1) given cotθ = 1/2√7, find sinθ and cosθ in quadrant I
2) given tanθ = √5, find secθ and cotθ in quadrant III
recall that
sinθ = y/r
cosθ = x/r
tanθ = y/x
r^2 = x^2+y^2
So, in QIII, where x and y are both negative,
tanθ = √5 = √5/1, so
y = -√5
x = -1
r = √6
cotθ = x/y = 1/√5
secθ = r/x = -√6
For #1, do you mean (1/2) √7 or 1/(2√7)?
In any case, follow the steps above.
for #1 i mean (1/2) √7
which 1/2 be the x and √7 be the y?
would*
so, cotθ = (√7)/2, meaning
x = √7
y = 2
since cotθ = x/y
To find sinθ, cosθ, secθ, and cotθ, we can first use the given information to find the value of tanθ. Then, we can use the trigonometric identity to find the other trigonometric functions.
1) Given cotθ = 1/(2√7), we can rewrite cotθ as 1/tanθ.
First, find the value of tanθ:
tanθ = 1/cotθ
Substituting cotθ = 1/(2√7):
tanθ = 1/(1/(2√7))
tanθ = 2√7
Now, we can use the Pythagorean identity to find sinθ and cosθ:
tanθ = sinθ/cosθ
Substituting tanθ = 2√7:
2√7 = sinθ/cosθ
Squaring both sides:
(2√7)^2 = (sinθ/cosθ)^2
28 = (sinθ)^2/(cosθ)^2
28(cosθ)^2 = (sinθ)^2
Since we are in quadrant I, both sinθ and cosθ are positive, so we can take the square root:
√28(cosθ) = sinθ
Now, to find the value of √28, we can simplify it further:
√28 = √(4 * 7) = 2√7
Now substitute √28 with 2√7:
2√7(cosθ) = sinθ
So sinθ = 2√7 and cosθ = √(1 - sin^2θ) = √(1 - (2√7)^2) = √(1 - 4 * 7) = √(1 - 28) = √(-27). However, since we are in quadrant I, where cosθ is positive, this is not a valid solution. Hence, we cannot find the value of cosθ in quadrant I using the given information.
2) Given tanθ = √5, we can find the value of cosθ using the Pythagorean identity:
tanθ = sinθ/cosθ
Substituting tanθ = √5:
√5 = sinθ/cosθ
To find secθ, we can use the reciprocal identity:
secθ = 1/cosθ
Since tanθ = sinθ/cosθ, we can rewrite it as:
cosθ = sinθ/tanθ
Substituting tanθ = √5:
cosθ = sinθ/√5
Squaring both sides:
(cosθ)^2 = (sinθ)^2/5
Using the Pythagorean identity:
(cosθ)^2 + (sinθ)^2 = 1
Substituting (sinθ)^2/5 for (cosθ)^2:
(sinθ)^2/5 + (sinθ)^2 = 1
Multiplying both sides by 5:
(sinθ)^2 + 5(sinθ)^2 = 5
Simplifying:
6(sinθ)^2 = 5
Dividing by 6:
(sinθ)^2 = 5/6
Taking the square root of both sides:
sinθ = ±√(5/6)
Since we are in quadrant III, sinθ is negative, so we take the negative square root:
sinθ = -√(5/6)
To find secθ, we can substitute sinθ into cosθ = sinθ/tanθ:
cosθ = (-√(5/6))/√5 = -√(1/6)
Finally, to find cotθ, we can use the reciprocal of tanθ:
cotθ = 1/tanθ = 1/√5