Before we delve into solving the sum of sin^2 values from 10° to 90°, let’s briefly review a crucial trigonometric identity that will be instrumental in our calculations:
Sine and Cosine Relationship:
One of the fundamental identities in trigonometry states that sin(θ) = cos (90° – θ). This identity essentially relates the sine of an angle to the cosine of its complementary angle (an angle that adds up to 90° with the given angle).
Calculating the Sum of Sin^2 Values:
Now, let’s proceed to calculate the sum of sin^2 values from 10° to 90° using the sine-cosine relationship:
Here we have a formula
SinꝊ = Cos (90 – Ꝋ)
Start with the expression sin^2(10°) + sin^2(20°) + sin^2(30°) + … + sin^2(90°).
Use the sine-cosine relationship to express sin^2(10°) in terms of cosines:
sin^2(10°) = cos (90° – 10°) = cos (80°)
Repeat this process for each angle in the series, always subtracting the angle from 90°:
sin^2(20°) = cos (90° – 20°) = cos (70°)
sin^2(30°) = cos (90° – 30°) = cos (60°)
…
sin^2(90°) = cos (90° – 90°) = cos (0°)
Now, our expression becomes:
cos^2(80°) + cos^2(70°) + cos^2(60°) + … + cos^2(0°)
The sum of the squares of the cosines of these angles is a well-known trigonometric identity:
cos^2(80°) + cos^2(70°) + cos^2(60°) + … + cos^2(0°) =
For Complete Solution watch the above video.
Conclusion:
In conclusion, our journey into the world of trigonometry has led us to a fascinating pattern—the sum of sin^2 values from 10° to 90°. By utilizing the sine-cosine relationship and recognizing complementary angles, we’ve uncovered this elegant solution. Trigonometry not only helps us understand the geometric relationships between angles but also reveals hidden patterns that are both intellectually stimulating and practically useful. Whether you’re a student exploring trigonometry or a professional applying it to real-world problems, these mathematical gems continue to enrich our understanding of the universe.
