Suppose we measure the lengths of the sides of this triangle. Here are some
realistic values:
Below is again the triangle from the above diagrams, except now the other
acute angle, B, is marked. Also marked are the sides that are adjacent and
opposite to angle B.
Here are the three trig functions for angle B:
sin(B) = opposite / hypotenuse = 6.00 cm / 7.21 cm = 0.8322
cos(B) = adjacent / hypotenuse = 4.00 cm / 7.21 cm = 0.5548
tan(B) = opposite / adjacent = 6.00 cm / 4.00 cm = 1.5000
If you look carefully you will notice that the sine of angle B is the same value
we calculated above for the cosine of angle A. You should also notice that the
cosine of angle B is equal to previous calculation for the sine of angle A.
This is because the opposite side for angle B is the adjacent side for angle A,
and because the adjacent side for angle B is the opposite side for angle A.
This is demonstrated in the following diagram:
We could summarize this relationship this way:
sin(A) = A's opposite / hypotenuse = 4.00 cm / 7.21 cm = 0.5548
cos(B) = B's adjacent / hypotenuse = 4.00 cm / 7.21 cm = 0.5548
cos(A) = A's adjacent / hypotenuse = 6.00 cm / 7.21 cm = 0.8322
sin(B) = B's opposite / hypotenuse = 6.00 cm / 7.21 cm = 0.8322
Here is an easy way to remember these relationships for trig functions and the
right triangle. Just write down this mnemonic:
SOH - CAH - TOA
It is pronounced "so - ka - toe - ah".
The SOH stands for "Sine of an angle is Opposite over Hypotenuse."
The CAH stands for "Cosine of an angle is Adjacent over Hypotenuse."
The TOA stands for "Tangent of an angle is Opposite over Adjacent."
Example. Indirect measurement. Trigonometry is used typically to
measure things that we cannot measure directly.
For example, to measure the height h of a flagpole, we could
measure a distance of, say, 100 feet from its base. From that point P,
we could then measure the angle required to sight the top . If that
angle (called the angle of elevation) turned out to be 37°, then
so that
h
= tan 37°
100
so that
so that
h = 100 × tan 37°.
From the Table, we find
tan 37° = .754
Therefore, on multiplying by 100,
h = 75.4 feet.
5
Example 1. In a right triangle, sin θ = 13. Sketch the triangle, place
the ratio numbers, and evaluate the remaining functions of θ.
To find the unknown side x, we have
x² + 5² = 13²
x² = 169 − 25 = 144
Therefore,
x=
= 12. (Topic 2: Radicals.)
We can now evaluate all six functions of θ:
5
csc θ = 5
13
sec θ = 12
13
5
cot θ = 5
sin θ = 13
12
cos θ = 13
12
tan θ = 12
Example 2. In a right triangle, sec θ = 4. Sketch the triangle, place
the ratio numbers, and evaluate the remaining functions of θ.
To say that sec θ = 4, is to say that the hypotenuse is
4
to the adjacent side in the ratio 4 : 1. (4 = 1)
To find the unknown side x, we have
x² + 1² = 4²
x² = 16 − 1 = 15.
Therefore,
x=
.
We can now evaluate all six functions of θ:
sin θ =
cos θ = 1
4
csc θ =
sec θ = 4
4
4
2
Problem 1. In a right triangle, cos θ = 5. Sketch the triangle and
evaluate sin θ.
Problem 2. cot θ =
. Sketch the triangle and evaluate csc θ.
Let's look at a simple right triangle, with one of the angles marked
The sides are named according to where the marked angle is.
.
The side called 'Hypotenuse' is always the long side across from the right angle.
The side called 'Opposite' is always across from, or 'opposite' to, the marked angle
The side called 'Adjacent' is the one next to, or 'adjacent' to, .
.
Now let's make the fractions from the named sides:
Here is the interesting part: each one of these fractions has its own name!
Here are the names of the fractions you can make from the sides of a right triangle:
Mathematicians being generally lazy sorts of people wanted an easy way to remember
these fraction names, and a quicker way to write them:
These can be remembered by using the first letter from each name, to make the acronym: