Evaluate each expression without using a calculator.
tan...

Question

Evaluate each expression without using a calculator.

tan (arcsin (3/5) + arccos (5/7))

Accepted Answer

Without using a calculator, determine the exact value of the expression 10 of art sign 7 25th plus art cosine to 11th. And we have four possible answers. They're all fractions with some radicals. We have to figure out. Now to solve this, we need to make use of the tangent some identity, which is from our textbook, we have the tangent of X plus Y. It equals two tangent X plus tangent Y divided by one minus tangent X multiplied by tangent Y. Now we notice we have a sun 7 25th and arc coun two elevens. Let's go ahead and modify this a little bit. Let's say X equals arc sine 7 25th and Y equals arc cosine to 11th. Now, we can make triangles to bind these values. You put X for the angle position, our arc sun is our opposite side divided by her hypotenuse, which means we have seven directly across and 25 as the hypotenuse. Similarly, for Kan, we will put Y in the angle position and have two for the adjacent side and 11 for the hypotenuse. Now, we just need to solve for X and Y, we'll use the Pythagorean theorem to do this, we get a squared plus B squared equals C squared, which means our missing side A will be the bottom of the tranquil. Get a square plus seven squared equals 25 squared. A square plus 49 equals 6, 25. And if we subtract 49 we get a square equals 576 taking the square root of both sides gives us a equals 24. Which means that is our X value. We can do the same for the Why? With two squared plus we win equals 11 squid. Actually, we won't use Y there, we'll use B two squared plus B squared equals 11 squared four plus B squared equals 121. If we subtract or from both sides, B squared equals 117, we take the square root of both sides to get B equals the square root of 117 which simplifies to three square roots of 13. This is our Y value and we can now set up our tangent some, we have TX which is equals to the tangent of our triangle seven divided by 24 ty. The same thing will be three square roots of 13 divided by two. We do now substitute these into our expression. We get seven divided by 24 plus three squares of 13, divided by two, all divided by one minus seven, divided by 24 multiplied by three square roots of 13, divided by two there's some algebra to do here. But once it's simplified, we end up with the following expression negative two multiplied by 3388 plus 6, 25 squared to 13, all divided by 1143. If we look at our possible answers, we can see the answer to our problem. Turns out to be answer. C OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

Evaluate each expression without using a calculator.
tan (arcsin (3/5) + arccos (5/7))

Key Concepts

Inverse Trigonometric Functions

Trigonometric Addition Formulas

Right Triangle Relationships

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