Blake is told that an angle has a cosine of −1/6√ and is in Quadrant III. Use the Pythagorean identity sin2(θ)+cos2(θ)=1 , the trigonometric identity tanθ=sinθcosθ , and the quadrant to find the tangent. (1 point)
To find the tangent, we can use the trigonometric identity tanθ=sinθ/cosθ.
Given that the angle is in Quadrant III, we know that both the sine and the cosine are negative.
We have the cosine given as -1/6√. To find the sine, we can use the Pythagorean identity sin^2(θ)+cos^2(θ)=1.
Let's assume the sine is represented by x:
x^2 + (-1/6√)^2 = 1
x^2 + 1/36 = 1
x^2 = 1 - 1/36
x^2 = 36/36 - 1/36
x^2 = 35/36
x = √(35/36)
x = √35/√36
x = √35/6
So, the sine is √35/6.
Using the trigonometric identity tanθ=sinθ/cosθ, we can substitute the values:
tanθ = (√35/6) / (-1/6√)
tanθ = (√35/6) * (-√/1)
tanθ = -√(35/1 * 1/36)
tanθ = -√(35/36)
tanθ = -√35/6
Therefore, the tangent of the angle is -√35/6.