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6.2 Right Angle Trigonometry
The angles of a right triangle determine the ratios of the sides. It does not matter the size of the
triangle, so long as the angles are the same, their ratios will be equivalent. If a triangle has three
sides, there exists six ways of representing their ratios. Let’s examine the six ratios along with
their corresponding trig function:
Sine: the ratio of the side opposite the given angle and the hypotenuse
Cosine: the ratio of the side adjacent the given angle and the hypotenuse
Tangent: the ratio of the side opposite the given angle and the side adjacent the given angle
Cotangent: the ratio of the side adjacent the given angle and side opposite the given angle
Secant: the ratio of the hypotenuse and the side adjacent the given angle
Cosecant: the ratio of the hypotenuse and side opposite the given angle
sin x =
opp.
hyp.
cos x =
adj.
hyp.
tan x =
opp.
adj.
cot x =
adj.
opp.
sec x =
hyp.
adj.
csc x =
hyp.
opp.
There is an acronym that is used to remember three of the above definitions: SOH-CAH-TOA
SOH: Sin x = Opp/Hyp
CAH: Cos x = Adj/Hyp
TOA: Tan x = Opp/Adj
The remaining three trig functions are reciprocals of the Sine, Cosine, and Tangent.
Example Based on the triangle given below, determine the values of the six trig
functions.
13
12
θ
5
sin θ = 12/13 cos θ = 5/13
tan θ = 12/5
cot θ = 5/12
sec θ = 13/5
Try the following:
10
6
18
23
θ
8
θ
11
Answers: 6/10; 8/10; 6/8; 8/6; 10/8; 10/6 (be sure to reduce fractions)
18/23; 11/23; 18/11; 11/18; 23/11; 23/18
csc θ = 13/12
You know from previous math courses that to find the missing side of a right triangle you
can use Pythagorean Theorem. Recall the formula to be a 2  b 2  c 2 where a and b are
the measures of the legs of the right triangle and c is the measure of the hypotenuse.
Let’s say we know two legs of a right triangle to be 3 and 4 and we want to know the
length of the hypotenuse. By applying the Pythagorean Theorem we see that
32  42  h 2 . Furthermore, we can state that 25 = h2 and that h = 5.
Example Given a point (2,1), draw a triangle with the positive x-axis and determine the
length of the hypotenuse. Determine the values of the six trig functions.
.(2,1)
.(2,1)1
θ
2
If we let a = 2 and b = 1, then a 2  b 2  c 2 becomes 4 + 1 = h2. Hence, the length of the
hypotenuse is 5 . Now we are ready to determine the values of the six trig functions.
sin θ = 1/ 5 cos θ = 2/ 5 tan θ = ½
cot θ = 2/1
sec θ =
5 /2 csc θ =
5 /1
Keep in mind the values should be reduced to the following:
sin θ =
5 /5 ; cos θ = 2 5 /5 ; tan θ = ½ ; cot θ = 2 ; sec θ =
5 /2 ; csc θ =
5
Try the following:
Given the point (5,2), draw a triangle with the positive x-axis, determine the length of the
hypotenuse, and the values of the six trig functions.
Answers: h =
29 ; sin θ = 2 29 /29 ; cos θ = 5 29 /29 ; tan θ = 2/5;
cot θ = 29 /5 ; sec θ = 29 /5 ; csc θ = 29 /2
With the use of the Pythagorean Theorem we can not only determine the length of the
hypotenuse, but we can also determine the length of any leg of a right triangle.
Example Let’s say we are given that cos  5/ 7 and we want to be able to determine
the values of the other trig functions. First, let’s draw a right triangle and label one acute
angle θ with its’ adjacent side length 5 and hypotenuse length 7. Let’s label the unknown
side b.
5
θ
7
b
We can apply the Pythagorean Theorem to determine the length of a missing side.
a 2  b2  c2
52 + b2 = 72
25 + b2 = 49
b2 = 24
b = 24 = 2 6
Now we’re ready to determine the values of the other five trig functions in reduced form.
sin θ = 2 6 /7 ; tan θ = 2 6 /5 ; cot θ = 5 6 /12 ; sec θ = 7/5 ; csc θ = 7 6 /12
Try the following:
Determine the values of the missing trig function given tan  6 /11.
Answers: h = 157 ; sin θ = 6 157 /157 ; cos θ = 11 157 /157;
cot θ = 11/6 ; sec θ = 157 /11 ; csc θ = 157 /6
There are two basic right triangles whose ratios assist us in determining the values of trig
functions. One of these is the 30-60-90 triangle. If you drew an equilateral triangle with
side lengths 2 and bisect one of the angles so that this line is perpendicular to the side
opposite the bisected angle, what happens to the measures of the angles and sides? Let’s
see:
60
2
2
60
60
2
30 30
2
2
60
60
1
1
How do you think we can determine the length of the altitude of this equilateral triangle?
First look at the equilateral triangle as two distinct right triangles. The length of 1 is the
length of one leg and the length of 2 is the length of the hypotenuse. We can apply the
Pythagorean Theorem to determine the length of the altitude, which is the length of the
other leg of the right triangle. The length turns out to be 3 . By applying the definitions
of the six trig functions we are able to determine the values for the angle measures 30
degrees and 60 degrees.
The other right triangle is 45-45-90 triangle. Just by knowing that the acute angles are
the same, this implies that the lengths of the legs opposite them are the same measure. So
If you let these legs be length 1, then by applying the Pythagorean Theorem we can
determine the length of the hypotenuse. The length of the hypotenuse turns out to be
2 . By applying the definitions of the six trig functions we able to determine the values
for the 45 degree angle.
θ
30
sin θ
½
cos θ
3 /2
45
60
2 /2
3 /2
2 /2
½
tan θ
1/ 3
1
3
By knowing these we can reciprocate the values to determine the values of the reciprocal
trig functions.
Notice: Examine the numerators under sine. Notice these numerators represent the
square roots of 1, 2, and 3, respectively. Now note the numerators under cosine. Notice
these numerators represent the square roots of 3, 2, and 1, respectively. Recall that
tangent is defined by sin x/ cos x, and that the numerators of sine and cosine are used to
yield the values under tangent.