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.

tan (π/4 + x)

Accepted Answer

Hey, everyone in this problem, we're asked to write the given trigonometric expression in terms of alpha only. And the expression we have is tangent of pi divided by three plus alpha. We have four answer choices. Option A, the square root of three plus tangent of alpha, all divided by one minus the square root of three multiplied by tangent of alpha. Option B is the same as option A. So the plus sign in the numerator is now a minus sign. Option C OK. This is the same as option A except both signs are switched. So in the numerator, we have a negative sign in the denominator, we have an addition sign. And in option D again, we have that same basic expression. But now we have plus signs in both the numerator and the denominator. All right. So we have tangent of two things added together. We wanna get this in terms of alpha only. So we wanna kind of break this tangent function up. So recall that we have the following trigonometric identity tangent of A plus B can be written as tangent of A plus tangent of B all divided by one minus tangent of bay tangent of feet on the left hand side is exactly what we have. OK. So we have that A is equal to pi divided by three and B is equal to alpha. OK? And then we can use this expression to break our function up or our expression up into this right hand side. Well, let's go ahead and do it tangent of pie divided by three plus alpha. It's gonna be equal to tangent of pi divided by three was tangent alpha, all divided by one minus tangent of pi divided by three multiplied by tangent of alpha. All right. So we've written this out now and now it's just a matter of simplifying. OK, we have these tangent of alpha terms. So it's a trigonometric function only in terms of alpha, which is exactly what we want. Everything else tangent of pi divided by three, we can get an exact value for that. So let's do that. Now, tangent of pi divided by three. Can I recall that we can write this as signed of pi divided by three, divided by co-signed of pi divided by three. Now sine of pi divided by three, this is a value that you're gonna wanna remember or be able to figure out quickly. OK? Whether you use an identity triangle or a unit circle, a sine of pi divided by three is the square root of three divided by two. And cosine of pi divided by three is one half. That means that tangent of pi divided by three is just going to be equal to the square root of three. Hey, those denominators of two are going to divide it. So we have now our value of tangent of pi divided by three. So we can get back to our full expression and simplify, OK, we have that tangent of pi divided by three plus alpha is going to be equal to the square root of three plus tangent of alpha divided by one minus the square root of three multiplied by tangent of alpha. OK. And this is exactly the expression. We want four answer choices if we compare what we found two of those answer choices. Can we have a plus term in the numerator, a subtraction in the denominator? And so our answer is going to match with option A, the square root of three plus tangent of alpha divided by one minus the square root of three multiplied by tangent of alpha. Thanks everyone for watching. I hope this video helped see you in the next one.

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

Key Concepts

Angle Sum Identity for Tangent

Value of Tangent at Special Angles

Simplification of Rational Expressions

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