given cotθ = (1/2) √7, find sinθ and cosθ in quadrant I
cotθ = x/y
would 1/2 be the x and √7 be the y?
To find sinθ and cosθ in quadrant I given cotθ, we can use the following trigonometric identities:
1. cotθ = cosθ / sinθ
From the given equation:
cotθ = (1/2) √7
We can rewrite it as:
cosθ / sinθ = (1/2) √7
Cross-multiplying:
2 cosθ = √7 sinθ
Dividing both sides by sinθ:
2 cosθ / sinθ = √7
Recall another trigonometric identity:
sin^2θ + cos^2θ = 1
Dividing both sides by sin^2θ:
1 + cot^2θ = csc^2θ
Substituting cotθ with the given value:
1 + (1/2)^2 √7^2 = csc^2θ
Simplifying:
1 + 1/4 * 7 = csc^2θ
1 + 7/4 = csc^2θ
15/4 = csc^2θ
Taking the square root of both sides:
√(15/4) = √(csc^2θ)
√15 / 2 = cscθ
Recall another trigonometric identity:
cscθ = 1 / sinθ
Therefore:
√15 / 2 = 1 / sinθ
Cross-multiplying:
√15 * sinθ = 2
Dividing both sides by √15:
sinθ = 2 / √15
To find cosθ, we can substitute the value of sinθ into the equation:
sin^2θ + cos^2θ = 1
(2 / √15)^2 + cos^2θ = 1
4/15 + cos^2θ = 1
cos^2θ = 1 - 4/15
cos^2θ = 11/15
Taking the square root of both sides:
cosθ = √(11/15)
Therefore, in quadrant I:
sinθ = 2 / √15
cosθ = √(11/15)