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:

a. sin(α + β)

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 A plus B under the circumstances where the sign of A equals seven 25th and A is between zero and pi divided by two. And the sign of B equals 9 40 firsts and B lies between pi divided by two. And pi we are given four answer choices. The first thing I like to write down is the formula that goes with the sign of A plus B. And that formula is the sign of A multiplied by the cosine of B plus the cosine of A multiplied by the sign of B. Now, I want to go through and see. Do I have all of those trig functions or do I need to discover what their values are? Well, I know I have the sign of A is seven 25th and I know that the sign of B is nine divided by 41. I do not have the cosine of either. So I will have to discover what that is. So recalling that sign is the ratio of Y divided by R I know I can use the Pythagorean theorem to find out what X is. So I can make cosine X divided by R. So X squared plus Y squared is gonna equal R squared. So X squared plus seven squared will equal 25 squared. So X squared plus 49 equals 625 subtracting 49 from both sides, X squared equals 576. And when I take the square root of both sides, X equals 24 I want to check and see where on our coordinate plane angle A lies to make sure that's a positive 24 and angle A is in quadrant one. So that is a positive 24. So this means the cosine of A is 24 divided by R which is 25. I'm gonna do very similar work to find out what the cosine of B is. So I again have the sign of B is nine divided by 41. So that's like Y divided by R and cosine is X divided by R. So again, I don't know what X is. So X squared plus Y squared, which is nine squared has to equal R squared. So 41 squared. So X squared plus 81 would equal 1681. Subtracting 81 from both sides, X squared equals 1600. Taking the square root of both sides. I have X equals to 40. Again, I need to check my location on the coordinate plane B is between pi divided by two and pi, which means that it's in quadrant two and the X values are negative and quadrant two. So this is a negative 40 as our X value. So the cosine of B is negative 40 divided by 41. So now that we have the values of all of the trig angles that are in our, our formula, we can plug it in. So we want to do the sign of A. So seven 25th multiplied by the cosine of B which is negative 40 divided by closed parentheses plus the cosine of A which we have as 24 25th multiplied by the sign of B which is 9 40 firsts. So I will do my multiplication. And when I do so I get negative 280 divided by 1025. And our next fraction we have divided by 125 or sorry, 1025. Since they have a common denominator, I'm going to add them together. And when I do so I get negative 64 divided by 1025. So this is the exact value of the sign of A plus B under these given circumstances. And that's answer choice. D have a nice day.

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

Key Concepts

Sum of Angles Formula for Sine

Determining Cosine from Sine and Quadrant

Understanding Angle Ranges and Quadrants

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