Given that sin^2ceta/1-cos ceta -sin^2ceta/1+cosceta=1 where 0degress is less than or equal to ceta and less than or equal to 360 degress. find the value of ceta
Isn't sin^2C=1-cos^2C= (1+cosC)(1-cosC) ?
If that is so, then it becomes
1+cosC-(1-cosC)=1
which solves very quickly.
write the two terms with a common denomimator. I will call "ceta" x,
sin^2 x [1/(1-cosx) - 1/(1+cos x)]
= sin^2x [(1 + cosx -1 +cosx)/(1-cos^2x)]
= 2 cos x = 1
The solutions are where cos x = 1/2
That would be at x = 60 degrees and 300 degrees.
To solve the given equation involving trigonometric functions, we can simplify it step by step:
1. Starting with the given equation:
sin^2(ceta)/(1-cos(ceta)) - sin^2(ceta)/(1+cos(ceta)) = 1
2. Since sin^2(ceta) is common to both terms, we can factor it out:
sin^2(ceta) * [(1+cos(ceta)) - (1-cos(ceta))] = 1 * (1-cos(ceta)) * (1+cos(ceta))
3. Simplifying further:
sin^2(ceta) * [1 + cos(ceta) - 1 + cos(ceta)] = 1 - cos^2(ceta)
sin^2(ceta) * [2cos(ceta)] = sin^2(ceta)
4. Now, we can divide both sides of the equation by sin^2(ceta):
2cos(ceta) = 1
5. Divide both sides by 2:
cos(ceta) = 1/2
6. To find the value of ceta, we need to evaluate the inverse cosine (arccos) of 1/2. This will give us the angle in radians:
ceta = arccos(1/2)
7. Evaluating arccos(1/2):
ceta = pi/3 + 2n*pi (where n is an integer)
Since the original problem states the range of ceta to be between 0 degrees and 360 degrees, we need to convert the angle from radians to degrees.
8. Convert pi/3 radians to degrees:
ceta = (pi/3) * (180/pi)
ceta = 60 degrees
Therefore, the value of ceta is 60 degrees.