In Exercises 7–14, use the given information to find the exact va...

Question

In Exercises 7–14, use the given information to find the exact value of each of the following:b. cos 2θ15sin θ = -------- , θ lies in quadrant II.17

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression. Using the provided data, we are looking for the value of the cosine of two beta where the sign of beta equals 12 37th and beta lies in quadrant two. We are given four answer choices. First thing to recall is our double angle identities. The double angle identity for cosine is that the cosine of two beta equals cosine squared beta minus sine squared beta. So we need to find a value for both cosine and sign. And then we're gonna plug it into this identity. We are told that the sign of beta is 12 37th. Recall that the ratio for sine is the Y value divided by the R value. And that if we were looking for the cosine of beta, this would be the X value divided by the R value. So we do not know our X value right now, we know that Y equals 12 and R equals 37. Since these ratios are based on the Pythagorean theorem, we can use the idea that X squared plus Y squared equals R squared to solve for X. So X squared plus Y squared. So 12 squared equals R squared to 37 squared. So we get X squared plus 144 equals 1369. I'm going to subtract 144 from both sides. So I have X squared equals 1225 to get the X by itself, I will do the opposite of a square which is a square root. So square root to both sides, we get that X is 35. Before we say conclusively that X is 35 we need to take into consideration that be uh excuse me, beta lies in quadrant two, in quadrant two, the X values are negative. So since this is in quadrant two, X equals negative 35. So now that we know that X equals negative 35 we can go back to writing the cosine of beta which will be the X value negative 35 divided by the R value 37. Now that we have both sign and cosine, we can plug it into our double angle identity. So we again are looking for the cosine of two beta and we need to do the cosine of beta squared. So negative 35 37th squared minus the sine squared of beta. So 12 37th squared, when I do out these squares, I am left with divided by minus 144 divided by 1369. What's very nice about this is we already have a common denominator so we can subtract our numerators. And when we do so, we will find that the cosine of two beta is 1081 divided by 1369. That is our exact value and it is answer choice B have a nice day.

In Exercises 7–14, use the given information to find the exact value of each of the following:b. cos 2θ15sin θ = -------- , θ lies in quadrant II.17

Key Concepts

Trigonometric Identities

Quadrants and Angle Properties

Finding Exact Values of Trigonometric Functions

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