**TRIGONOMETRIC RATIOS OF ANGLES OF ANY MAGNITUDE - PART 1
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu.**

**Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!**

**1. Find coterminal angles in degrees and radians of given angles and vice versa.**

**2. ****Memorize when trigonometric ratios have positive and negative values.
3. Memorize the values of trigonometric ratios of quadrantal angles.**

In the previous lecture, we limited our discussion to values of trigonometric ratios of acute positive angles. In this lesson, we will begin a discussion of trigonometric ratios of angles of any magnitude, positive and negative. However, we before we do so we must introduce the concept of

coterminal angles.

Coterminal Angles - see #1 through 5 in the "Examples" documentAngles with the same initial and terminal side are called

coterminal angles. For example, the angles with measure ofand are coterminal.0^{o}0and

We find the

coterminal anglesof an angle by adding to it integer multiples ofor360^{o}depending on the angle measure. We usually express angles coterminal to angle in general terms as follows:

+ 360^{o}or in radiansk+kwherekis any integerNOTE: The letter

is often used to indicate the integer, however, other letters can be used as well.kPlease examine the picture below! We can see that

added to360^{o}^{}results in an angle that has the same initial and terminal side, namely135^{o }. We can say that495^{o}is coterminal with495^{o}.135^{o}

YOU MUST MEMORIZE HOW TO FIND COTERMINAL ANGLES OF GIVEN ANGLES AND VICE VERSA!

Positive and Negative Values of Trigonometric Ratios - see #6 and 7in the "Examples" documentUsing the calculator, we quickly notice that the values of trigonometric ratios can be positive or negative when the magnitude of an angle is larger than

. For example,90^{o}is approximately equal tosin 200^{o}while0.342is approximately equal tosin (300^{o}).0.866So when are trigonometric ratios positive and when are they negative?

There is actually a pattern! It is summarized in the table below and it was developed in the documents found under the link "Point of Interest 2" in the

Learning Materials#4 in theMyOpenMathcourse.The pattern above must be memorized! Following is a handy memorization aid we call

"All Students Take Calculus". It lets us know in which quadrant the value of the trigonometric ratio of an angle is positive.

Values of Trigonometric Ratios of Quadrantal AnglesWe already mentioned the

quadrantal angles,0^{o},90^{o},180^{o}, and270^{o}and we were told that they are often used in trigonometry, physics, and engineering.360^{o}Of course, we can find the values of trigonometric ratios of

quadrantal angleswith a calculator. However, they appear often in mathematics, physics, and engineering, therefore, it is common practice to require trigonometry students to memorize them.In the table below find the values of the trigonometric ratios of the

quadrantal angles. If you are interested where these values come from, please read the documents found under the link "Point of Interest 3" in theLearning Materials#4 in theMyOpenMathcourse.

YOU MUST MEMORIZE THE VALUES OF ALL TRIGONOMETRIC RATIOS GIVEN THE FOLLOWING ANGLES!

Memorization Hint:Only memorize the values for sine and cosine. Please note the special pattern of 0's, and 1's. Use theReciprocalandQuotient Identitiesto find the values for the remaining trigonometric ratios.Some students who have had trigonometry in the past might have memorized the values in the table above using the

unit circle.The

of the points on the circle above are the values ofx-coordinatesof quadrantal angles and thecosineare the values ofy-coordinates. The values of thesinetangentcan be calculated using theQuotient Identity.

Some Calculator "Odditites"- see #8, 9, and 10in the "Examples" documentIn the table above, some of the values of trigonometric ratios of quadrantal angles are undefined. When we evaluate these ratios with a calculator, most will tell us "domain error." Some calculators show the infinity symbol .

The biggest issue arises when we want to evaluate the cotangent of some

quadrantal angles. When we use theReciprocal Identitymost calculator will incorrectly state "domain error" even though the value of the cotangent is actually 0.Therefore, it is best to use theQuotient Identitywhen evaluating the cotangent with the calculator!

Strategy for Finding EXACT Values of Trigonometric Ratios of Integer Multiples of Quadrantal Angles WITHOUT a Calculator- see #11 through 14in the "Examples" documentOf course, we can find these values with a calculator. However, they appear often in mathematics, physics, and engineering, therefore, it is common practice to require trigonometry students to find them without a calculator..

Assume that angle has a magnitude of

90^{o}kwhere180^{o}270^{o}360^{o}is an integer.kStep 1:

Find the angle coterminal with angle where

< <0^{o}36or0^{o}< < .0Step 2:

Use the fact that the values of trigonometric ratios of coterminal angles are identical.

Reminder: