Find the values of the six trigonometric functions of θ. (If an answer is undefined, enter UNDEFINED.)
Function Value: csc(θ) = 4
Constraint: cot(θ) < 0
1) sin(θ)=
2) cos(θ)=
3) tan(θ)=
4) csc(θ)=
5) sec(θ)=
6) cot(θ)=
1) To find sin(θ), we can use the reciprocal identity of csc(θ). The reciprocal of csc(θ) is sin(θ), so sin(θ) = 1/csc(θ). Therefore,
sin(θ) = 1/4
2) To find cos(θ), we can use the Pythagorean identity. The Pythagorean identity is sin²θ + cos²θ = 1. Rearranging the equation, we get cos²θ = 1 - sin²θ. Substituting sin(θ) = 1/4, we can find cos(θ) as follows:
cos²θ = 1 - (1/4)²
cos²θ = 1 - 1/16
cos²θ = 15/16
Taking the square root of both sides, cos(θ) = ±√(15/16). Since the constraint cot(θ) < 0, θ must be in the third quadrant where cos(θ) is negative. Therefore,
cos(θ) = -√(15/16) = -√15/4
3) To find tan(θ), we can use the identity tan(θ) = sin(θ)/cos(θ). Substituting the values we found in steps 1 and 2:
tan(θ) = sin(θ)/cos(θ)
tan(θ) = (1/4)/(-√15/4)
tan(θ) = -1/√15
4) Given csc(θ) = 4, we already know this value.
csc(θ) = 4
5) To find sec(θ), we can use the reciprocal identity of cos(θ). The reciprocal of cos(θ) is sec(θ), so sec(θ) = 1/cos(θ). Therefore,
sec(θ) = 1/(-√15/4)
sec(θ) = -4/√15
6) Given the constraint cot(θ) < 0, we know that cot(θ) is negative. The reciprocal identity of tan(θ) is cot(θ), so cot(θ) = 1/tan(θ). Therefore,
cot(θ) = 1/(-1/√15)
cot(θ) = -√15
To find the values of the six trigonometric functions of θ, we need to use the given function value and constraint.
1) To find sin(θ), we know that sin(θ) is equal to 1 divided by csc(θ). Since csc(θ) is given as 4, we can calculate sin(θ) as 1/4.
2) To find cos(θ), we know that cos(θ) is equal to 1 divided by sec(θ). However, sec(θ) is not provided directly, so we need to find it using another trigonometric function. We can use the Pythagorean identity to find sec(θ).
The Pythagorean identity states: sin²(θ) + cos²(θ) = 1
Rearranging the equation, we get: cos²(θ) = 1 - sin²(θ)
Plugging in the value of sin(θ) we found earlier (1/4), we have: cos²(θ) = 1 - (1/4) = 3/4
Since cos(θ) is positive (based on the given constraint), we take the positive square root of 3/4, which is √3/2.
3) To find tan(θ), we can use the identities:
tan(θ) = sin(θ) / cos(θ)
Plugging in the values we found earlier, we have: tan(θ) = (1/4) / (√3/2)
To simplify this, we multiply the numerator and denominator by the reciprocal of the denominator: tan(θ) = (1/4) * (2/√3)
Simplifying further, we get: tan(θ) = 2/4√3 = 1/2√3 = (√3) / (2*3) = √3 / 6
4) Given: csc(θ) = 4
5) To find sec(θ), we can use the identity: sec(θ) = 1 / cos(θ). Using the value we found for cos(θ) earlier (√3/2), we have: sec(θ) = 1 / (√3/2)
To simplify this, we multiply the numerator and denominator by the reciprocal of the denominator: sec(θ) = (1 * 2) / √3
Simplifying further, we get: sec(θ) = 2 / √3 = (2√3) / (3√3) = 2√3 / 3
6) Given: cot(θ) < 0
1. csc A = 4.
1/sinA = 4,
sinA = 1/4 = Y/r,
2. x^2 + y^2 = 4^2.
x^2 + 1^2 = 16,
X = sqrt(15).
Cos A = X/r = sqrt(15)/4.
3. Tan A = Y/X = 1/sqrt(15).
4. CSC A = r/Y = 4/1 = 4.
5.
6.