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Math 113
Right Triangle Trigonometry Handout
B
(length of hypotenuse) - c
a - (length of side
opposite θ )
θ
A
C
b
(length of side adjacent to θ )
Pythagorean’s Theorem: for triangles with a right angle ( side 2 + side 2 = hypotenuse 2 )
a 2 + b2 = c2
The definitions of the six trigonometric functions of the acute angle θ are as follows:
sin θ =
a length of side opposite θ
=
c
length of hypotenuse
csc θ =
c
length of hypotenuse
=
a length of side opposite θ
cos θ =
b length of side adjacent to θ
=
c
length of hypotenuse
sec θ =
c
length of hypotenuse
=
b length of side adjacent to θ
tan θ =
length of side opposite θ
a
=
b length of side adjacent to θ
cot θ =
b length of side adjacent to θ
=
a
length of side opposite θ
Example: Find the value of each of the six trigonometric functions of θ in the figure below.
B
a=3
c
θ
A
b=4
C
Solution: In order to evaluate all six trigonometric functions,
we need to know the length of all sides of the triangle. Since
the lengths for sides a and b are given, we can use
Pythagorean’s Theorem, c 2 = a 2 + b 2 , to find the length of
side c.
c 2 = a 2 + b 2 = 32 + 42 = 9 + 16 = 25
c 2 = 25
c = 25 = 5
So, we know that a = 3, b = 4, and c = 5. We can now use the above definitions of the trigonometric
functions to evaluate them for the angle θ .
sin θ =
a
cos θ =
b
tan θ =
a
c
c
b
=
=
=
3
hypotenuse 5
opposite
adjacent
hypotenuse
opposite
adjacent
=
=
c
csc θ =
=
4
5
sec θ =
3
4
cot θ =
a
c
b
b
a
=
=
=
hypotenuse
opposite
hypotenuse
adjacent
adjacent
opposite
=
=
=
5
3
5
4
4
3
Two special Right­triangles 1. The “ 45D − 45D − 90D ” right triangle.
We can construct a right triangle with a 45D angle. The triangle has two 45D angles. Therefore, the
triangle is isosceles – that is, it has two sides of the same length. Assume that each leg of the triangle
has length 1. We can find the length of the hypotenuse using Pythagorean’s Theorem.
(length of hypotenuse) 2 = 12 + 12
(length of hypotenuse) 2 = 1 + 1
2
(length of hypotenuse) 2 = 2
1
(length of hypotenuse) = 2
45D
1
Now that we know the lengths of the sides of this right triangle, we can find the six trigonometric
function values for the angle θ = 45D .
sin 45D =
cos 45D =
tan 45D =
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
=
=
1
csc 45D =
2
=
1
sec 45D =
2
1
1
cot 45D =
hypotenuse
opposite
hypotenuse
adjacent
adjacent
opposite
=
1
1
=
=
2
1
2
1
2. The “ 30D − 60D − 90D ” right triangle.
There are two other angles that occur frequently in trigonometry, 30D and 60D . We can find the values
of the trigonometric functions for these angles using a right triangle. To form this right triangle, draw an
equilateral triangle-that is a triangle with all sides the same length. Assume that each side has a length
equal to 2. If we draw a line right down the middle of this triangle bisecting the top angle and dividing
the base into two equal parts, then we will have a right triangle. See the figure below.
We can find the length of the missing side, a,
using Pythagorean’s Theorem.
22 = 12 + a 2
30D
2
2
4 = 1 + a2
a
4 − 1 = a2
60D
3 = a2
60D
1
3=a
1
So, our triangle has sides with lengths, 1, 2, and 3 . Using this right triangle we can find the function values
for both 30D and 60D . Fill in the blanks below.
opposite
sin 30D =
hypotenuse
30D
3
2
cos 30D =
adjacent
hypotenuse
= __________
= __________
csc 30D =
sec 30D =
hypotenuse
opposite
hypotenuse
adjacent
= __________
= __________
60D
1
tan 30D =
sin 60D =
cos 60D =
tan 60D =
opposite
adjacent
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
= __________
cot 30D =
adjacent
opposite
= __________
csc 60D =
hypotenuse
= __________
sec 60D =
hypotenuse
cot 60D =
adjacent
= __________
opposite
adjacent
opposite
= __________
= __________
= __________
= __________