In Exercises 35–38, find the exact value of the following unde...

Question

In Exercises 35–38, find the exact value of the following under the given conditions:

d. sin 2α

sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression. The sign of two B under the given circumstances where the sign of A equals seven 25th and A is located between zero and pi divided by two. The sign of B equals nine divided by 41 and B is located between pi divided by two and pi we are given four answer choices. First, I wanna look up what are my options for the sign of two B? What identity works best for me? And I think the one that's gonna work best for me is that it equals two sign B cosine B. I think this works because we know that the sign of B is nine divided by 41. And in knowing this, I know this is Y divided by R. So I can find the cosine of B because I know that is X divided by R and I can use the nine and the 41 in the Pythagorean theorem to find the value for X. So X squared plus Y squared equals R squared X is what we don't know. So X squared stays X squared, Y squared. Is gonna be nine squared and R squares will be 41 squared. So doing this out, we have X squared plus 81 equals squared, which is 1681. Subtracting 81 from both sides, X squared equals taking the square root of both sides. We get that X equals 40. Now checking before I say that that's my final value. We see that B is located in quadrant two and the X values in quadrant two are negative. So this is gonna be negative 40. So now we have enough to write the cosine of B is negative divided by 41. So knowing this information, I'm gonna plug both the sign of B and the cosine of B into the formula sign of two B equals two sine B cosine B. So two multiplied by the sign of B which is nine divided by 41 multiplied by the cosine of B. So negative 40 divided by and doing that multiplication, I know 41 multiplied by 41 is 1681 for my denominator for my numerator two times or two multiplied by nine, multiplied by negative 40 comes out to a negative 720. And a quick glance to see if there's anything that can be reduced and it does not appear to have any common factors that I can cancel out. So the exact value of The sign of two B is negative 720 divided by 1681. That's answer choice. A have a nice day.

In Exercises 35–38, find the exact value of the following under the given conditions:
d. sin 2α
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

Key Concepts

Double-Angle Identity for Sine

Using the Pythagorean Identity to Find Cosine

Understanding Angle Quadrants and Their Significance

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