
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.
Example 2
Definition of cosine ratio
Substitution
Simplify.
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.
Example 3
Solve the right triangle. Round side measures to
the nearest hundredth and angle measures to the
nearest degree.
Example 3
Step 1
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 3
Step 2
Find mB using complementary angles.
mA + mB = 90
30 + mB ≈ 90
mB ≈ 60
Definition of
complementary
angles
mA ≈ 30
Subtract 30 from
each side.
So, the measure of B is about 60.
Example 3
Step 3
Find AB by using the Pythagorean Theorem.
(AC)2 + (BC)2 = (AB)2
Pythagorean Theorem
72 + 42
= (AB)2
Substitution
65
= (AB)2
Simplify.
Take the positive
square root of each
side.
8.06
≈ AB
Use a calculator.
Example 3
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 4
• 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° =
opposite
adjacent
h
tan 59° =
45
Write the ratio
Substitute values
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.
Example 5
• 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. Use correct
adj. ratio
tan x° = 15.7 Substitute
59
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 tan 19° =
point in its descent.
Round your answer to
tan 19° =
the nearest tenth.
distance to runway
5 miles
opp. Use correct
adj. ratio
h
Substitute
5
values
5 Isolate h by
5 tan 19° = h
5
multiplying by 5.
 The shuttle’s altitude is
1.7 ≈ h Approximate using
about 1.7 miles.
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?
75˚
building
20
x
Using the 75˚ angle as a reference, we know hypotenuse and adjacent side.
adj
Use
hyp
cos
cos 75˚ =
x
20
About 5 ft.
20 (cos 75˚) = x
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˚ angle as a reference, we know opposite and adjacent side.
opp
Use
adj
tan
tan 62˚ =
x
18
18 (tan 62˚) = x
18 (1.8807) = x x ≈ 33.9
About 34 m
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˚ angle as a reference, we know hypotenuse and opposite side.
Use
opp
hyp
sin
sin 55˚ =
x
85
About 70 m
85 (sin 55˚) = x
85 (.8192) = x
x ≈ 69.6