In Exercises 63–82, use a sketch to find the exact value of each ...

Question

In Exercises 63–82, use a sketch to find the exact value of each expression.
_
sin (cos⁻¹ √2/2)

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the following trigonometric expression by sketching the expression we are given is the sign of the ar cosine of radical three divided by two. We have four answer choices. The first thing I'm going to do is draw my X and Y axis. So I have somewhere to sketch my angle. After that, I'm gonna take what is in the parentheses. So the a cosign of radical three divided by two and I'm gonna set it equal to an angle. I'm calling that angle alpha from here. Now that I have an equation, I'm gonna take the cosine of both sides which will give me that the cosine of alpha equals the square root of three divided by two. Since cosine is positive, we are gonna say that this angle lies in quadrant one. So in standard position, we'll have an angle, something like this. It might not be exactly this, but I'm gonna use it to draw my right triangle from there. I recall that the cosine of an angle is the X value divided by the R value. So in this example, X is radical three R is chip, we want to find what Y equals. So we're gonna use the Pythagorean theorem X squared plus Y squared equals R squared and find out what the Y value is. So X squared will have radical three in parentheses, squared plus Y squared equals R squared. So two squared, the squared of three squared is three plus Y squared equals four, subtracting three from both sides, we get that Y squared equals one, taking the square root of both sides. Y will equal one. Since we're in the first quadrant, we know Y is positive. So that will be a positive one. Labeling this all in my right triangle, alpha is the inside angle X is the horizontal. So that's radical three Y is the vertical. So that is one and our diagonal or our hypotenuse is R which is a value of two from there. I can use the knowledge that the sign of an angle is the Y value divided by the R value to find our final value. So why is one R is two? So this means that the sign of alpha is one half and since alpha represents the arc cosine of negative, I'm sorry, the a cosine of radical three divided by two, we know that one half is our exact value. And that's answer choice. A have a nice day.

In Exercises 63–82, use a sketch to find the exact value of each expression. _ sin (cos⁻¹ √2/2)

Key Concepts

Inverse Trigonometric Functions

Trigonometric Ratios

Unit Circle

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