An equation of the terminal side of an angle θ in standard positi...

Question

An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ .  See Example 3.   
                                                                                                                                                                                                                                                                                                                                                                                   
  x = 0 , y ≥ 0

Accepted Answer

Welcome back. I am so glad you're here. We are asked to draw the sketch of the least positive angle beta whose equation of the terminal side in standard position is given below. Also find the values of the sine of beta, the cosine of beta and the tangent of beta. Our terminal sides equations are X equals zero and Y is less than or equal to zero. Then we are given a coordinate plane. We have a vertical Y axis and a horizontal X axis. The range is from negative five to positive five. And the domain of what's shown is from negative five to positive five as well. All right. So graphing our angle beta, it's going to be in standard position, meaning its initial side is heading along the positive part of the X axis toward positive infinity. And then we're going to head counter clockwise. But how far do we go? Well, when it says least positive, it means we're not going to keep going forever and ever, we are going to go just the first round and we're given two conditions, we're told that X is equal to zero. So that means we're going to be along the Y axis where X is equal to zero. But are we in the po positive part or the negative part? We're also told that Y is less than or equal to zero. So that means we're going to be in the negative part of the Y axis. So as we are heading counterclockwise from our initial side, because we're going to the least positive angle beta, we are only going as far as 200 70 degrees and heading with our terminal side along the negative part of the Y axis. So there is our beta. Now we need to determine the sign of beta, the cosign of beta and the tangent of beta. Now, when we have a quadrant angle, we recall from previous lessons and from the tables in the textbooks that these are given to us in the tables, these exact values. But if you didn't have those tables available, the way you would determine this is figuring out your X, your Y and you are. And then we know that the sign of any angle is going to be Y divided by R. We know that the cosine of any angle is X divided by R and we know that the tangent of any angle is Y divided by X. So what are our XY and R values? Well, we can choose any point along that negative part of the Y axis for our X and Y values. So I'm just going to choose the point at zero. That's always going to be our X value and negative one for our, our Y value. So just plugging those in X is equal to zero, always Y is any negative number. We'll just go with negative one. And then our R value, we recall from previous lessons, the equation R squared equals X squared plus Y squared. And plugging in for X and Y, we've got zero squared for the X squared plus negative one squared for the Y squared, zero squared is zero and negative one squared is just one. So R squared equals one, taking the square root of both sides. We have an R value equal to plus or minus the square root of one is one. However, R is a distance, it's always going to be greater than zero. So we can disregard that negative value and R is just equal to one. So that now that we have our XY and R values, we can plug those in sol for sine, cosine and tangent. All right. For the sine that's Y divided by RY is negative one, divided by R is one negative one, divided by one is negative one. So the sine of beta is equal to negative one for cosine X in the numerator is zero, divided by R is still 10, divided by one is zero. The cosine of beta is zero. And for a tangent that's Y divided by XY is negative one divided by X is zero, anything divided by zero is undefined. So that's where they got those values in the tables. In the textbook. For the final one, the tangent of beta is undefined. And there we go. Well done. We'll catch you on the next one.

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An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . See Example 3. x = 0 , y ≥ 0