Given sin θ = −1/2 and sec θ > 0, find the exact value of tan 2θ.
Radical 3
Radical 3/2
Negative radical 3
Negative radical 3/2
since cosθ > 0, we are in QIV.
cosθ = √3/2
even without figuring it, we have θ = -π/6, so tan -π/3 = -√3
or, we have
sin2θ = 2(-1/2)(√3/2) = -√3/2
cos2θ = 2(3/4)-1 = 1/2
tan2θ = -√3
or, since tanθ = -1/√3,
tan2θ = 2(-1/√3)/(1-1/3) = -√3
To find the exact value of tan 2θ, we can use the trigonometric identity for tangent of double angle:
tan 2θ = 2tan θ / (1 - tan² θ)
Given that sin θ = -1/2 and sec θ > 0, we can find the values of the trigonometric functions involved.
First, we know that sin θ = -1/2. Since sin θ = opposite/hypotenuse, we can imagine a right triangle with the opposite side length as -1 and the hypotenuse as 2.
Using the Pythagorean theorem:
(opposite)² + (adjacent)² = (hypotenuse)²
(-1)² + (adjacent)² = 2²
1 + (adjacent)² = 4
(adjacent)² = 3
So, the adjacent side has a length of √3.
We are also given that sec θ > 0. Since sec θ = 1/cos θ, this implies that cos θ > 0. In the given triangle, the adjacent side is positive. Therefore, cos θ = adjacent/hypotenuse = √3/2.
Now, we can determine the value of tan θ using the identity tan θ = sin θ / cos θ:
tan θ = (-1/2) / (√3/2) = -1/√3 = -√3/3.
Finally, we can substitute this value into the formula for tan 2θ:
tan 2θ = 2tan θ / (1 - tan² θ)
= 2(-√3/3) / (1 - (-√3/3)²)
= -2√3 / (1 - 3/3)
= -2√3 / (1 - 1)
= -2√3 / 0
Since we cannot divide by zero, the value of tan 2θ is undefined. None of the answer choices provided are correct.