Find the value of each of the remaining trig functions.
Cscè=-4, ð<è<3ð/2
To find the values of the remaining trigonometric functions given the value of csc(θ), we can use the reciprocal identities and the Pythagorean identity.
Given: csc(θ) = -4
First, we can find the value of sine (sin(θ)). Recall that the reciprocal of csc is sin:
csc(θ) = 1/sin(θ)
So, we have:
1/sin(θ) = -4
To solve for sin(θ), we can take the reciprocal of both sides:
sin(θ) = 1/(-4)
Therefore, sin(θ) = -1/4.
Next, we can use the Pythagorean identity to find the value of cosine (cos(θ)). The Pythagorean identity states:
sin^2(θ) + cos^2(θ) = 1
Since we know sin(θ) = -1/4, we can substitute this value into the equation:
(-1/4)^2 + cos^2(θ) = 1
Simplifying the equation:
1/16 + cos^2(θ) = 1
We can solve for cos(θ):
cos^2(θ) = 1 - 1/16
cos^2(θ) = 15/16
Taking the square root of both sides, remember to consider both the positive and negative square root:
cos(θ) = ± √(15/16)
Since θ is in the third quadrant (θ < 3π/2), cosine is negative, so we have:
cos(θ) = -√(15/16) = -√15/4
Lastly, we can find the remaining trigonometric functions using the previously found values of sine and cosine:
tan(θ) = sin(θ)/cos(θ) = (-1/4) / (-√15/4) = 1/√15
cot(θ) = 1/tan(θ) = √15
sec(θ) = 1/cos(θ) = 1 / (-√(15/16)) = -4/√15
Therefore, the values of the remaining trigonometric functions are:
sin(θ) = -1/4
cos(θ) = -√15/4
tan(θ) = 1/√15
cot(θ) = √15
sec(θ) = -4/√15
To find the values of the remaining trigonometric functions given that csc(θ) = -4 and the angle θ is between θ and 3π/2, we can use the reciprocal identities.
Reciprocal identities:
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
cot(θ) = 1/tan(θ)
Given that csc(θ) = -4, we can rewrite it as 1/sin(θ) = -4. Solving for sin(θ), we get:
sin(θ) = 1/(-4)
sin(θ) = -1/4
Now, to find the value of the remaining trigonometric functions, we can use the Pythagorean identity:
sin^2(θ) + cos^2(θ) = 1
Substituting sin(θ) = -1/4 into the equation:
(-1/4)^2 + cos^2(θ) = 1
1/16 + cos^2(θ) = 1
cos^2(θ) = 1 - 1/16
cos^2(θ) = 16/16 - 1/16
cos^2(θ) = 15/16
Taking the square root of both sides:
cos(θ) = ± sqrt(15)/4
Since the angle θ is between θ and 3π/2, the cosine function will be negative in that quadrant, so:
cos(θ) = -sqrt(15)/4
Now, we can use the remaining reciprocal identities to find the values of sec(θ), tan(θ), and cot(θ):
sec(θ) = 1/cos(θ)
sec(θ) = 1/(-sqrt(15)/4)
sec(θ) = -4/sqrt(15)
tan(θ) = sin(θ)/cos(θ)
tan(θ) = (-1/4) / (-sqrt(15)/4)
tan(θ) = 1/sqrt(15)
cot(θ) = 1/tan(θ)
cot(θ) = 1 / (1/sqrt(15))
cot(θ) = sqrt(15)
So, the values of the remaining trigonometric functions are:
sec(θ) = -4/sqrt(15)
tan(θ) = 1/sqrt(15)
cot(θ) = sqrt(15)