If tan theta equal - square root of 22 divided by 11 and pie/2 is less than theta and theta is less than pies, what is the cos of theta in simplified rationalized form
Negative square root of 22*
Grrrr! pi, not pie!!! Pie is what you eat.
Other than that, why not use real math symbols, instead of all those messy words??
tan θ = -√22/11 and π/2 < π < π
Draw the triangle in standard position.
tanθ = y/x = √22/-11
sinθ = y/r
cosθ = x/r
You will see that
y = √22
x = -11
r = √143
Now you can easily evaluate cosθ and sinθ
To find the value of cos(theta), we will use the relationship between cosine and tangent:
cos(theta) = 1 / sqrt(1 + tan^2(theta))
Given that tan(theta) = -sqrt(22)/11, we can substitute this value into the formula:
cos(theta) = 1 / sqrt(1 + (-sqrt(22)/11)^2)
Now, let's simplify this expression by evaluating the square:
cos(theta) = 1 / sqrt(1 + 22/121)
To combine the terms in the square root, we need to find a common denominator:
cos(theta) = 1 / sqrt((121 + 22) / 121)
cos(theta) = 1 / sqrt(143 / 121)
Next, we can simplify the square root by taking the square root of the numerator and the denominator:
cos(theta) = 1 / (sqrt(143) / sqrt(121))
cos(theta) = 1 / (sqrt(143) / 11)
To rationalize the denominator, we can multiply both the numerator and denominator by the conjugate of sqrt(143) / 11, which is sqrt(143) / 11:
cos(theta) = (1 / (sqrt(143) / 11)) * (sqrt(143) / 11)
cos(theta) = sqrt(143) / 11 * 11 / sqrt(143)
The square roots cancel each other out:
cos(theta) = 1
Therefore, the cos(theta) in simplified rationalized form is 1.