In Exercises 29–51, find the exact value of each expression. Do n...

Question

In Exercises 29–51, find the exact value of each expression. Do not use a calculator.

sin(cos⁻¹ 3/5)

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression, the cosine of the arc sign of 63 divided by 65 without using a calculator. Our answer choices are a 65 divided by B 16, divided by 65 C 31 divided by 65 D three radical seven. So I'm gonna start with what's in my parentheses. So the arc sign of divided by 65 I'm gonna set that equal to an angle theta and then I'm going to take the sign of both sides and get that the sign of theta will equal 63 divided by 65. At this point. I recall that if I were to think about sketching this or if I were drawing this on a right triangle, that this is the ratio of Y divided by R. So to find the cosine of theta, which is gonna give me my ultimate answer. I need to do X divided by R. So I'm going to use the values from the sign to find the X value and I'll use the Pythagorean theorem. So Y squared or X squared plus Y squared. Will equal R squared, X squared is my unknown plus Y squared. So 63 squared has to equal 65 squared. So X squared plus will equal 4225. And we say without using a calculator, you can use a calculator to square your numbers. We just mean we want to make sure we have an exact fraction at the end. I'm gonna subtract 3969 from both sides. So X squared will equal taking the square root of both sides, X would equal plus or minus since this appears to be sinus positive. And that we know that the range of the arc sign function is between negative pi divided by two and pi divided by two. We can say we are in quadrant one and cosine is positive and quadrant one. So our X value can be positive. So we now know that X is 16, Y is 63 R is 65. So now to find the cosine of the, we do X divided by R so 16 divided by 65. And that's our answer 16 divided by 65 which is answer choice B have a nice day.

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In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin(cos⁻¹ 3/5)