Determine the value of y = sin ( − π 2 ) + 50 y=\sin\left(-\frac{\pi}{2}\right)+50 y = sin ( − 2 π ) + 50 without using a calculator or the unit circle.

Let's see how we can solve this problem. So here we're asked to determine the value of y is equal to the sine of negative pi over two plus 50, and we're told that we cannot use a calculator nor can we use the unit circle. So how could we really go about solving this problem? Well, it turns out there's another tool that we have in the toolbox, and that is going to be graphing the function for the sine of x, because we know what these graphs look like, so we can use that graph to figure out what this value is and thus determine the value of this whole thing. So what I'm first going to do is draw a graph for y is equal to the sine of x. Now recall that the sine of x starts right here in the middle of our graph. Then what we do is we go up here and we reach a peak when we get to pi over two, and then we're going to dip down and cross through pi on the x axis. Then we're going to reach a valley at three pi over two, and then we're gonna keep waving as we go to the right. Now likewise, this graph is also going to have this waving pattern as we go to the left. So we're going to go down here and reach a valley at negative pi over two, we're going to cross through negative pi, then we're going to reach a peak at negative three pi over two, and then this graph is going to keep waving. So this is what the graph is going to look like for the sine of x, we should be familiar with this. And what we can do is use this graph to determine what this value is for the sine of negative pi over two. Now what we can see here is that for negative pi over two, we can just go to this on the x axis, which is right there. And notice that the output that we have on this curve is right here at negative one. So because we're at negative one, that that means this entire sign value is going to evaluate to negative one. So we have negative one plus 50, and negative one plus 50 is 49. So the answer to this problem is y is equal to 49. And notice how we were able to find this without using a calculator or the unit circle, we could just recognize how the graph behaves to evaluate this answer. So that's how you solve this problem. Thanks for watching.

Determine the value of y = sin ⁡ ( − π 2 ) + 50 y=\sin\left(-\frac{\pi}{2}\right)+50 y = sin ( − 2 π ) + 50 without using a calculator or the unit circle.

y = 50 + 3 2 y=50+\frac{\sqrt3}{2} y = 50 + 2 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

12. Trigonometric Functions

Graphs of Sine & Cosine

Business Calculus

12. Trigonometric Functions

Graphs of Sine & Cosine

Struggling with Business Calculus?

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Multiple Choice

Determine the value of $y=\sin\left(-\frac{\pi}{2}\right)+50$ without using a calculator or the unit circle.

$y=50$

$y=51$

$y=49$

$y=50+\frac{\sqrt3}{2}$

Verified step by step guidance

Step 1: Begin by analyzing the given function y = sin(-π/2) + 50. The sine function is periodic and odd, meaning sin(-x) = -sin(x). This property will help simplify the expression.

Step 2: Use the property of the sine function to rewrite sin(-π/2) as -sin(π/2). The value of sin(π/2) is 1, so sin(-π/2) becomes -1.

Step 3: Substitute the value of sin(-π/2) into the equation. The equation now becomes y = -1 + 50.

Step 4: Simplify the expression by performing the addition. Combine -1 and 50 to find the value of y.

Step 5: The final simplified expression for y is y = 49. This matches the correct answer provided in the problem.

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Struggling with Business Calculus?

Determine the value of y=sin(−π2)+50y=\sin\left(-\frac{\pi}{2}\right)+50y=sin(−2π)+50 without using a calculator or the unit circle.

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Graphs of Sine & Cosine practice set

Determine the value of $y=\sin\left(-\frac{\pi}{2}\right)+50$ without using a calculator or the unit circle.