Download Section 10.6: Right Triangle Trigonometry §1 Ratios The word

Section 10.6: Right Triangle Trigonometry
§1 Ratios
The word trigonometry means the measurement of triangles. Right triangle forms the basis of trigonometry.
We’ve already gone over one important property or right triangles, the Pythagorean Theorem. Another
important property we’ve learned about triangles in general is that similar triangles have corresponding sides. It
turns out that in right triangles, the lengths of the sides actually depend on the measure of the angles in the
triangle! This is a pretty neat property, because it helps us find the lengths of the sides and the measures of the
angles using ratios.
Let’s first go over some terminology. In relation to the angle, we need to be able to first identify the side that is
OPPOSITE the angle, and the side that is ADJACENT to the angle. The hypotenuse is always the side that is
longest and opposite the right angle.
There are three trigonometric ratios – sine (abbreviated as sin), cosine (abbreviated as cos) and tangent
(abbreviated as tan). Hence if we refer to the picture above, we can set up the following three ratios:
Note that in any of the ratios, there are three quantities – two sides and an angle. As long as we are given two of
these quantities, we can always find the third one.
We can use a helpful mnemonic device to help memorize this. It is SOHCAHTOA. It means that the sin of the
angle is equal to the opposite side divided by the hypotenuse. The cosine of the angle is equal to the adjacent
side divided by the hypotenuse. The tangent of the angle is equal to the opposite side divided by the adjacent
side.
Look at the following triangle.
6
8
6
, cos A 
and tan A  . Of course we
8
10
8
3
4
3
always reduce fractions so we get that sin A  , cos A  and tan A  .
4
5
4
We can set up the three trig ratios for angle A. We get that sin A 
Now let’s actually try to find the numeric values. Look at the following triangle.
Note that we are given the measure of angle A and the length of side b. How can we use trig ratios to find the
lengths of side a and of the hypotenuse?
We can see that in relation to angle A, tan A is equal to the opposite side divided by the adjacent side, or
a
a
. Hence we set up a trig ratio to find side a as tan 61 
. There is only one unknown here. We can
b
10
use a calculator to find tan 61. We can solve for a to get that a  10  tan 61 . The answer is a  18.04 . Now we
tan A 
can use another trig function to find the hypotenuse. Once again, in relation to angle A, cos A is equal to the
adjacent side divided by the hypotenuse. Hence cos A 
side b is, we get that cos 61 
c
b
. Since we know what angle A is and we know what
c
10
. From here, we can do a little bit of algebra to solve for c to get that
c
10
. The answer is 20.63.
cos 61
The best way to approach these problems is to first identify what parts of the triangle you are given and what
you are being asked to find. You will always need to set up a trig ratio, so make sure you remember SOHCAHTOA
and how to determine which sides and angles to use.
PRACTICE
1) Find the length of sides b and c in the following figure:
§2 Finding Angles
Let’s say we are given the lengths of two sides of a right triangle. We can work backwards to find the angles! For
example, look at the following figure:
We are given the length of sides a and b. How can find the measure of angles A and B?
We use trig ratios of course! In relation to side A, we are given the opposite side and the adjacent side. Which
trig ratio should be used? We would use the tangent of angle A. Hence we get that tan A 
tan A 
a
, so in this case
b
10
. From here, we use inverse trigonometric key to find the measure of angle A. Since we know that
24
10/24 = 5/12, we can find the angle by finding the inverse trigonometric value of 5/12. We have to use a
calculator for this. If you examine your calculator closely, you should find the sin button. Right above it you
should see the words sin 1 , either in yellow or some other color. This means that it is the secondary function
for the button labeled sin. Hence we would first need to hit the 2nd key and then the sin key. The answer you
should get is 24.62 degrees. Make sure you get comfortable using your calculators! It does take some practice
getting used to the trig functions and the inverse trig functions.
PRACTICE
2) Find the missing lengths in the following figure: