1. What is the exact value of
csc^2 210degrees - cot^2 210degrees?
2. Can you make a generalization about the value of csc^2 theta - cot^2 theta?
I'm not sure what to do, I'm familiar with the terms but not sure how to go about solving the question.
1/sin^2 210
well 210 is 180 + 30
so it is 30 degrees below the y axis in quadrant 3
sin 210 = -1/2
1/(-1/2)^2 = 4
That should get you started
1/sin^2 - cos^2/sin^2
(1-cos^2)/sin^2
sin^2/sin^2
1
To find the exact value of csc^2 210 degrees - cot^2 210 degrees, we can start by finding the individual values of csc 210 degrees and cot 210 degrees.
1. Start by finding the value of csc 210 degrees:
- The cosecant (csc) function is equal to 1 divided by the sine (sin) function: csc θ = 1/sin θ.
- We know that 210 degrees is in the third quadrant, where the sine function is negative. So, sin 210 degrees = -sqrt(3)/2.
- Therefore, csc 210 degrees = 1 / sin 210 degrees = 1 / (-sqrt(3)/2) = -2/sqrt(3).
2. Next, find the value of cot 210 degrees:
- The cotangent (cot) function is equal to 1 divided by the tangent (tan) function: cot θ = 1/tan θ.
- We know that 210 degrees is in the third quadrant, where the tangent function is negative. So, tan 210 degrees = -sqrt(3).
- Therefore, cot 210 degrees = 1 / tan 210 degrees = 1 / (-sqrt(3)) = -1/sqrt(3).
Now, substitute the values we found back into the original expression:
csc^2 210 degrees - cot^2 210 degrees = (-2/sqrt(3))^2 - (-1/sqrt(3))^2
Simplifying this expression:
(-2/sqrt(3))^2 = 4/3
(-1/sqrt(3))^2 = 1/3
Substituting the simplified values back in:
csc^2 210 degrees - cot^2 210 degrees = 4/3 - 1/3 = 3/3 = 1
Therefore, the exact value of csc^2 210 degrees - cot^2 210 degrees is 1.
Regarding the generalization about the value of csc^2 theta - cot^2 theta:
By using similar steps as above, you can find that regardless of the specific value of theta, the expression csc^2 theta - cot^2 theta will always simplify to 1. In other words, for any angle theta, the value of csc^2 theta - cot^2 theta is 1.