Use the formula for the cosine of the difference of two angles to...

Question

Use the formula for the cosine of the difference of two angles to solve Exercises 1–12.

In Exercises 1–4, find the exact value of each expression.

(  2π          π   )
cos   ------- ﹣ ------
       (   3            6   )

Accepted Answer

Hello, today we're going to be determining the exact value of the trigonometric expression. So what we are given is cosine five pi over six minus pi over three. Now we'll need to expand the expression by using the trigonometric identity. There is a difference identity for cosine that states that if we have cosine of A minus B, this is equal to cosine of a multiplied by cosine of B plus sign of A multiplied by sign of B. So we'll need to use this trigonometric identity in order to evaluate the trigonometric expression, but we'll need to identify our A and B values. Well, we have cosine five pi over six minus pi over three which matches the form of cosine A minus B here five pi over six is in the spot of A. So we're going to allow A to equal to five pi over six and pi over three is in the spot of B. So we're going to allow B to equal to pi over three. So if we were to use the following trigonometric identity, we can rewrite the expression as cosine of A which is cosine five pi over six, multiplied by cosine of B, which is cosine pi over three plus sign of A, which is sine of five pi over six, multiplied by sign of B which will be S sign of pi over three. We'll then need to evaluate each of the trigger metric values in order to find the exact value. And we can do that by using a unit circle. So let's draw a unit circle. Now, what we'll need to do is we'll need to identify the locations of pi over three and five pi over six pi over three exists in the first quadrant. And the terminal point at pi over three is given to us as one half comma square root 3/2. Next, the location of five pi over six will be in quadrant two. And the terminal point at five pi over six is given to us as negative square root 3/2 comma 1/2. Now recall that any terminal point on the unit circle is defined as cosine comma S sign. What this means is that cosine is the X values of the terminal points and sine is the Y values of the terminal points. So let's start by evaluating cosine five pi over six. What this means is that we want the X value of the terminal point located at five pi over six. And the X value of that terminal point is given to us as negative square root 3/2. So that means cosine five pi over six will evaluate to give us negative square root 3/2. Then we'll want to evaluate cosine pi over three. And that means we need the X value of the terminal point at pi over three that X value is one half. So cosine power over three, evaluates to one half. Next, let's evaluate sine of five pi over six sine of five pi over six will be the Y value of the terminal point F five pi over six. And that is one half and sine pi over three is the Y value of the terminal point at pi over three. And that is given to us as square over 3/2. So that means we can rewrite the equation as negative square root 3/2, multiplied by one half plus one half multiplied by positive square root 3/2, negative square root 3/2, multiplied by a half will give us negative square root 3/ and one half, multiplied by square root 3/2 will give us positive square root 3/4. And if we were to add negative square 3/4 and positive square with 3/4, the final value we get for the expression is zero. What this means is that cosine of five pi over six minus pi over three has an exact value of zero. And with that being said, the answer to this problem is going to be a. 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.

Use the formula for the cosine of the difference of two angles to solve Exercises 1–12. In Exercises 1–4, find the exact value of each expression. ( 2π π ) cos ------- ﹣ ------ ( 3 6 )

Key Concepts

Cosine of the Difference of Two Angles

Unit Circle

Exact Values of Trigonometric Functions

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