Use the identities for the cosine of a sum or difference to wr...

Question

Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.

cos(θ - 270°)

Accepted Answer

Hello, today we'll be simplifying the following trigonometric expression in terms of the, and we'll be using the difference of cosine to help us do that. So what is the difference of cosine? Well, the difference of cosine is given to us as cosine of A minus B is equal to cosine of A multiplied by cosine of B plus sine of A multiplied by S of B. So in order to use this identity, we'll first need to figure out the values of A and B if we look back at the trigonometric expression given to us 180 degrees is in the spot of A. So we're going to allow A to equal to 180 degrees. Furthermore, theta sits in the spot of B. So we're going to allow B to equal to theta. So using the difference of cosine, we can expand the trigonometric expression cosine of 180 degrees minus the as cosine of A which is 180 degrees. So cosine of 180 multiplied by cosine of B which is the, so cosine theta plus sine of A which is S of 180 degrees multiplied by S of B which will give a sign of the. Now, in order to simplify this equation, we're going to have to evaluate cosine of 180 degrees and sine of 180 degrees. In order to do this, we're going to need the help of a unit circle. So what we'll need to do is we'll need to locate the 180 degree angle on the unit circle. The 180 degree angle is located on the left most horizontal axis of the unit circle. And the terminal point at this angle is given to us as negative 10. Now recall that cosine represents any X value on the unit circle and sine represents any Y value of the unit circle. So what this means is that cosine of 180 degrees is equal to the X value of the terminal point of the angle 180 degrees on the unit circle. The X value of that terminal point is negative one. So cosine of 180 degrees will evaluate to give us negative one. Furthermore, since sine is going to be the Y value of the terminal point at 180 degrees of the unit circle that Y value is going to be zero. So sine of 180 degrees will evaluate to give us zero. We can simplify the equation to be negative one multiplied by cosine of theta plus zero, multiplied by sign of the. Now anything multiplied by zero will give us zero. So zero, multiplied by sine theta will simplify to be zero, negative one multiplied by cosine of theta will simplify to give us negative cosine of theta. So this equation will simplify to be negative cosine of theta plus zero, which itself will simplify to be negative cosine of theta. And this is going to be the simplified form of cosine of 180 degrees minus theta. 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 use the difference formula for cosign. And I'll go ahead and see you while in the next video.

Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.
cos(θ - 270°)

Key Concepts

Cosine of a Difference Identity

Values of Trigonometric Functions at Special Angles

Simplification of Trigonometric Expressions

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