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The Three Basic Trigonometric Ratios
Trigonometry is defined as the branch of mathematics that deals with relationsbetween sides and angles of triangles. The word trigonometry come from the Greekword trigonon "triangle" (from tri- "three" + gonia "angle") and word metron "ameasure".
Trigonometry has its foundations in two geometric concepts: The PythagoreanTheorem and Similar Triangles.
The Pythagorean Theorem
If you are given a right angle triangle, the square of the longest side(hypotenuse) is equal to the sum of the squares of the other two sides.
The Pythagorean Theorem is usually written as: 2 2 2c a b
Similar Triangles
If two triangles ABC and DEF whose corresponding angles are equal, then
AB BC ACDE EF DF
or , ,AB DE AB DE AC DFBC EF AC DF BC EF
The ratios of specific sides on the Right Angle Triangle have special names. The samenamed ratio on all other Right Angle Triangles all have the same value.
A
B C
x
y z
D
E F
x
y z
ca
b
Definition of the Three Basic Trigonometric Functions
Sine Cosine Tangent
sinOpposite O
Hypotenuse H cos
Adjacent AHypotenuse H
tanOpposite OAdjacent A
SOH CAH TOA
Note: The three orange words on the bottom row, SOH-CAH-TOA produce aword that can assist you in remembering the definitions of Sine, Cosine,and Tangent.
Note: The side opposite the angle θ is called Opposite, the side joining theangle with the hypotenuse is called the Adjacent. The θ angle cannot be the right angle.
Example
Solve for the length of x and y in the following diagram.
Solution:
Both triangles are similar ( AA
The corresponding sides have t
Thus, 40 14014040
3.5
x
x
HypotenuseOpposite
Adjacent
4
28
)
he following ratio:40 285 x
0
x5
The symbol represents similarity.
Example
Determine the value of x and y to one decimal place.
Solution:
Triangle ABC ADE AA
Let’s find the ratio of ABC to ADE
14 9.5 23.514 14
ADAB
Now,
15 23.514
23.5 2108.9
DE ADBC AB
xxx
, and
21 23.521 14
294 14 493.514 199.5
14.25
AE ADAC AB
y
yyy
Example
Evaluate the three trigonometric ratios of the angle as shown below.
Solution:
5sin 0.38461312cos 0.9231135tan 0.4167
12
OHAHOA
135
12
14 21
15
9.5x
y
A
B
C
D
E
Example
Solve (find the value of all sides and angles) the following triangle
Solution:
The missing angle is 90 28 62
Side “a”: 7.2sin 28
7.20.46947156
15.3
OH a
aa
Side “b”:
22 2
2 2
2
7.2
15.3 51.84
234.09 51.84182.2513.5
a b
b
b
b
Example
From the top of a building 80m high, the angle of depression to the base of a secondbuilding is 32. From the same point, the angle of elevation to the top of the secondbuilding is 10. Calculate the height of the second building.
28
7.2
a
b
Solution:
We need to determine the value of “a”, but to do this we first need to find thedistance RP.
In PST ,
80tan 32
80tan 32
128.0268
OA ST
ST
Now, in PQR ,
tan 10128.0268tan 10 128.0268
22.575
a
a
The height of the building is then 80 22.575 102.575 m
Example
A tree casts a shadow 3 times its height. Find the elevation of the sun
Solution:
Let represent the angle of elevation of the Sun.
Then
1
1tan3 3
1tan 303
hh
The angle of elevation is 30
By Alternate AngleTheorem , 32PST
Value of b, the heightof first building
3h
h
A
B C
The special angle
triangle 1, 2, 3
Example
In ABC , we have 90C , c=6 cm and a=4 cm. Find
a) the length of bb) the values of sin(A), cos(A), and tan(A)c) the values of sin(B), cos(B), and tan(B)
Solution:
By using the Pythagorean Theorem, we can determine b
2 2 2
2
2
6 4
36 16
36 1620
20
2 5
b
b
b
b
b
For Angle A, the adjacent side measures 2 5 and the opposite side measures 4
sin
4623
oppositeAhypotenuse
cos
2 565
3
adjacentAhypotenuse
tan
42 525
2 55
oppositeAadjacent
6 cm
A
C B
b
4 cm
Opposite
4
Adj
ace
nt
Hypotenuse62 5
A
C B
20 4 5
4 5
2 5
For Angle B, the adjacent side measures 4 and the opposite side measures 2 5
sin
2 565
3
oppositeB
hypotenuse
cos
4623
adjacentB
hypotenuse
tan
2 545
2
oppositeB
adjacent
Note: 2sin cos3
A B and 5cos sin3
A B
Example
A satellite is orbiting 172 miles above the earth’s surface. (See diagram) When it isdirectly above the point T, the angle S is found to be 73.5°. Find the radius of theearth (correct to three significant figures).
Solution:
Let the radius of the earth be r miles.
Then:
sin 73.5172
172 sin 73.5
sin 73.5 172 sin 73.5
172 sin 73.5 sin 73.5
1 sin 73.5
172 sin 73.51 sin 73.5
4004.8
rr
r r
r r
r r
r
r
r
Hence the radius of the earth is approximately 4000 miles (correct to threesignificant figures.)
Adjacent
4
Op
posi
te
Hypotenuse62 5
A
C B
Radius, r
r
T
73.5°
172 mi
C
S
(r + 112) mi
Example
An airplane is flying at an altitude of 5325 ft, directly above a straight stretch ofhighway along which a car and a bus are traveling towards each other on oppositesides of the highway. The vehicles are on opposite sides of the airplane, the car atan angle of depression of 41.6° and the bus at an angle of depression of 57.1° fromthe plane. How far apart in miles are the vehicles correct to 3 decimal places?
Solution:
Let the distance the car is to the left of the point directly below the plane be x ft, andthe distance the bus is to the right of the point directly below the plane be y ft
Then the car and the bus are (x + y) ft apart.
We will treat each Right triangle separately and use basic trigonometric ratios tosolve for x and y. Since we have the measure of the angle and its adjacent side, andwe wish to find the opposite side, we use the tangent ratio.
tan 41.653255325 tan 41.6
4727.75623165
x
x
tan 57.153255325 tan 57.1
8231.19677122
y
y
4727.75623165 8231.1967712212958.9530029
x yft
12958.95300295280
2.454347159642.454 mile
Therefore the vehicles are approximately 2.454 miles apart
41.6° 57.1°
5325 ft
yx
There are 5280 ftin a mile
Example
From a point 244.0 meters from the base of a building, the angle of elevation to thetop of the building is 15.8°. The angle of elevation from the same point to the tip of aantenna on top of this building is 19.5°. What is the height of the antenna to 2decimal places?
Solution:
Let the height of the building be h meters an
Then:
tan 15.824469.045040
h
h
and
tan 19.5244
236 tan 19.
86.40493
x h
x h
x h
Solving for x:
86.4049386.40493 69.0450417.35989117.36
x h
Hence the pole is approximately 17.36 m tall
15.8°19.5°
244 m
x
h
d the height of the an
5
0
tenna be x meters.
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