Sin 225 - sin 330 bölü (kesir halde) cos
The expression you’ve shared is:
\frac{\sin 225^\circ - \sin 330^\circ}{\cos^2 210^\circ + \cot 135^\circ}
Let’s evaluate the trigonometric values step by step.
1. Evaluate the Numerator:
-
\sin 225^\circ:
- 225^\circ = 180^\circ + 45^\circ, so we’re in the third quadrant where sine is negative.
- Thus, \sin 225^\circ = -\sin 45^\circ = -\frac{\sqrt{2}}{2}.
-
\sin 330^\circ:
- 330^\circ = 360^\circ - 30^\circ, so we’re in the fourth quadrant where sine is negative.
- Thus, \sin 330^\circ = -\sin 30^\circ = -\frac{1}{2}.
-
Difference:
- \sin 225^\circ - \sin 330^\circ = -\frac{\sqrt{2}}{2} - \left(-\frac{1}{2}\right) = -\frac{\sqrt{2}}{2} + \frac{1}{2}.
2. Evaluate the Denominator:
-
\cos^2 210^\circ:
- 210^\circ = 180^\circ + 30^\circ, so we’re in the third quadrant where cosine is negative.
- Thus, \cos 210^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}.
- Therefore, \cos^2 210^\circ = \left(-\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}.
-
\cot 135^\circ:
- 135^\circ = 180^\circ - 45^\circ, so we’re in the second quadrant where cotangent is negative.
- Thus, \cot 135^\circ = -\cot 45^\circ = -1.
-
Sum:
- \cos^2 210^\circ + \cot 135^\circ = \frac{3}{4} + (-1) = \frac{3}{4} - \frac{4}{4} = -\frac{1}{4}.
3. Combine the Results:
Now, substitute these results back into the expression:
\frac{-\frac{\sqrt{2}}{2} + \frac{1}{2}}{-\frac{1}{4}}
Simplify:
- Calculate the numerator: -\frac{\sqrt{2}}{2} + \frac{1}{2} = \frac{1 - \sqrt{2}}{2}.
- Divide by the denominator: \frac{\frac{1 - \sqrt{2}}{2}}{-\frac{1}{4}} = (1 - \sqrt{2}) \cdot -2 = -2(1 - \sqrt{2}).
Thus, the simplified result is:
2(\sqrt{2} - 1)
This is the final simplified answer.