suppose tan theta equal 4 and theta lies in the third quadrant. find the values of sec theta
sec^2θ = 1+tan^2θ = 17
since secθ < 0 in QIII,
secθ = -√17
To find the value of sec theta when tan theta is given, we can use the relationship between secant and tangent.
The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be represented as the reciprocal of the cotangent function.
In the third quadrant, the value of tangent is positive, while the values of cosine and sine are negative. Therefore, we can determine the values of cosine and sine based on the given information.
Given that tan theta = 4, we can determine the values of cosine and sine using Pythagorean identity:
tan^2 theta = sin^2 theta / cos^2 theta
Since tan theta = 4, we can square both sides of the equation:
(4)^2 = sin^2 theta / cos^2 theta
16 = sin^2 theta / cos^2 theta
Now, let's consider the relationship between secant and cosine:
sec theta = 1 / cos theta
We can find cos theta based on the equation derived from Pythagorean identity:
cos^2 theta = 1 / (1 + tan^2 theta)
cos^2 theta = 1 / (1 + 4^2)
cos^2 theta = 1 / 17
Taking the square root of both sides:
cos theta = ± √(1/17)
Now, let's calculate sec theta:
sec theta = 1 / cos theta
If cos theta = √(1/17), then sec theta = 1 / √(1/17).
Simplifying further, sec theta = √17 / 1.
Since theta lies in the third quadrant, both sine and cosine are negative. Therefore, the final value of sec theta is - √17.
Hence, the value of sec theta when tan theta equals 4 and theta lies in the third quadrant is - √17.