# Express cos(2 tan^-1(x/4)) and sin(2tan^-1(x/4) as an algebraic expression in x

• MHB
• Elissa89
In summary, we discussed how to express cos(2 tan^-1(x/4)) and sin(2tan^-1(x/4)) as algebraic expressions in x. We used the identity formulas for double angles and substituted in the values for cos(t) and sin(t) in terms of x to simplify the expressions. We also clarified that 2tan^-1(x/4) is not the same as 2tan(theta)=x/4 and explained the difference.

#### Elissa89

So my professor gave us a study guide for the final but no there is no answer key. Could someone check my answers please?

Express cos(2 tan^-1(x/4)) and sin(2tan^-1(x/4) as an algebraic expression in x

I got:

cos(theta)=8*sqrt(x^2+64)/x^2+64

sin(theta)=x*sqrt(x^2+64)/x^2+64

Elissa89 said:
So my professor gave us a study guide for the final but no there is no answer key. Could someone check my answers please?

Express cos(2 tan^-1(x/4)) and sin(2tan^-1(x/4) as an algebraic expression in x

I got:

cos(theta)=8*sqrt(x^2+64)/x^2+64

sin(theta)=x*sqrt(x^2+64)/x^2+64

Let $t = \tan^{-1}\left(\dfrac{x}{4}\right) \implies \tan{t} = \dfrac{x}{4}, \, \cos{t} = \dfrac{4}{\sqrt{x^2+16}}, \, \sin{t} = \dfrac{x}{\sqrt{x^2+16}}$$\cos(2t) = 2\cos^2{t}-1 \sin(2t) = 2\sin{t}\cos{t} take it from here? skeeter said: Let t = \tan^{-1}\left(\dfrac{x}{4}\right) \implies \tan{t} = \dfrac{x}{4}, \, \cos{t} = \dfrac{4}{\sqrt{x^2+16}}, \, \sin{t} = \dfrac{x}{\sqrt{x^2+16}}$$\cos(2t) = 2\cos^2{t}-1$

$\sin(2t) = 2\sin{t}\cos{t}$

take it from here?

but its 2*tan^-1(x/4). Isn't that the same as 2*tan(theta)=x/4. So wouldn't I divide both sides by 2 and get x/8 and go from there?

Elissa89 said:
but its 2*tan^-1(x/4). Isn't that the same as 2*tan(theta)=x/4. So wouldn't I divide both sides by 2 and get x/8 and go from there?

no.

$\theta = 2\tan^{-1}\left(\dfrac{x}{4}\right) \implies \dfrac{\theta}{2} = \tan^{-1}\left(\dfrac{x}{4}\right) \implies \dfrac{x}{4} = \tan\left(\dfrac{\theta}{2}\right)$

note $\tan^{-1}\left(\dfrac{x}{4}\right)$ is an angle and $2 \tan^{-1}\left(\dfrac{x}{4}\right)$ is double that angle

## 1. What is the algebraic expression for cos(2tan^-1(x/4))?

The algebraic expression for cos(2tan^-1(x/4)) is (1 - x^2/16) / (1 + x^2/16).

## 2. How can sin(2tan^-1(x/4)) be expressed algebraically?

Sin(2tan^-1(x/4)) can be expressed algebraically as (2x) / (1 + x^2/16).

## 3. Can the expressions be simplified further?

Yes, the expressions can be simplified by factoring out a common factor of (1 + x^2/16) from both the numerator and denominator.

## 4. What is the domain of the expressions?

The domain of both expressions is all real numbers, as there are no restrictions on the input value x.

## 5. Can these expressions be graphed?

Yes, these expressions can be graphed using a graphing calculator or online graphing tool. The graphs will show a periodic function with a range of -1 to 1.