In Exercises 1–8, a point on the terminal side of angle θ is g...

Question

In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ. (3, 7)

Accepted Answer

Hello, today we're gonna be determining the values of the six trigonometric functions for alpha with the given terminal point. Now the six trigonometric functions rely on the values of a right triangle. So we can use our terminal point to create a right triangle. Let's start by drawing an axis. Now, the terminal point given to us is five comma 17. This is a point that can be found within the first quadrant of the axis. So five comma 17 is gonna be located roughly about here and we can use this point to make a right triangle. Alpha is gonna be located in the center of this triangle. And now what we need to do is we need to identify the lengths of the three sides of the triangle. Well, the X value is given to us as five. So the length of this triangle is going to equal to five and the height of the triangle is given to us as 17. So we can label the height as 17. Now we need to label the length of the hypotenuse. We are not given the length of the hypo news, but we do have an equation that can help us find it. We can use the Pythagorean theorem, which is given to us as hypotenuse squared is equal to X squared plus Y squared. We know that the X value is five and the Y value is 17. So if we plug this into our equation, we get hypotenuse squared is equal to five squared plus 17 squared. Five squared is going to give us the value of 17 squared is going to give us the value of 25 plus 289 is going to give us 314. So H squared is equal to 314. Finally, if we take the square root of both sides of this equation, we get H is equal to the square root of 314. And what this means is that the hypotenuse is going to have a length of the square root 314. So now that we have a right triangle with the three sides labeled, we can go ahead and use this to find the values of the six trigonometric functions for alpha. Now, the six trigonometric functions have the given definitions. Sine is equal to the Y value over the hypotenuse cosine is equal to the XX value over the hypotenuse and tangent is equal to the Y value over the X value sequent is the inverse of sign. And that is going to give us hypotenuse over the Y value. Second is the inverse of cosine, which is hypotenuse over the X value. And cotangent is the inverse of tangent, which is the X value over the Y value. We can go ahead and plug in the values of our right triangle into the identities of the six trigonometric functions to figure out their values. Let's go ahead and start with sine again. Sine is defined as our Y value over the hypotenuse that is going to give us 17 over the square root of 314. But note that we cannot keep a square root in the denominator of our answer. So we're going to need to rationalize this value by multiplying the numerator and denominator by the square root of 314. And doing so is going to give us a sign value of 17 times the square root of 314 over 314. Now, let's go ahead and find the cosine value cosine is defined as the X value over the H value and that is going to give us five over the square root of 314 just like Sine. We're going to need to rationalize this quantity by multiplying the numerator and denominator by the square root of 314. And that is going to give us a cosine value of five times the square root of 314 over 314. Now, let's look for tangent tangent is defined as our Y value over the X value. And that is going to give us 17/5, which is in its simplest form. Now, let's look for the coin value. Coin is defined as our hypotenuse over the Y value, which is going to give us the square root of 314 over 17 second is defined as the hypotenuse over the X value. And that is going to be squared at 314 over five. And cotangent is defined as our X value over the Y value which is going to be 5/17. So just to summarize, we have the sine value equal to 17 times the square root 14, over 314 we have a cosine value of five times the square root 314. Over 314 we have tangent is equal to 17/5 coin is equal to the square root of 314. Over 17, seeking is equal to the square root of 314 over five and cotangent is equal to 5/17. These are going to be the six trigonometric values for the angle alpha with the given terminal point. And this is going to be the answer to our problem. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ. (3, 7)

Key Concepts

Coordinates and the Terminal Side of an Angle

Definition of the Six Trigonometric Functions

Calculating the Radius (r) from Coordinates

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