**TRIGONOMETRIC RATIOS OF ANGLES OF ANY MAGNITUDE - PART 2
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 reference angles in degrees and radians of given angles.**

We are going to continue our discussion of trigonometric ratios of angles of any magnitude. Besides the

quadrantal anglesand their integer multiples, there is is one other set of angles that appear often in mathematics, physics, and engineering and those are the integer multiples of the "special" angles,30^{o}, and45^{o}.60^{o}It is common practice to teach trigonometry students how to find the EXACT values of the trigonometric ratios of integer multiples of special angles without using the calculator.

The following integer multiples occur frequently.

You must memorize them. You must also memorize their radian equivalents.Often used integer multiples of

30^{o}

and5(30^{o}) = 150^{o}and7(30^{o}) = 210^{o}11(30^{o}) = 330^{o}NOTE:

Often used integer multiples of

45^{o}

and3(45^{o}) = 135^{o}and5(45^{o}) = 225^{o}7(45^{o}) = 315^{o}NOTE:

Often used integer multiples of

60^{o}

and2(60^{o}) = 120^{o}and4(60^{o}) = 240^{o}5(60^{o}) = 300^{o}Please note that

,120^{o}, and240^{o}are also integer multiples of300^{o}. However, any time30^{o}divides evenly into an angle measure, it is considered an integer multiple of60^{o}.60^{o}NOTE:

Reference Angles- see #1, 2, and 3in the "Examples" documentBefore we find the EXACT values of their trigonometric ratios without using the calculator, a discussion of

reference anglesis necessary.

Reference anglesare positive acute (less than90) angles that lie between the terminal side of some angle and the horizontal axis in a coordinate system.^{o}

How to find a Reference Anglefor some Angle :Assume that is positive and between

and0^{o }(or360^{o}and ).01. If is a first-quadrant angle, then the

reference angleequals.2. If is a second-quadrant angle, then the

reference angleequalsor .180^{o}3. If is a third-quadrant angle, then the

reference angleequalsor .180^{o}4. If is a fourth-quadrant angle, then the

reference angleequalsor .360^{o}

Four Characteristics of Reference Angles:The follwoing characteristics of

reference angleswill be stated without a formal proof. They should be evident from the discussion found under the link "Point of Interest 2" in theLearning Materials#4 in theMyOpenMathcourse.1. The trigonometric ratio of an angle and that of its

reference anglehave the same ABSOLUTE value.2. Negative angles have the same

reference anglesas their positive counterparts.3. Coterminal angles have the same

reference angle.4. Quadrantal angles do not have

reference angles.

Strategy for Finding EXACT Values of Trigonometric Ratios of Integer Multiples of Special Angles WITHOUT a Calculator- see #4 through 14in the "Examples" documentAssume that angle is an integer multiple of a "special" angle.

Step 1:

Find the

reference angleof the given angle . Might have to use one or more of the four characteristics ofreference angles.Step 2:

Find the value of the given trigonometric ratio of the

reference anglefrom Step 1.Step 3:

Find the value of the trigonometric ratio of the given angle with the help of the following:

- the value found in Step 2
- one or more of the four characteristics of
reference angles- "All Students Take Calculus" on the given trigonometric ratio of angle
NOTE: Some students who have had trigonometry in the past might have memorized the following unit circle.

It is NOT a good idea to memorize the coordinates of the points in the picture above!UsingReference Anglesis necessary for later concepts as well and anyone who refuses to learn it now will be in trouble later on! So please, do not use your unit circle coordinates.