Use the appropriate reciprocal identity to find each function ...

Question

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1.

sin θ , given that csc θ = √24/3

Accepted Answer

Hello, everyone. We are asked to use the reciprocal identity to determine the value of the following trigonometric function. So we have given that the CO C can of theta equals the square root of 20 divided by eight. And we're asked to find the sign of theta. Our answer choices are a in parentheses, four radical five divided by five B, four radical five C in parentheses, two radical five divided by five D squared of five in parentheses divided by five. So first, I'm gonna rewrite this as a fraction. So I know the of theta is radical divided by eight. So I recall that the sign of any angle. So the sign of the is equal to one divided by the can of theta. So if we are given the Cantu theta is the squared of divided by eight, one divided by that is essentially flipping it over. So we can say it's one divided by the square root of 20 divided by eight. If I want to write that out as a fraction, I'd have one divided by radical 20 divided by eight, which is the same as multiplying by the reciprocal. So one multiplied by eight, divided by the square of 20. So the sign of theta equals eight divided by the square root of 20 we cannot leave the square root of 20 in the denominator. However, we need to rationalize this. So before we rationalize it, I'm gonna simplify my denominator. So off to the side, I'm gonna simplify the square root of 20. I recall that I can break this into factors. So I could say the squared of 20 equals the square of four multiplied by the squared of five, the square of four is two. So the square to 20 could be rewritten as two radical five. So I'm rewriting this as eight divided by two radical five, I'm gonna reduce the eight and the 22 goes into two once goes into times. So I now have four divided by the square to five. To rationalize my denominator. I multiplied both the numerator and the denominator by the square of five. In my numerator, I get four radical five in my denominator, the square of five multiplied by the square of five is like the squared of 25 which is five. So after rationalizing my denominator, we can say the sign of theta equals four radical five divided by five, which is answer choice. A have a nice day.

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1.
sin θ , given that csc θ = √24/3

Key Concepts

Reciprocal Identities

Rationalizing the Denominator

Simplifying Radicals

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