**
THE SIX TRIGONOMETRIC RATIOS**

Example 1:

Given the following triangle, find the numeric values of the sine, cosine, and tangent ratios of angles

andA. Express your answers both as a fraction and a decimal.B

Ratios for Angle

:ARatios for Angle

:B

Please note that the location of the "side opposite" and the" side adjacent" changes with the location of the angle in the triangle. The numeric values for the trigonometric ratios of angle A are different from the numeric values of the trigonometric ratios of angle B.

Example 2:

Given , use appropriate trigometric ratios to find the exact value of the sine and tangent ratios of angle.

Knowing that , we also know the following:

(side adjacent to the angle ) is units in lengthadj

(hypotenuse) ishypunits in length2We can now find

(side opposite the angle ) by using the Pythagorean Theorem NOTE: Unless you are told otherwise,oppis always the hypotenuse of a right triangle andcandaare the legs.bLet's find

(side opposite the angle). Let's assume that it isoppin thebPythagorean Theorem!Now we'll use the

Square Root Propertyto find the length of:b = opp

Since we are working with a right triangle whose sides are never negative, we find

b = opp=Now we are ready to find the sine and cosine ratios given

= ,opp= , andadj=hyp.2Reminder:

Example 3:

Given a right triangle for which and , use the

ReciprocalandQuotient Identitiesto find the remaining four trigonometric ratios.Let's use the

Reciprocal Identitiesand to find the ratios for cosecant and secant.Then and .

Please note that there is a radical in the denominator of the cosecant. While this is usually a "no-no" in algebra, in trigonometry it is okay to leave the radical in the denominator.

If you do want to rationalize the denominator, you must multiply the numerator and denominator by as follows:

Let's use the

Quotient Identitiesand to find the ratios for tangent and cosecant.Then and .

Please note that there is a radical in the denominator of the cotangent. If you do want to rationalize the denominator, you must multiply the numerator and denominator by as follows:

Example 4:

Given a right triangle for which , find the ratios of the other five trigonometric ratios.

Knowing that , we can make the following picture:

We can find the length of the hypotenuse

by using thecPythagorean Theorem.Using the

Square Root Property, we can solve foras follows:cSince we are working with a right triangle whose sides are never negative, we can state that

.c = 5

Now we can find the other trigonometric ratios.