If tanθ=3√3 , which of the following is a possible coordinate pair? (1 point) Responses (−12,−3√2) left parenthesis negative Start Fraction 1 over 2 End Fraction comma negative Start Fraction Start Root 3 End Root over 2 End Fraction right parenthesis (−3√2,12) left parenthesis negative Start Fraction Start Root 3 End Root over 2 End Fraction comma Start Fraction 1 over 2 End Fraction right parenthesis (3√2,12) left parenthesis Start Fraction Start Root 3 End Root over 2 End Fraction comma Start Fraction 1 over 2 End Fraction right parenthesis (12,3√2)
To find the possible coordinate pairs given that tanθ=3√3, we need to find the values of sinθ and cosθ.
By using the Pythagorean identity, we have sin²θ + cos²θ = 1.
Since tanθ = 3√3 = sinθ/cosθ, we can square both sides to get:
(3√3)² = (sinθ/cosθ)²
27 = sin²θ/cos²θ
Since sin²θ + cos²θ = 1, we can substitute sin²θ with 1 - cos²θ in the equation above:
27 = (1 - cos²θ)/cos²θ
Simplifying the equation, we have:
27cos²θ = 1 - cos²θ
28cos²θ = 1
cos²θ = 1/28
cosθ = ±√(1/28)
cosθ = ±√(1/4*7)
cosθ = ±(1/2)√(1/7)
cosθ cannot be negative since the tangent is positive, so cosθ = (1/2)√(1/7).
Since tanθ = 3√3 = sinθ/cosθ, we can find sinθ:
sinθ = tanθ * cosθ
sinθ = 3√3 * (1/2)√(1/7)
sinθ = (3/2)√(3/7).
Now we can check which of the given coordinate pairs satisfy tanθ = 3√3.
Checking each coordinate pair:
1) (-12, -3√2):
tanθ = y/x = -3√2 / -12 = √2/4
tanθ is not equal to 3√3, so (-12, -3√2) is not a possible coordinate pair.
2) (-3√2, 12):
tanθ = y/x = 12 / (-3√2) = -2√2
tanθ is not equal to 3√3, so (-3√2, 12) is not a possible coordinate pair.
3) (3√2, 12):
tanθ = y/x = 12 / (3√2) = 2√2
tanθ is not equal to 3√3, so (3√2, 12) is not a possible coordinate pair.
4) (12, 3√2):
tanθ = y/x = 3√2 / 12 = √2/4
tanθ is not equal to 3√3, so (12, 3√2) is not a possible coordinate pair.
Therefore, none of the given coordinate pairs is a possible solution.