In Exercises 49–59, find the exact value of each expression. D...

Question

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin(-𝜋/3)

Accepted Answer

Hello, today we're gonna be determining the exact value of the following expression. So what we are given is sign of negative pi over six. Now, there's one simplification that we can do to the following expression, we can use the rule of signs for sign in order to rewrite the expression, the rule of Stein state that if you have a negative angle inside the sine function, you can move the negative sign to the front of the sine function and rewrite it as negative sign of X. So if we use the rule signs on a given expression, we can rewrite sign of negative pi over six as negative sign of pi over six. Now, what we wanna do is we want to determine the value of sine pi over six. In order to do this, we're going to have to locate the angle pi over six on the unit circle. Now, if you're ever unsure of where a radon is located on the unit circle, you can always convert the radiant to degrees. In order to do that, you take the angle in radiance and multiply it by the quantity 180 over pi doing So allows us to reduce the pies in the numerator and denominator to just one. And taking the product of these values gives us 100 80 times one, which is 100 and 80/6 times one, which gives us six and 100 and 80 divided by six, gives us the value of 30. So what this means is that pi over six is equivalent to degrees. So we just need to locate 30 degrees on the unit circle. So the question is where is 30 degrees on the unit circle while 30 degrees is located in the first quadrant of the unit circle. And the terminal point at this angle is given to us as square root 3/ comma one half. Now keep in mind that every point that is found along the unit circle is defined as cosine comma sine. What this means is that cosine represents every X value for any terminal point on the unit circle. And sine represents any Y value for any terminal point on the unit circle. So in this case here, because we know that the terminal point for 30 degrees is squared root 3/ comma one half, we just need to take a look at the Y value for sine of Pi over six. And in this case here, the Y value is going to be one half. So what this tells us is that Sine of Pi over six is equal to 1/2. But keep in mind that we rewrote the original statement as negative sign of pi over six. So if we substitute one half for negative sign pi over six, what we are left with is negative one times the quantity one half and multiplying these quantities together is going to give us negative 1/2. So what this means is that sign of negative pi over six is equal to negative 1/2. And with that being said, the answer to this problem is going to be c so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin(-𝜋/3)

Key Concepts

Unit Circle and Angle Measurement

Sine Function and Its Properties

Exact Values of Special Angles

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