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 (7π/6)

Accepted Answer

Welcome back everyone. In this problem, we want to solve for the exact value of the trigonometric expression of the secant of 4 thirds of pi. For our answer choices, a is negative 2, b is 2 divided by the square root of 3, c is 2, and d is negative 2 divided by the square root of 3. Now, what do we know here? Well, we know that we're trying to find the secant of an angle And recall that the secant of an angle is equal to the reciprocal of the cosine of that angle. So that tells us that for our angle, the secant of 4 thirds of Pi, then, that would be equal to the reciprocal of the cosine of 4 thirds of Pi. What else do we know? Well, we know that we're working with an angle that's greater than pi and there's also another property for the cosine that tells us that if we have an angle that's xradians greater than pi, then it would be equivalent to the negative value of the cosine of x. In other words, if we can find out how much greater than pi is 4 thirds of pi in this case, then, we can use the reference angle and make it negative to figure out the exact value for our expression. So, let's go ahead and do that. So, basically, what we're doing is we want to find out how much greater than pi is 4 thirds of pi. So that tells us then that 4 thirds of pi equals pi plus x. That's our unknown value, how much it's greater by. And that means it's greater by 4 thirds of pi minus pi. Pi would be the same thing as 3 thirds of pi and 4 thirds of pi minus 3 thirds of pi equals a third of pi. So, therefore, know that we have that value. That tells us that the cosine of 4 thirds of pi equals the cosine of. Sorry. Pi plus a third of pi, which would be equal to the negative value of the cosine of a third of Pi based on our property. So now we need to think what is the cosine of a third of Pi? Well, recall that we know that value. We know that that value equals a half. So now, all we need to do is to substitute it into our expression, One divided by the cosine of 4 thirds of Pi to help us to solve. Because now that would mean that one divided by the cosine of worth thirds of PI would be equal to remember it's equal to the negative value. So that's one divided by the negative value of the cosine of a third of bi pi, which is a half. That's one divided by negative a half. So that would be the negative value of 1 multiplied by 2 divided by 1, which would have been equal to negative 2. Therefore, the exact value of our problem is negative 2, our trigonometric expression. And when we look back on our answer choices, that would have been answer choice a. Thanks a lot for watching everyone. I hope this video helped.

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

sec (7π/6)

Key Concepts

Trigonometric Functions

Unit Circle

Angle Measurement in Radians

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