**TRIGONOMETRIC RATIOS OF ACUTE ANGLES
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 values of trigonometric ratios of acute angles on the calculator in degrees and radians

2.Memorize the EXACT values of trigonometric ratios of the "special" angles30^{o}, 45^{o},and.60^{o}

Finding Values of Trigonometric Ratios using a Calculator - see #1 through 9in the "Examples" documentIn the previous lecture, we found values of the six trigonometric ratios of an angle using the sides of a right triangle. You may have noticed that we did not need actual angle measures to do this.

In this lecture, we will find the value of each trigonometric ratio when the angle measure is known. For example,

sinor54^{o}(radians).cos 1.3We will most often use a calculator to find values of trigonometric ratios given angles. It utilizes a program using concepts from advanced calculus.

Let's look at two different

Scientific Calculators, theTI-30X IISand theTI-30XS. If you have a different calculator YOU must know how to use it.

Both calculators have a

,sin, andcosbutton. They do NOT have a button fortan,csc, andsec!!! We MUST find the values of secant, cosecant, and cotangent on the calculator by using thecotReciprocaland/orQuotient Identities.NOTE: The , , and buttons on the calculators

do NOTfind the values of cosecant, secant, and cotangent !!! These buttons calculate the inverse sine, inverse cosine, and inverse tangent which will be discussed later.

A calculator must be told to evaluate radians or degrees. Using the wrong mode will give you false information!!!In the

TI-30X IIScalculator you must locate thebutton. It is in the first button row under the display window. Press it and observe the modesDRG,DEG, orRADin the display window. Use the arrow buttons (located under the display window) to highlight the desired mode. Then press theGRDbutton to exitEnter.In the lower right-hand corner of the display window you will then see what mode your calculator is in.In the

TI-30XScalculator you must locate thebutton. It is in the first button row under the display window. Press it and observe the modesmode,DEG, orRADin the display window. Use the arrow buttons (located under the display window) to highlight the desired mode. Then press theGRDbutton. Finally, press theenterclearbutton to exitmode.In the upper right-hand corner of the display window you will then see what mode your calculator is in.

EXACT Values of Trigonometric Ratios of "Special" Angles- see #10 through 14in the "Examples" documentMathematicians, physicists, and engineers are obsessed with using EXACT values as much as possible. While most values of trigonometric ratios cannot be expressed as EXACT numbers, the values of trigonometric ratios of the "special" angles

,30^{o}, and45^{o}can. These EXACT numbers are used almost exclusively in application problems.60^{o}Since these values appear so often, it is common practice to force trigonometry students to memorize the values of trigonometric ratios of special angles instead of using the calculator to find them.

Please note that is considered an EXACT number, while its decimal approximation

is not considered to be exact.1.732In the table below find the EXACT values of the trigonometric ratios of these "special" angles. If you are interested where these values came from, please read the documents found under the link "Point of Interest 1" in the

Learning Materials#4 in theMyOpenMathcourse.

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

Memorization Hint:Only memorize the values for sine, cosine, and tangent. Then use theReciprocaland/orQuotient Identitiesto find the remaining values.Please observe the following decimal approximations for the EXACT values of trigonometric ratios of the "special" angles

,30^{o}, and45^{o}.60^{o}

Some students who have had trigonometry in the past might have memorized the values of the trigonometric ratios of the "special" angles using the

unit circle.The

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