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. cos(-35𝜋 / 6)

Accepted Answer

Hello everybody. Everything right today today, we're going to be looking at this question that states determine the exact value of the following expression without the help of the calculator where the expression is cosine of negative 41 pi divided by six. Now the answer choices are a screed of three divided by two B negative scored of three divided by two C one half and D zero. Now, the first thing I wanna do here, we have a negative angle, sorry angle is negative 41 pi divided by six. And the first thing we wanna do is rewrite this and rewrite our angle between zero is less than beta is less than two pie. Now, basically what this means is we want to write it within the positive coordinate system because because right now we are with a negative status. So we're in the negative coordinate system. So we wanna be in a positive coordinate system. Now to do that, we just have negative 41 pi divided by six plus eight pi and that will give us seven pi divided by six. Now, where do we obtain this information? So imagine well, with trigonometric functions, we are always working in a coordinate system with the unit circle. So if we are at negative 41 pi divided by six, we went around that circle multiple times in the negative side. And so counterclockwise, we want to be moving clockwise in a positive way. So we add eight pi clockwise to go back to the positive side. And we end up at seven pi divided by six. And that's the angle that the data that we will care about and work with. Now. Now for seven pi divided by six, where does that lie? That's our final point. So we want to know where we are seven pi divided by six is less than pi which is but is uh is greater than pie but less than two pie. Now pie is the left side of or I'm sorry, it's greater than pie, but less than three pi divided by two. Now pi is the left side of the X axis and three pi over two is the bottom half of the Y axis. So we are somewhere in quadrant three and that will be where we are at right now. Now we have different reference angle rules for different quadrants. And to continue forward with this question, we want to know what our reference angle is. Now in quadrant three, specifically, we have that the reference angle is equal to the given angle minus pi. So we'll have the given angle is the angle we're working with is seven pi divided by six and we'll have that minus pi. So we'll have that the reference angle is equal to pi divided by six. Now we move forward and we see that cosine is negative in quadrants three and that's something we should not. So in quadrant three, cosine is negative. And now we look at the values that we know for the unit circle. So according to the unit circle, and you might be asking why is cosine negative? Well, when we're talking about cosine and sine, when we're talking about cosine, specifically, we're referring to the X component of our graph. And where is the X component negative? It's to the left of the Y axis. So quadrant two and three. So if cosine is associated with the X component, we know that cosine is negative and two in quad two and three. And since we're in quantum three, cosine is negative, now, according to the unit circle at pi divided by six, we have a point of square root of three divided by two comma one half. So therefore, cosine of negative four pi divided by six is equal to negative cosine of pi divided by six. And I put negative cosine because we said cosine wouldn't be negative. So if we keep in mind that cosine, we're talking about the X component, we have a point with the X component being squared of three divided by two. So our cosine will be negative squared of three divided by two. And that will be our final answer for this question. And as you can see, it's very important to know how to calculate reference angles and understand how the unit circle as a whole works. So I hope that was helpful. And until next time.

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cos(-35𝜋 / 6)

Key Concepts

Unit Circle and Angle Measurement

Coterminal Angles

Exact Values of Trigonometric Functions

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