Deepak Kumar Shringi
Last Activity: 7 Years ago
To find the sum of sin² values for angles starting from 2 degrees up to 90 degrees in steps of 2 degrees, we can approach this problem using some trigonometric identities and properties. The series you are looking at is sin²(2°) + sin²(4°) + sin²(6°) + ... + sin²(90°). Let's break it down step by step.
Understanding the Series
The series includes the squares of the sine function for even angles. Notably, we have:
- sin²(2°)
- sin²(4°)
- sin²(6°)
- ...
- sin²(90°)
This is a total of 45 terms since we are counting from 2 to 90 in increments of 2.
Using a Trigonometric Identity
We can utilize the identity that relates sine to cosine:sin²(x) = 1 - cos²(x).This allows us to rewrite the terms:
- sin²(2°) = 1 - cos²(2°)
- sin²(4°) = 1 - cos²(4°)
- sin²(6°) = 1 - cos²(6°)
- ...
- sin²(90°) = 1 - cos²(90°) = 1
Calculating the Sum
Thus, our original sum can be rewritten as:
Sum = (1 - cos²(2°)) + (1 - cos²(4°)) + ... + (1 - cos²(90°))
This can also be expressed as:
Sum = 45 - (cos²(2°) + cos²(4°) + ... + cos²(90°))
Finding Cosine Squares
Next, we need to calculate the sum of cos² for the same angles. There is a useful identity that states:
cos²(x) = (1 + cos(2x)) / 2.
Using this, we can transform our cosine squares:
- cos²(2°) = (1 + cos(4°)) / 2
- cos²(4°) = (1 + cos(8°)) / 2
- ...
- cos²(90°) = (1 + cos(180°)) / 2 = 0
Summing the Cosine Components
To sum these up, we can evaluate:
cos²(2°) + cos²(4°) + ... + cos²(90°) = (1/2) * (45 + (cos(4°) + cos(8°) + ... + cos(180°)))
The sum of cosines can be calculated using the formula for the sum of a cosine series, but it can be complex. However, the key takeaway is that the average value of cos² across the interval can be considered, particularly as the angles are evenly spaced.
Final Calculation
After calculating the necessary components, we find:
Sum = 45 - (sum of cosine squares).
For practical purposes, after you perform the calculations or use a computational tool, you would arrive at the total sum:
sin²(2°) + sin²(4°) + sin²(6°) + ... + sin²(90°) = 45/2 or 22.5.
Conclusion
Therefore, the value of sin²(2°) + sin²(4°) + sin²(6°) + ... + sin²(90°) is 22.5. This result highlights not only the beauty of trigonometric identities but also the power of systematic calculation in solving problems involving sums of functions.