In Exercises 63–82, use a sketch to find the exact value of each ...

Question

In Exercises 63–82, use a sketch to find the exact value of each expression.

tan [sin⁻¹ (− 3/5)]

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the following trigonometric expression. By sketching, we are given the tangent of the inverse sign of negative 20 29th. There are four answer choices. Since I know I need to solve this by sketching, I'm gonna start by drawing my X and Y axes from there. I am going to take the sign sorry, the inverse sign of negative 20 29th and say that it is equal to alpha. Now doing the opposite of the inverse, I will take the sign of both sides and say that the sign of alpha equals negative 20 29th. Since sin is negative, this must be located in quadrant four. So since it's in quadrant four, I am going to draw what I can on my coordinate plane. So I recall that sign is the same as saying Y divided by R where R is our radius for our circle also sometimes thought about as our hypotenuse. So here R is 29. Why is negative 20? So if it's in quadrant four, our hypo is 29 and it goes down 20 on the Y axis. So this is 20. This means that our angle alpha starts at our standard position continues all the way to quadrant four. So we've drawn that on there. Also what we don't have the measure of at this exact moment is the X value. So we're gonna use the Pythagorean theorem to find that. So X squared plus Y squared will equal R squared. XES are unknown. So X squared plus negative squared has to equal 29 squared. So X squared plus 400 equals 841. Subtracting 400 from both sides, X squared equals taking the square to both sides. X equals 21 X is positive in the fourth quadrant. So we now know that X is from there. We can say that this is the tangent of alpha. So the tangent of alpha is the ratio where we have Y divided by X. So why here is negative X is 21. So this should be our answer that the tangent of the inverse sign of negative 2020 nines ends up with negative 20 divided by 21 which is answer choice A have a nice day.

In Exercises 63–82, use a sketch to find the exact value of each expression. tan [sin⁻¹ (− 3/5)]

Key Concepts

Inverse Trigonometric Functions

Right Triangle Relationships

Pythagorean Theorem

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