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Right Angle Trigonometry
I.
Basic Facts and Definitions
1.
Right angle – angle measuring 90
2.
Straight angle – angle measuring 180
3.
Acute angle – angle measuring between 0 and 90
4.
Complementary angles – two angles whose sum is 90
5.
Supplementary angles – two angles whose sum is 180
6.
Right triangle – triangle with a right angle
7.
Isosceles triangle – a triangle with exactly two sides equal
8.
Equilateral triangle – a triangle with all three sides equal
9.
The sum of the angles of a triangle is 180 .
10. In general, capital letters refer to angles while small letters refer
to the sides of a triangle. For example, side a is opposite angle A
.
II.
Right Triangle Facts and Examples
1.
Hypotenuse – the side opposite the right angle, side c .
2.
Pythagorean Theorem - a 2  b 2  c 2
3.
A and B are complementary.
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Page 1 of 12
III.
Examples:
1. In a right triangle, the hypotenuse is 10 inches and one side is 8
inches. What is the length of the other side?
Solution:
B
a 2  b2  c 2
82  b 2  10 2
64  b 2  100
a 8
b  36
2
c  10
b6
C
b?
A
2. In a right triangle ABC , if A  23 , what is the measure of B ?
Solution:
The two acute angles in a right triangle are
complementary.
A  B  90
23  B  90
B  67
IV.
B
C
Similar Triangles:
a. Two triangles are similar if the angles of one triangle are equal to
the corresponding angles of the other. In similar triangles, ratios of
corresponding sides are equal.
A
Conditions for Similar Triangles (
ABC ~ EGF )
1. Corresponding angles in similar triangles are
equal:
A  E
B  F
C  G
2. Ratios of corresponding sides are equal:
AC AB BC


EG EF FG
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Example 1:
AE  50 meters EF  22 m and AB  100 m
Find the length of side BC .
A
Notice that ABC and AEF are similar
since corresponding angles are equal. (There
is a right angle at both F and C , A is the
same in both triangles and B equals the
acute angle at E .)
50
F
E
22
50
C
B
Solution:
AE EF
50
22


so
AB BC
100 BC
By cross multiplying we get:
50( BC )  22(100)
Therefore BC  44 meters.
AE  50 meters EF  22 m and AB  100 m
Find the length of side BC .
Example 2:
All 45  45  90 triangles are similar to one
another. Two sides are of equal length and the
hypotenuse is 2 times the length of each of the
equal sides.
All 30  60  90 triangles are similar to one
another. The shortest side of length a is opposite
the smallest angle ( 30 ). The hypotenuse is twice
the length of the shortest side. The side opposite
the 60 has a length 3 times the shorter leg.
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Page 3 of 12
Problem: Find the lengths of the legs of a 30  60  90 triangle if the
hypotenuse is 8 meters.
Solution: 1) If 2a  8 , then a  4 meters and 2)
V.
3a  3(4)  4 3 meters.
The Six Trigonometric Ratios for Acute Angles
B
c
a
C
A
b
opposite
a

hypotenuse c
adjacent
b

cosine A  cos A 
hypotenuse c
opposite a

tangent A  tan A 
adjacent b
1
hypotenuse c


sin A
opposite
a
1
hypotenuse c


secant A  sec A 
cos A
adjacent
b
1
adjacent b


cotangent A  cot A 
tan A opposite a
sine A  sin A 
cosecant A  csc A 
TRIG TRICK: A good way to remember the trig ratios is to use the mnemonic
SOH CAH TOA!
SOH CAH TOA
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n
t
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p
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t
e
n
u
s
e
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a
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Page 4 of 12
Example 1:
Find the six trigonometric ratios for the acute angle B .
Solution:
opposite
b
 . Using the above definitions, the rest are:
hypotenuse c
a
b
c
c
a
cos B  , tan B  , csc B  , sec B  , cot B 
c
a
b
a
b
sin B 
Example 2:
In the right ABC , a  1 and b  3 . Determine the six trigonometric ratios for
B .
Solution:
Use Pythagorean Theorem:
c2  a2  b2
c 2  12  32
c 2  10 2
c   10
(Since length is positive, we will only use c  10 .)
sin B 
opp
3
3 10


hyp
10
10
cos B 
adj
1
10


hyp
10 10
tan B 
opp 3
 3
adj 1
csc B 
hyp
10

opp
3
sec B 
hyp
10

 10
adj
1
cot B 
adj 1

opp 3
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VI.
Special Cases
a. Trigonometric values of 30 and 60 (Use the 30  60  90
triangle from pg. 3.)
sin 30 
30
c2
a 3
60
60
b 1
1
2
3
2
1
3
tan 30 

3
3
cos 30 
csc 30 
3
2
1
cos 60 
2
3
tan 60 
 3
1
sin 60 
2
2
1
csc 60 
2

3
sec 30 
2
2 3

3
3
sec 60 
cot 30 
3
 3
1
cot 60 
2 3
3
2
2
1
1
3

3
3
b. Trigonometric values of 45 (Use the 45  45  90 triangle from
pg. 3.)
a 1
45
c 2
45
b 1
sin 45 
1
2

2
2
csc 45 
cos 45 
1
2

2
2
sec 45 
1
tan 45   1
1
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2
 2
1
2
 2
1
1
cot 45   1
1
Page 6 of 12
VII.
Converting Minutes and Seconds to Decimal Form
(Necessary for most calculator use in evaluating trig values)
1.
2.
To convert from seconds to a decimal part of a minute, divide the
number of seconds by 60.
To convert from minutes to a decimal part of a degree, divide the
number of minutes by 60.
Example 1: Convert 6447 to degrees using decimals.

Solution:
 47 
6447  64   
 60 
6447  64  .783  64.783
Example 2: Convert 151210 to degrees using decimals.
Solution:
151210  1512  10
 10 
151210  1512   
 60 
151210  1512.167


 12.167 
151210  15  

 60 
151210  15  .203  15.203
VIII. Right Triangle Trigonometry Problems
To Solve Right Triangle Problems:
There are six parts to any triangle; 3 sides and 3 angles. Each trig formula
(ex: sin A = a/c) contains three parts; one acute angle and two sides. If you
know values for two of the three parts then you can solve for the third
unknown part using the following method:
1. Draw a right triangle. Label the known parts with the given values and
indicate the unknown part(s) with letters.
2. To find an unknown part, choose a trig formula which involves the
unknown part and the two known parts.
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Page 7 of 12
Example 1: A right triangle has a  38 and B  61 . Find the length of side b .
Solution:
Which trig formulas involve an acute angle (B) and the
side opposite (b) and the side adjacent (a) to the angle?
Since both tangent and cotangent do, either could be
used to solve this problem. We will use tangent.
A
b
C
opp b
b
b
 so, tan 61 
or 1.8040 
.
adj a
38
38
Therefore b  68.6 .
tan B 
61
a  38
B
(Always check your answer by comparing size of angle and length of side; the
longer side is always opposite the larger angle.)
IX.
Angles of Elevation and Depression
Angle of depression
Angle of elevation
Example:
From a point 124 feet from the foot of a tower and on the same level, the angle of
elevation of the tower is 3620 . Find the height of the tower.
Solution:
h
 tan 3620  tan(36.333)
24
h  124(0.7355)
h
3620
124 ft.
h  91.2 ft.
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Practice Problems:
1. In right triangle ABC , if c  39 inches and b  36 inches, find a .
2. Find the length of side AC . Note: This problem and diagram corresponds
to finding the height of a street light pole ( AC ) if a 6 ft. man ( EF ) casts a
shadow ( BF ) of 15 ft. and the pole casts a shadow ( BC ) of 45 ft.
A
E
6
C
30
15
B
F
3. Evaluate:
F
13
12
G
5
E
a) sin E = _____________
b) tan E = _____________
c) cos F = _____________
d) sec F = _____________
4. Evaluate: (Draw reference 30  60  90 and 45  45  90 triangles)
a) sin 30 = _____________
b) tan 60 = _____________
c) sec 60 = _____________
d) tan 45 = _____________
e) csc 45 = _____________
f) cot 30 = _____________
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5. Evaluate:
B
10
8
C
A
a) tan A = _____________
b) csc B = _____________
c) cot A = _____________
d) sec B = _____________
6. Convert to decimal notation using a calculator:
a) 7606
b) 452713
Evaluate, using a calculator:
c) sin 5248
d) cot 3942
7. Label the sides and remaining angles of right triangle ABC , using A , B ,
a , b and c . If a  43 and A  37 , find the values of the remaining parts.
C
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8. Given right triangle ABC with b  0.622 and A  5140 , find c . Draw a
diagram.
9. From a cliff 140 feet above the shore line, an observer notes that the angle
of depression of a ship is 2130 . Find the distance from the ship to a point
on the shore directly below the observer.
cliff
ship
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Answers to Right Triangle Trigonometry:
1. a  15 inches (use Pythagorean Theorem)
2.
AC BC

EF BF
3. a) sin E 
AC 6

45 15
12
13
b) tan E 
AC  18
12
5
c) cos F 
4. (see part E of the handout)=
a) sin 30  1
b) tan 60  3
2
d) tan 45  1
e) csc 45  2
12
13
7. B  53
b) 45.454
b  57.06
13
12
c) sec 60  2
f) cot 30  3
5. b  6 (use Pythagorean Theorem)
8 4
5
3
a) tan A  
b) csc B 
c) cot A 
6 3
3
4
6. a) 76.1
d) sec F 
c) .7965
d) sec B 
5
4
d) 1.2045
c  71.45
B
c
a
C
b
8. c  1 (Use cos A 
A
b
c
or sec A  to solve for unknown c )
c
b
140
x
x  355.41 ft.
9. tan 2130 
(Angle of depression)
cliff
ship
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