Evaluate each expression without using a calculator.
sin...

Question

Evaluate each expression without using a calculator.

sin (2 cos⁻¹ (1/5))

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the expression without using a calculator. Our expression is the sign of two inverse cosine of 3/11. Our answer choices are answer choice. A 24 square root seven divided by 121 answer choice B negative three square root seven divided by 11 answer choice C negative 24 divided by 121 and answer choice D 103 divided by 121. All right. So we've got the sign of two multiplied by something and that something is the inverse cosine of three divided by 11. And we recall from previous lessons that when we have an inverse trigonometric function that's really equal to an angle. So another way to think about that would be what is the angle when we have the inverse or excuse me, the cosine of that angle will be equal to three divided by 11. So we can actually just substitute in that angle theta for the inverse cosine of three divided by 11. And then bringing down the rest, what we have is the sine of two theta. And that is a double angle. We recall from previous lessons, the double angle identity from the textbook, they'd say the sign of two A equals two sine A cosine A and here our A is equal to our angle theta. And we already know what the cosine of our angle is. That's equal to three divided by 11. So we're a good way there. We can set our s of two theta equal to two sine theta cosine theta or bringing that down. That's too sine theta. And the cosine theta is three divided by 11. We just need to figure out what the sine of theta is. And we're pretty close. We recall from previous lessons that the cosine of an angle theta is equal to X divided by R and that the sign of an angle theta is equal to Y divided by R. We know that three divided by 11 equals X divided by R, which means our X value is three and our R value is 11. We just need to figure out our Y value and we can plug that in and we'll have our sign of data. One note is that we know that cosine is equal to a positive value in both the first quadrant and the fourth quadrant. If we draw a real quick sketch of a coordinate plane, we've got a vertical Y axis, a horizontal X axis. Those come together at the origin in the middle cosine is positive in the first and fourth quadrants. But we're actually talking about the inverse cosign, which has a limited range. So the range of an inverse cosign, we recall from previous lessons and from the tables in the textbook, from zero to P, that's our range for the inverse cosine zero to pi. Well, pi is the negative part of the X axis from zero to pi is the 1st and 2nd quadrants. So we can ignore any inkling that we might be in the fourth quadrant, which means our Y values are only going to be positive because we're in the first quadrant, our cosine is equal to a positive value. We're in quadrant one and our Y values are going to be positive there. So now we can figure out our Y, we recall from previous lessons that R equals the square root of X squared plus Y squared, R equals 11 that equals the square root of X squared is three squared plus, we don't know why yet. So that's just Y squared. We can bring this down and say that 11 equals the square root of nine or three squared is nine plus Y squared. Now, to solve this, we'll square both sides. And on the left 11 squared is 121 that equals nine plus Y squared. Subtracting nine from both sides, we have 112 equals Y squared, taking the square root of both sides will have Y equal two. And if you can do this without using a calculator, that's fantastic. Y will equal plus or minus four square root seven. But note, we said that Y is going to be positive because we're in quadrant one. So this is just going to be a positive four square root seven. So our sign of theta which is Y divided by R is going to be four square seven. That's our Y divided by R which is 11. And now we can plug that in for the sign of theta. So bringing this down, we have two multiplied by, instead of sine theta, that's four square root seven divided by 11 multiplied by the cosine of theta which is three divided by 11. Now, if we multiply our numerators straight across, we're going to have 24 square root seven divided by. Now, it would be really cool if we could have had that equal 3 65 24 7365 but 11 multiplied by 11 is just 121. And that is our expression simplified. We look at our answer choices for 24 square seven divided by 121. And that matches with answer choice. A well done. We'll catch you on the next one.

Evaluate each expression without using a calculator.
sin (2 cos⁻¹ (1/5))

Key Concepts

Inverse Trigonometric Functions

Double Angle Formulas

Pythagorean Identity

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