Sketch an angle θ in standard position such that θ has the least ...

Question

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of  θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4.                                                                                                                                                                                                                                                                               
                                                                                                                                                                                                                                                                                                                                                                                   
  (0, ―3)

Accepted Answer

Hey, everyone here, we are asked to plot the least positive measure of the given angle in standard position, such that the given point is on the terminal side. Also, we need to write the value of all trigonometric functions for the given angle. And we need to perform at rationalization wherever necessary. Here, we are given the point as the ordered pair zero and negative seven. So to begin plotting the least positive measure of the angle, we first want to plot our given point. So we plot the 0.0 and negative seven on our graph here. And now we want to sketch in the terminal side. So the terminal side is an arrow which starts at the origin and extends along the X axis. Next, we need to sketch the terminal side which we are told passes through or intersects our given point. So our terminal side just follows from the origin along the Y axis and again crosses through or intersects our given point. And now we want to draw in our angle theta. So our angle theta starts at the initial side and moves counterclockwise to the terminal side. And now we can move on to determining the value of all of the trigonometric functions for our given angle beta by first identifying our values of XY and R. So we are given the 0.0 and negative seven. So that tells us that X is equal to zero and Y is equal to negative seven. And so now we also need to find the value of R and so we need to recall the formula for R, which is that R is equal to the square root of X squared plus Y squared. And now just substituting in our given values into this formula, we have R is equal to the square root of zero squared plus the quantity of negative seven squared. And now simplifying we have the square root of 49. And simplifying further, we get that R is equal to seven. And now with our values for XY and R clearly identified, we can begin finding the values of our functions. And so we can begin with sign of theta. And so we just need to recall that sign of theta is the ratio Y divided by R. And so now substituting in our identified values, we have negative seven divided by seven and simplifying we just have negative one. So sine of theta is negative one. Now moving on, we can find cosine of theta and recalling cosine of theta is equivalent to the ratio X divided by R. Once again, substituting in our values we have zero divided by seven. And simplifying we just have zero. Now moving on to tangent of theta, we need to recall that tangent is equivalent to Y divided by X. And now substituting in our values, we have negative seven divided by zero. And here we see that we are dividing by zero. So this statement or this expression is undefined. So tangent of theta is undefined. Now we can move on to cotangent of theta. And we just need to recall that cotangent of theta is the ratio X divided by Y substituting in our values, we have zero divided by negative seven which just simplifies to zero. Next, we move on to second of theta and second of theta is equivalent to R divided by X. Once again, substituting in our values, we have seven divided by zero. And just like with tangent of theta, with sin of theta, we are dividing by zero again. So that means that second of theta is undefined and now we can move on to our last trigonometric function which is co sequent of theta and sinking of theta. As we recall is equivalent to the ratio are divided by Y substituting in our values, we have seven divided by negative seven. And simplifying we just get negative one. So with this, we have found the values for all of our six trigonometric functions and we have successfully plotted the least positive measure of our given angle in standard position. Thanks for watching, I hope you found this video helpful and I'll see you again next time.

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (0, ―3)

Key Concepts

Standard Position of an Angle

Terminal Side of an Angle

Trigonometric Functions

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