Use half angle identities to find exact value.
Given tan theta = -sqrt 2 / 2 and theta lies in quadrant IV, find cos theta / 2.
so, go find the appropriate half-angle rule. What do you get?
Just to get you started, for that θ in QIV,
sinθ = -1/√2
cosθ = 1/√2
To find the exact value of cos(theta/2), we need to utilize the half angle identity for cosine. The half angle identity for cosine states that:
cos(theta/2) = ± sqrt((1 + cos(theta)) / 2)
Given that theta lies in quadrant IV, where the tangent is negative, we can determine that sine and cosine are both positive. Therefore, to find the exact value of cos(theta/2), we will use the positive sign in the half angle identity.
First, we need to find the value of cos(theta). We can use the given value of tan(theta) to find this.
Since tangent is the ratio of sine to cosine, we have:
tan(theta) = sin(theta) / cos(theta)
Given that tan(theta) = -sqrt(2) / 2, we can substitute this value into the equation:
(-sqrt(2) / 2) = sin(theta) / cos(theta)
Next, we can solve for sin(theta) using the Pythagorean identity:
sin^2(theta) + cos^2(theta) = 1
Rearranging this equation, we have:
sin^2(theta) = 1 - cos^2(theta)
Substituting this equation into the first equation, we get:
(-sqrt(2) / 2) = (sqrt(1 - cos^2(theta))) / cos(theta)
Now we can solve for cos(theta).
Multiplying both sides by cos(theta) to eliminate the denominator, we have:
(-sqrt(2) / 2) * cos(theta) = sqrt(1 - cos^2(theta))
Square both sides of this equation, we get:
(cos^2(theta) * 2 / 4) = 1 - cos^2(theta)
Simplifying this equation, we have:
2cos^2(theta) = 4 - 4cos^2(theta)
Combining like terms, we get:
6cos^2(theta) = 4
Dividing both sides by 6, we have:
cos^2(theta) = 4/6
Simplifying further, we get:
cos^2(theta) = 2/3
Taking the square root of both sides of this equation, we have:
cos(theta) = ± sqrt(2/3)
Since theta lies in quadrant IV, cosine is positive. Therefore, we have:
cos(theta) = sqrt(2/3)
Now we can substitute this value into the half angle identity for cosine:
cos(theta/2) = sqrt((1 + cos(theta)) / 2)
cos(theta/2) = sqrt((1 + sqrt(2/3)) / 2)
This is the exact value of cos(theta/2) given the information provided.