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 3 pi over 2 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 pitwo. Then going to cross through pi, and then reach a valley at 3 pi over 2, 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 2, and then you're going to cross through negative pi and reach a peak at negative 3 pi over 2, 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 2 times the sine of negative 3 pi over 2, plus 10. Now because I just drew a graph for the sine of x, what I can do is evaluate this graph at negative 3 pi over 2. And looking at this graph, if I go to negative 3 pi over 2, I see that we're at a high value here, which is an output of 1. So that means what we're going to end up having is y is equal to negative 2 times the sine of negative 3 pi over 2, which we just figured out is 1 plus 10. Now negative 2 times 1 is negative 2, and then adding 10 to negative 2 will give you 8. So that means the solution for y is positive 8 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.

10. Graphing Trigonometric Functions

Graphs of the Sine and Cosine Functions

Precalculus

10. Graphing Trigonometric Functions

Graphs of the Sine and Cosine Functions

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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

Recognize that the problem involves evaluating the sine function at a specific angle, which is $-\frac{3\pi}{2}$.

Recall the property of the sine function: $\sin(-x) = -\sin(x)$. This means $\sin\left(-\frac{3\pi}{2}\right) = -\sin\left(\frac{3\pi}{2}\right)$.

Understand that $\frac{3\pi}{2}$ is an angle that corresponds to the negative y-axis on the unit circle, where the sine value is $-1$.

Apply the property from step 2: $\sin\left(-\frac{3\pi}{2}\right) = -(-1) = 1$.

Substitute the value of $\sin\left(-\frac{3\pi}{2}\right)$ into the original expression: $y = -2 \cdot 1 + 10$. Simplify this expression to find the value of $y$.