if x=4 tan theta, express cos(2 theta) as a function of x
cos 2t = 2cos^2(t) - 1 = 2/sec^2(t) - 1
sec^2 = 1 + tan^2
2/(1 + tan^2(t)) - 1
= [2 - (1 + tan^2 t)]/(1 + tan^2 t)
x = 4tan t
tan t = x/4
[2 - ( 1 + x^2 / 16)]/(1 + x^2 / 16)
(32 - 1 - x^2)/(16 + x^2)
(31 - x^2)/(16 + x^2)
The answer above looked wrong to me, and it didn't take long to see that it was so.
cos 2t = 2 cos^2 t - 1
= 2/sec^2 t - 1
= (2 - sec^2 t) / sec^2 t
= (2 - (1 + tan^2 t))/(1 + tan^2 t)
= (1 - tan^2 t)/(1 + tan^2 t)
= (1 - x^2/16)/(1 + x^2/16)
= (16 - x^2)/(16 + x^2)
To express cos(2θ) as a function of x, first, we need to find cos(θ). We know that x = 4 tan(θ).
Dividing both sides of the equation by 4, we get:
x/4 = tan(θ)
Now, we can use the trigonometric identity:
tan^2(θ) + 1 = sec^2(θ)
Substituting x/4 for tan(θ), we get:
(x/4)^2 + 1 = sec^2(θ)
Simplifying this equation:
x^2/16 + 1 = sec^2(θ)
Now, we can rewrite sec^2(θ) as 1/cos^2(θ):
x^2/16 + 1 = 1/cos^2(θ)
Solving for cos^2(θ):
cos^2(θ) = 1 / (x^2/16 + 1)
Next, to find cos(2θ), we use the identity:
cos(2θ) = cos^2(θ) - sin^2(θ)
Since we only have cos^2(θ), we need to find sin^2(θ). Using the Pythagorean identity:
sin^2(θ) + cos^2(θ) = 1
Rearranging the equation:
sin^2(θ) = 1 - cos^2(θ)
Now, substituting the value of cos^2(θ) we found earlier:
sin^2(θ) = 1 - (1/(x^2/16 + 1))
Finally, we can substitute the values of sin^2(θ) and cos^2(θ) in the expression for cos(2θ):
cos(2θ) = cos^2(θ) - sin^2(θ)
cos(2θ) = [1 / (x^2/16 + 1)] - [1 - (1/(x^2/16 + 1))]
Simplifying further, we get:
cos(2θ) = [1 / (x^2/16 + 1)] - [1 - (1/(x^2/16 + 1))]
To express cos(2 theta) as a function of x, we need to find an expression that relates cos(2 theta) to x using the given information x = 4 tan(theta).
First, recall the double angle formula for cosine:
cos(2 theta) = cos^2(theta) - sin^2(theta)
To use this formula, we need to find sin(theta). We can do that by using the trigonometric identity:
sin^2(theta) + cos^2(theta) = 1
Rearranging the identity, we get:
sin^2(theta) = 1 - cos^2(theta)
Substituting this expression for sin^2(theta) into the double angle formula for cosine, we have:
cos(2 theta) = cos^2(theta) - (1 - cos^2(theta))
Expanding the expression, we get:
cos(2 theta) = cos^2(theta) - 1 + cos^2(theta)
Combining like terms, we find:
cos(2 theta) = 2 cos^2(theta) - 1
Now, recall the given information x = 4 tan(theta).
Using the trigonometric identity, tan(theta) = sin(theta)/cos(theta), we can rearrange it to solve for sin(theta):
sin(theta) = tan(theta) * cos(theta) = (x/4) * cos(theta)
Substituting this expression for sin(theta) into the expression for cos(2 theta), we get:
cos(2 theta) = 2 cos^2(theta) - 1
= 2 * (1 - sin^2(theta)) - 1
= 2 * (1 - (x^2/16) * cos^2(theta)) - 1
Simplifying the expression:
cos(2 theta) = 2 - (x^2/8) * cos^2(theta)
Therefore, we have expressed cos(2 theta) as a function of x:
cos(2 theta) = 2 - (x^2/8) * cos^2(theta)