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

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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.
  • #1
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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
 
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  • #2
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?
 
  • #3
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?
 
  • #4
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.

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