Write each function as an expression involving functions of θ&...

Question

Write each function as an expression involving functions of θ or x alone. See Example 2.

sin (3π/4 - x)

Accepted Answer

Hey, everyone in this problem, we're asked to write the given trigonometric expression in terms of alpha only. So we have sine of two pi divided by three minus alpha. We want to take this and write it so that our trigonometric functions are only in terms of alpha. We have four answer choices. Option A, the square root of three multiplied by cosine of alpha plus sine of alpha all divided by two. Option B is the same as A but we have minus instead of plus in the numerator. Option C cosine of alpha plus the square root of three multiplied by sine of alpha all divided by two. And then option D is the same as C but again, with a negative A minus sine instead of A plus sign in that numerator. All right. So we have sign of something minus something. Now recall we have these trigonometric identities for the difference. So we can write sign of A minus B A sign of A multiplied by cosine FB minus cosine of a multiplied by sine of B. OK. So the A term is what we have is two pi divided by three. The B term is our alpha. OK? And the left hand side sign of A minus B is exactly what we were given. OK. So if we take this expression, we can write it as the expression on the right hand side. Now we have to figure out if that's going to help us. Well, we're gonna get sine of two pi divided by three. And that's a value. We can figure out what that value is. Then we get cosine of alpha, we get cosine of two pi divided by three. Again, we can figure out what that value is and then we have sine of alpha. So all of our trigonometric functions will only be in terms of alpha, which is exactly what we want. So this is the identity that we're gonna use. So we are gonna have our, a value two pi divided by three and our B value of alpha, we start with our original expressions of two pi divided by three minus alpha. And we can write this as sign of two pi divided by three, multiplied by cosine alpha minus cosine of two pi divided by three, multiplied by sine of alpha. All right. So what we need to do is figure out these values of sine of pi divided by three and cosine of or sorry sine of two pi divided by three and cosine of two pi divided by three. So let's go ahead and draw these out to get a better sense. OK. Because they are not angles between zero and 90 degrees. This thing is bigger than 90 degrees. It's bigger than pi divided by two. So it might not be as intuitive or might not be of value that you've remembered. OK. So let's try this out. We have our quadrants and this thing is going to lie in quadrant two and we have this angle that it makes with the positive x axis of two pi divided by three. Now, because we're in quadrant two, we know that sine is going to be positive and cosine is going to be negative. All right now, to determine the values, we can look at our reference angle and the reference angle, it's gonna be this angle that goes from our line we've drawn at two pi divided by three to the negative X axis. We're gonna call this the prime. OK. So our reference angle the prime, well, this is gonna be equal to pi minus two pi divided by three maybe because if we go all the way from the positive X axis to the negative X axis, that's pi and then we're subtracting that two pi divided by three that we already have to get the remainder. OK. So this gives us a reference angle of pi divided by three, no, between the reference angle and knowing what quadrant we're in and the sign we can write out these values. So sign of two pi divided by three. OK. Well, we know that sine of pi divided by three is a square root of three divided by two. We have that reference angle. We're in a quadrant where sine is positive. So we're just gonna get the square root of three divided by two. Now, for cosine, it's very similar cosine of two pi divided by three, we know that with that reference angle, we get a value cosine of pi divided by three of one half. But because we're in quadrant two cosine is negative and so we get a value of negative one half, all right. So we have the value of sine and cosine now at that particular angle and we can simplify our expression. So now we get that signed of two pi divided by three minus alpha is going to be equal to the square root of three divided by two, multiplied by cosine of alpha minus negative one half multiplied by sine alpha. OK. We have a denominator of two in both of these terms. So we can write this as one single fraction just like the answer choices. Did you can write this as a square root of three cosine of alpha, we have minus negative half that gives us a positive. So we have plus sign of alpha all divided by two. And that is our expression written so that these trigonometric functions are in terms of alpha only. Everything else is just multiples um scalar multiples being multiplied together. OK. Let's compare this to our answer choices. And what we can see is that the answer we found corresponds with option A, the square root of three multiplied by cosine of alpha, left sine of alpha all divided by two. That's it for this one. Thanks everyone for watching. I hope this video helped.

Write each function as an expression involving functions of θ or x alone. See Example 2.
sin (3π/4 - x)

Key Concepts

Angle Difference Identity for Sine

Exact Values of Trigonometric Functions at Special Angles

Function Notation and Variable Isolation

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