Rotational Trigonometry:Trig at a PointDr. ShildneckFall, 2014

The Trigonometric Ratios

We all know “Soh-Cah-Toa.” The three basic trigonometric functions are the sine, cosine and tangent. But, there are three more trig functions called the “reciprocal functions.”

Recall from before,Soh-Cah-Toa gives us

The reciprocal functions, named the cotangent, secant, and cosecant, are

Example 1

Given the following triangle, find the values of all six trigonometric ratios for θ.

θ

13 5

12

Trigonometry of Any AngleLet θ be any angle in standard position and a point P(x, y) be a point on the

terminal side of θ. Let r be the non-zero distance from the origin to P.

x

y r

How could you use this informationto find the distance r?

The PYTHAGOREAN THEOREM,of course!

So… 2 2 2r x y

2 2r x y P(x,y)

Note: Since x2 and y2 are always positive, the square root of their sum is always positive. Therefore, r is always positive.

Trigonometry of Any AngleFurthermore, we can look at the right triangle created by the terminal side (r),

And the x-axis to determine the values of the six trigonometric functions of θ.

x

y r

Now, for that angle of reference, then:

sin csc

cos sec

tan cot

y rr yx rr xy xx y

P(x,y)

y is the _________________

x is the _________________

r is the _________________

opposite

adjacent

hypotenuse

(hyp)

(opp)

(adj)

Example 2Let the point (-4, 5) be a point on the terminal side of an angle in standard position. Find the values of all six trigonometric functions for the angle.

Example 3Let P be a point on the terminal side of standard of θ in Quadrant IV, such that .

Find the values of all six trigonometric functions for the angle.

4sin

7

The Trigonometric Values of the Quadrantal AnglesRecall that the quadrantal angles are those that are made up of the positive and negative axes.

2

The angles are multiples of 90o or radians.

To determine the trigonometric values of any of these angles, use the definition of the trig functions at specific points.

1) Choose ANY point on the axis that makes the terminal side of the angle.2) Determine the distance to that point (r).3) Plug in the correct values for x, y, and/or r in the formulas we discovered earlier.

Example 4Find the values of each of the following.

sin90Let’s call this pointP(0, 5)

1. What is r (the distance from (0,0) to P)?

2. What the x-value of the point?

3. What the y-value of the point?

4. How do you find the sine? sin90 yr

55

1

5

5

0

We could pick ANY number onthis part of the axis.

Example 5Find the values of each of the following.

cos270

Example 6Find the values of each of the following.

tan180

Example 7Find the values of each of the following.

sin

Example 8Find the values of each of the following.

tan2

ASSIGNMENT

Page 251 #1-8, 33-40