Evaluating trigonometric functions Without using a calculator,...

Question

Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

sec (5π/2)

Accepted Answer

Hello. Today, we're going to be solving for the exact value of the following trigonometric expression. So what we are given is secant of 23 pi divided by 2. Now one thing to note is that this angle of theta lies outside the region or beyond one rotation of the unit circle. Normally, we would like to work with an angle in standard form where it stays within the rotation of 1 unit circle between 0 and 2 pi. So what we can do is we can subtract the period of secant from 23 pitwo in order to get an equivalent angle that lies within one rotation of the unit circle. Now recall that the period of secant is 2pi. So what we're going to do is we're going to subtract 2pi from 23pitwo until we get an angle that lies between 0 and 2pi. So, we have 23pi divided by 2 minuteus 2pi. This difference will give us the value of 19 pi divided by 2. Next, we're going to take 19 pi divided by 2 and subtract another period which is 2 pi. 19 pi divided by 2 minus 2 pi will give us the value of 15pi divided by 2. Since this angle is still outside 2pi, we're going to take 15pi divided by 2 and subtract another increment of 2pi. Doing this will give us the value of 11 pi divided by 2. Then we're going to take 11 pi divided by 2 and subtract another 2 pi from this value, and this will give us the value of 7 pi divided by 2. And finally, 7pi divided by 2 minuteus 2pi will give us the value of 3pi divided by 2. If we take a look at the unit circle, 3 pi divided by 2 is the 270 degree mark between 0 and 2 pi. So 3 pi divided by 2 is an equivalent angle to 23 pi divided by 2, and we can rewrite the original expression as secant of 3pitwo. Now what we want to do is we want to solve for the exact value of secant at this angle. Well, recall that the terminal point at the angle 3pitwo is given to us as 0 comma negative 1. And recall that secant is the inverse of cosine. So secant is equal to the hypotenuse of a right triangle divided by the x value of the right triangle But at the angle 3 PI divided by 2 the x value at this angle is given to us as 0 and the hypotenuse will always have a measure of 1 since any radius along the unit circle is going to be 1. So we can write secant of 3pitwo is equal to 1 divided by the x value of the terminal point at 3pitwo, which is 0. So we have 1 divided by 0, which will give us an undefined value. So what this means is that the original expression secant of 23pi divided by 2 is going to give us an undefined value. And with that being said, the answer to this problem is going to be d. So I hope this video helps you in understanding how to approach this problem, and I'll go ahead and see you all in the next video.

Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

sec (5π/2)

Key Concepts

Trigonometric Functions

Unit Circle

Angle Measurement and Periodicity

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