Determine the value of y = − 2 ⋅ sin ( − 3 π 2 ) + 10 y=-2\cdot\sin\left(-\frac{3\pi}{2}\right)+10 y = − 2 ⋅ sin ( − 2 3 π ) + 10 without using a calculator or the unit circle.

Hey everyone. Let's see if we can solve this problem. So here we're asked to determine the value y is equal to negative two times the sine of negative three pi over two plus 10, and we are not allowed to use a calculator or the unit circle to solve this problem. So how can we go about doing this? Well, when I see these types of problems, I think the best strategy is generally going to be to try graphing this. And what I'm going to graph is this trig function that I see. Now I'm going to draw the graph for y is equal to the sine of x. This isn't what we're looking for, but this is going to help us to see where our values lie, and thus, for allow us to evaluate this problem. So looking at the sine of x, recall that the graph starts here at the center. And then what you do is you go up and you reach a peak when you get to pi over two. You're then going to cross through pi and then reach a valley at three pi over two, and then this will continue waving as we go to the right. Now likewise, you can draw this graph to the left as well. You're going to reach a valley at negative pi over two, and then you're going to cross through negative pi and reach a peak at negative three pi over two, and then this graph will keep waving as you go to the left. So this is what the graph for y is equal to the sine of x looks like. But this is not what we're looking for, but rather we're looking for y equals negative two times the sine of negative three pi over two plus 10. Now because I just drew a graph for the sine of x, what I can do is evaluate this graph at negative three pi over two. And looking at this graph, if I go to negative three pi over two, I see that we're at a high value here, which is an output of one. So that means what we're going to end up having is y is equal to negative two times the sine of negative three pi over two, which we just figured out is one plus 10. Now negative two times one is negative two, and then adding 10 to negative two will give you eight. So that means the solution for y is positive eight when we evaluate this equation above. So that's how you can go about solving this problem. Thanks for watching. Let me know if you have any questions.

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 $y=-2\cdot\sin\left(-\frac{3\pi}{2}\right)+10$ without using a calculator or the unit circle.

$y=8$

$y=10$

$y=-2$

$y=12$

Verified step by step guidance

Step 1: Start by analyzing the given equation y = -2 * sin(-3π/2) + 10. The goal is to evaluate the sine function and then substitute its value into the equation.

Step 2: Recall the periodic properties of the sine function. The sine function is odd, meaning sin(-x) = -sin(x). Therefore, sin(-3π/2) = -sin(3π/2).

Step 3: Use the graph provided to determine the value of sin(3π/2). From the graph, sin(3π/2) corresponds to the y-coordinate at the angle 3π/2, which is -1.

Step 4: Substitute sin(3π/2) = -1 into the equation sin(-3π/2) = -(-1). This simplifies to sin(-3π/2) = 1.

Step 5: Substitute sin(-3π/2) = 1 into the original equation y = -2 * sin(-3π/2) + 10. This becomes y = -2 * 1 + 10. Simplify the expression to find the value of y.