Evaluate each expression without using a calculator.
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Question

Evaluate each expression without using a calculator.

tan (arccos 3/4)

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression without using a calculator. We are given the tangent of the arc cosine of 3/11. Our answer choices are a three radical 11 divided by seven B, four radical three divided by seven C, three radical seven, divided by 11 D four radical seven divided by three. So first, I'm rewriting my expression. So I have the tangent of the arc cosine of 3/11. And now I want to take this apart a little bit. I'm gonna take what's in my parenthesis, the arc cosine of 3/11. I'm gonna rewrite that as the inverse cosine of 3/11. And I'm going to set that equal to some unknown angle. I'm calling it theta. This means I can rewrite this as the cosine of theta equals 3/11 from there. I'm gonna use a little right triangle trig to find the other side to help me find the tangent. So I draw myself a little right triangle. I'm putting theta in the bottom right corner and I recall that cosine is the adjacent sides measure divided by the hypotenuse measure. So the bottom of my rate triangle has a measure of three hypotenuse has a measure of 11. And I'm going to call my unknown side A which is a cross from theta. Now, how does this help me? Well, if I say that the arc cosine of 3/11 equals theta, I now could say that I need to find the tangent of theta. And recalling that tangent when I'm using right triangle, trigo trigonometry is the opposite divided by the adjacent side. I now need to find the opposite side and I will have my answer. So going back to my right triangle, I'm gonna use the Pythagorean theorem which states that A squared plus B squared equals C squared. So the sum of the squares of the legs is equal to the square of the hypotenuse A squared is my unknown plus three squared equals 11 squared. So A squared plus nine will equal 121. Subtracting nine from both sides. A squared equals 112. I'll take the square to both sides. A will equal plus or minus the square root of 112. And that will reduce to plus or minus four radical seven. Thinking about my inverse cosine, I know that this must be in quadrant one or quadrant two. But since 3/11 is greater than zero, it can't be in quadrant two. It must be in quadrant one, meaning this value will be positive. So it'll be a positive four radical seven for a, again, our A is our opposite side. So if I go back to trying to find the tangent of theta, the opposite side, which we just said has a measure of four radical seven divided by the adjacent side, which has a measure of three. Well, give us our answer. Our answer here is four radical seven divided by three. And that's answer choice. D have a nice day.

Evaluate each expression without using a calculator.

tan (arccos 3/4)

Key Concepts

Inverse Trigonometric Functions

Trigonometric Ratios

Pythagorean Theorem

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