find sin of theta if cos is less than 0 and cot equals 3
cosθ < 0 means θ is in 2nd or 3rd quadrant
cotθ > 0 means θ in 1st or 3rd quadrant.
So, θ is in 3rd quadrant
cot 18.4° = 3
so, θ = 180 + 18.4 = 198.4°
sin 198.4° = 0.316
or, more directly, if cotθ = 3, sinθ = -1/√10 = 0.316
Oops. Both those values should be -0.316
cos ( theta ) = + OR - sqrt [ 1 - sin ^ 2 ( theta ) ]
cot( theta ) = cos ( theta ) / sin ( theta )
if cos ( theta ) < 0 that mean cos ( theta ) is negative
if cot ( theta ) = 3 and cos ( theta ) is negative sin ( theta ) must be negative.
So:
cos ( theta ) / sin ( theta ) = 3
sqrt [ ( 1 - sin ^ 2 ( theta ) ] / sin ( theta ) = 3
[ ( 1 - sin ^ 2 ( theta ) ] / sin ^ 2 ( theta ) = 9
1 - sin ^ 2 ( theta ) = 9 sin ^ 2 ( theta )
1 = 9 sin ^ 2 ( theta ) + sin ^ 2 ( theta )
1 = 10 sin ^ 2 ( theta )
10 sin ^ 2 ( theta ) = 1 Divide both sides with 10
sin ^ 2 ( theta ) = 1 / 10
sin ( theta ) = + OR - sqrt ( 1 / 10 )
If sin ( theta ) is negative solution is :
sin ( theta ) = - 1 / sqrt ( 10 )
To find the value of sin(theta), we will use the given information that cos(theta) < 0 and cot(theta) = 3.
First, let's understand the meaning of the given information:
- cos(theta) < 0 implies that the cosine function is negative in the particular quadrant where theta lies.
- cot(theta) = 3 means that the cotangent function is equal to 3, which can be represented as cot(theta) = adjacent side length / opposite side length = 3.
Using these details, we can determine the quadrant in which theta resides and then find the value of sin(theta).
Since the cosine function is negative (cos(theta) < 0) and cot(theta) = 3, we know that theta is in either the second or third quadrant.
In the second quadrant, both sine and cosine functions are positive, so it does not satisfy the given condition of cos(theta) < 0.
In the third quadrant, sine is positive, but cosine is negative. Therefore, we can conclude that theta is in the third quadrant.
In the third quadrant, sine is positive, so sin(theta) > 0.
Now, the task is to find the value of sin(theta) in the third quadrant.
Using the Pythagorean identity, sin^2(theta) + cos^2(theta) = 1, we can solve for sin(theta).
Since we know that cos(theta) < 0, we can express it as -√(1 - sin^2(theta)) = 1 - sin^2(theta) = 1 - [sin(theta)]^2.
Based on cot(theta) = 3 (which implies adjacent/opposite = 3), we can form a right triangle. Let's assign a value of 3 to the adjacent side length and a value of 1 to the opposite side length (since cot(theta) = adjacent/opposite = 3/1) to solve for sin(theta).
Using the Pythagorean theorem, we can calculate the hypotenuse:
hypotenuse^2 = adjacent^2 + opposite^2
hypotenuse^2 = 3^2 + 1^2
hypotenuse^2 = 9 + 1
hypotenuse^2 = 10
hypotenuse = √10
In the third quadrant, the sine function is positive. Since we have a positive value for the opposite side (equal to 1), we can conclude that sin(theta) = opposite/hypotenuse = 1/√10.
Thus, the value of sin(theta) when cos(theta) < 0 and cot(theta) = 3 is sin(theta) = 1/√10.