Friday, August 10, 2012

Trigonometry: Evaluating Base Angles - The Hand Method

Notice: On 12/6/2012, I uploaded a YouTube video that corresponds to this blog post. You can find it by following this link.

Most Trigonometry students are encouraged (and sometimes required) to memorize the "base angle" evaluations for the three fundamental trigonometric functions. These five core angles (in degrees) are 0, 30, 45, 60, and 90. The three fundamental trig functions are sine, cosine, and tangent. So students spend hours attempting to memorize 15 values... or they don't memorize them and do poorly on the test.

Of course, a calculator will give you these values quite easily. Memorizing them is useful though because (1) these angles show up frequently in introductory trigonometric and calculus courses and (2) having the answers memorized saves time on exams and homework. But memorization is a chore and is difficult for most students. That's why I really love this "hand method" of remembering (or rather, determining) these values. I did not create this--I learned it while taking Pre-Calculus my junior year of high school, and I've been using it ever since!

Each finger on the left hand with the palm facing towards you (thumb up, pinkie down) represents one of the five base or core angles. Visualize your hand as the first quadrant--the pinkie is in the 0 degree position and the thumb is in the 90 degree position. Then it is easy to see that the ringer finger is the 30 degree finger, the middle is the 45 degree finger, and the pointer represents the 60 degree finger. Here is a visual with angles in degrees and radians. Knowing the fingers in both measurements will save you the time it would take to convert all radian measures into degrees!
Once you have the fingers identified, we can begin evaluating the base angles, those found in the first quadrant. We will learn later in the post how to evaluate angles that are constant multiples of these base angles.

First, let's look at evaluating the cosine or sine of one of these core angles. In my visual below, I've chosen to evaluate at 60 degrees. Step 1 is to identify the finger that corresponds to the angle. Here, the pointer finger represents 60 degrees. Step 2 is to lower that finger, just fold it down, as in the image below. To find the cosine of this angle, simply count the number of fingers still standing above the lowered finger. Then, take the square root of this number and divide by 2. This is the cosine of the angle. To find the sine of this angle, count the number of fingers still standing below the lowered finger, which is 3 in this case. Take the square root of this number and divide by 2.
Easy enough, right? Evaluating tangent is simple too. You only have to do one extra thing. After lowering the required finger, flip your hand over so that the pinkie is on top and the thumb is on bottom. The fingers still represent the same angle values. Avoid confusion by lowering your finger first, then flipping your hand. After flipping your hand over, to find the tangent of the angle, count the number of fingers above (which is 3 here) and the number of fingers below (which is 1). Take the square root of the number above and divide by the square root of the number below.
See, I told you it was just as easy! Only one extra step, flipping the hand over. You can quickly evaluate any of these base angles at any of the three fundamental trig functions using this method. I've compiled a few for you to try. Evaluate these trig functions at the given angles using the hand method. Don't convert between radians and degrees--practice learning the fingers in both units. After you've evaluated all of them, use a calculator to check your answers (your calculator will give decimal approximations; enter your answer to see if the decimals match). Make sure your calculator is in the right mode for the type of angle you are evaluating!
You can also see the answers quickly by clicking here.

It is also possible to evaluate the trig functions at any constant multiple of the five core angles using this method. There is just an extra step involved, and that is to determine the sign (positive or negative) the answer will have. You know that all angles can be found on the unit circle. The unit circle contains 360 degrees (or 2pi radians). Angles between 0 and 90 degrees are in quadrant I of the standard x,y-plane; angles between 90 and 180 are in quadrant II; angles between 180 and 270 are in quadrant III; and angles between 270 and 360 are in quadrant IV.

But there are angles greater than 360 and angles less than 0. We discussed coterminal angles in another post. Review How to Find a Coterminal Angle in that post.

The first step to evaluating a trig function at a multiple of one of the five core angles is to determine which core angle is multiplied. To do this, find the angle that is coterminal with the given angle. For example, given the angle -405, the angle coterminal with this is 315 degrees. This angle is a multiple of 45 degrees: 315 = 7 * 45. Find this by trial and error--divide the angle by each of the five base angles. Whichever base angle gives a whole quotient, that is the base angle you are looking for.

The second step is to evaluate the trig function at the base angle you found.

The third step is to determine the quadrant of the original angle. Since -405 is coterminal with 315, and since 315 lies in the fourth quadrant, I know that -405 lies in quadrant IV.

Now, you can determine the sign of your final answer. Refer to the diagram below:
This little phrase will help you remember which trig functions are positive in which quadrants: All Students Take Calculus. "All" stands for, well, all. That is, in quadrant I, all of the trig functions are positive. "Students" stands for sine; so only the sine function is positive in the second quadrant. "Take" stands for tangent; so only the tangent function is positive in the third quadrant. And "Calculus" stands for cosine; so only cosine is positive in the fourth quadrant.

In my example, -405 degrees lies in the fourth quadrant where only cosine is positive. If I'm evaluating cosine at this particular angle, the answer is positive; if I'm evaluating sine or tangent at this angle, the answer is negative.

Here are a few examples:

In each instance, the steps are the same. First, identify the base angle by finding the angle that is coterminal to the given angle. Divide the given angle by each of the core angles to determine which angle to evaluate. Then determine which quadrant the angle is in and use it to determine the sign of the final answer.

Practice this method of angle evaluation enough, and it will be very helpful to you in future classes. After trigonometry, you will probably use it in calculus and maybe other subjects as well. You can impress your teachers or classmates with the speed at which you can answer evaluation questions. Of course, it only works for multiples of the five core angles, but you will see that these angles appear frequently in homework and test questions because trig functions evaluated at these angles yield nice numbers.

Edit: This post was edited on 8/16/2012 to correct a mistake in the "Examples" photo.
Edit #2: This post was edited on 12/6/2012 to add the notice at the top.