8-2 Trigonometric Ratios
Trigonometric ratio—a ratio of two sides of a triangle.
Sine—of an angle is the ration of the length or the leg opposite the angle to the length of the
hypotenuse
Cosine—of an angle is the ratio of the length of the leg adjacent to the angle to the length of the
hypotenuse.
Adjacent means beside. In this case, it is the leg which helps to make the angle.
The hypotenuse is always the longest side of a triangle. So the denominator of a sine or cosine
ratio is always greater than the numerator.
The sines and cosines of acute angles are always between 0 and 1. WHY?
The sine of <A is congruent to the cosine of < C.
The cosine of <A is congruent to the sine of <C.
We can use sine or cosine to find the length of a leg or hypotenuse using the other side and the
angle measure.
How to find the sine on the TI-83, 84: Press the sin button, then put the angle measurement in.
Close the parenthesis and hit enter. We will go out 4 decimal places, so go out to the 5th
If you want to find the other side, you can set up another trig equation, or use the Pythagorean
Theorem to solve (remember, this theorem can always be your friend!!!).
Tangent—in a right triangle, the tangent of either acute angle is the ratio of the length of the
leg opposite the angle to the length of the leg adjacent to the angle.
Adjacent means beside. In this case, it is the leg which helps to make the angle.
Tan = length of the leg opposite <A
length of the leg adjacent to <A
The ratio is always expressed in the reduced form. 8/10 should be reduced to 4/5. You can
leave fractions as improper, as long as they are reduced. Example: 4/3 15/9 = 5/3.
Helpful hint: The tangent ratios of the two acute angles are reciprocals of each other. If one
ratio is 2/3, the ratio for the other angle is 3/2.
Tangent can be less than, equal to, or greater than one. It is the only trig function that can be
less than, equal to, or greater than one. !!
The tangent of an angle in a right triangle depends only on the measure of the angle, not the
size of the triangle.
How to find the tangent on the TI-83, 84: Press the tan button, then put the angle measurement
in. Close the parenthesis and hit enter.
SOH CAH TOA
Sine
Opposite
Hypotenuse
Cosine
Adjacent
Hypotenuse
Tangent
Opposite
Adjacent