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:

a. sin 2θ

15
sin 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 told that we want to find the sign of two beta where sign of beta equals 12 37th and beta lies in quadrant two, we are given four answer choices. So starting with, I know that the sign of two beta has a double angle identity. And I recall that that identity is two multiplied by the sign of beta multiplied by the cosine of beta. So this is what we're going to plug our values into to get the exact value. We are told that the sign of beta equals 12 37th. I recall that when I'm thinking about sign ratios within a coordinate plane that this ratio is the ratio of Y to R and that the cosine of beta would then be the ratio of X to R. So I know that Y is gonna equal R is gonna equal 37. But I need to find out what X equals these values are based on the idea of the Pythagorean theorem. So I can use the X squared plus Y squared equals R squared. Plug in the values were given and solve for X. So that's what I'm going to do. So X squared plus squared will equal 37 squared. So X squared plus 144 will equal 1369. Subtracting 144 from both sides, I get the X squared equals 1225. And the opposite of a square is a square root. So square root of both sides X equals 35 before I can say that that is exactly my X value. I need to take into account that this lies in quadrant two thinking of the values in quadrant two, we have negative X values and positive Y values. So since this is in quadrant two, X equals negative but now that we know this, we can find out what is the cosine of beta, it's gonna be our X value of negative 35 divided by our R value of 37. So we now know both the sign and the cosine of beta. So we're going back to our double angle identity. So the sign of two beta equals two multiplied by the sign of beta. So 12 37th multiplied by the cosine of beta, which we just found as negative 35 37th multiplying my fractions. When I multiply straight across the top, the numerator is negative and the denominator comes out to 1369. So the sign of two beta equals negative 840 divided by 1369 which is answer choice A have a nice day.

In Exercises 7–14, use the given information to find the exact value of each of the following: a. sin 2θ 15 sin θ = -------- , θ lies in quadrant II. 17

Key Concepts

Sine Function and Its Values

Double Angle Formula for Sine

Pythagorean Identity

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