What is the value of sin 26° + sin 212° + sin 218° + … + sin 284° + sin 290°?
8
Understanding the Sum of Sine Squares Series
The question asks for the value of the sum of squares of sine functions for a specific series of angles: \(\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 84^\circ + \sin^2 90^\circ\).
The angles in the series start at \(6^\circ\), increase by \(6^\circ\) each time, and end at \(90^\circ\). The angles are \(6^\circ, 12^\circ, 18^\circ, 24^\circ, 30^\circ, 36^\circ, 42^\circ, 48^\circ, 54^\circ, 60^\circ, 66^\circ, 72^\circ, 78^\circ, 84^\circ, 90^\circ\). There are 15 terms in this series.
Applying Trigonometric Identities for Sine Squares
To find the sum, we can use the fundamental trigonometric identity: \(\sin^2 \theta + \cos^2 \theta = 1\). We also know that the cosine of an angle is equal to the sine of its complementary angle, i.e., \(\cos \theta = \sin (90^\circ - \theta)\).
Combining these identities, we get: \(\sin^2 \theta + \sin^2 (90^\circ - \theta) = \sin^2 \theta + \cos^2 \theta = 1\). This identity is very useful when dealing with sums of sine squares where the angles are complementary.
Grouping Terms and Calculating the Sum
Let's look at the angles in our series and see if we can form pairs that add up to \(90^\circ\). The angles are \(6^\circ, 12^\circ, \dots, 84^\circ, 90^\circ\).
- \(6^\circ\) and \(84^\circ\) are complementary (\(6^\circ + 84^\circ = 90^\circ\))
- \(12^\circ\) and \(78^\circ\) are complementary (\(12^\circ + 78^\circ = 90^\circ\))
- \(18^\circ\) and \(72^\circ\) are complementary (\(18^\circ + 72^\circ = 90^\circ\))
- \(24^\circ\) and \(66^\circ\) are complementary (\(24^\circ + 66^\circ = 90^\circ\))
- \(30^\circ\) and \(60^\circ\) are complementary (\(30^\circ + 60^\circ = 90^\circ\))
- \(36^\circ\) and \(54^\circ\) are complementary (\(36^\circ + 54^\circ = 90^\circ\))
- \(42^\circ\) and \(48^\circ\) are complementary (\(42^\circ + 48^\circ = 90^\circ\))
Using the identity \(\sin^2 \theta + \sin^2 (90^\circ - \theta) = 1\), each of these pairs sums to 1:
- \(\sin^2 6^\circ + \sin^2 84^\circ = 1\)
- \(\sin^2 12^\circ + \sin^2 78^\circ = 1\)
- \(\sin^2 18^\circ + \sin^2 72^\circ = 1\)
- \(\sin^2 24^\circ + \sin^2 66^\circ = 1\)
- \(\sin^2 30^\circ + \sin^2 60^\circ = 1\)
- \(\sin^2 36^\circ + \sin^2 54^\circ = 1\)
- \(\sin^2 42^\circ + \sin^2 48^\circ = 1\)
There are 7 such pairs. These 7 pairs account for 14 terms in the series (\(2 \times 7 = 14\)). The total number of terms in the series is 15. The term that is left is \(\sin^2 90^\circ\).
The sum of the series can be written as the sum of these pairs plus the remaining term:
Sum = \((\sin^2 6^\circ + \sin^2 84^\circ) + (\sin^2 12^\circ + \sin^2 78^\circ) + \dots + (\sin^2 42^\circ + \sin^2 48^\circ) + \sin^2 90^\circ\)
Sum = \((1) + (1) + (1) + (1) + (1) + (1) + (1) + \sin^2 90^\circ\)
Sum = \(7 \times 1 + \sin^2 90^\circ\)
We know that \(\sin 90^\circ = 1\). Therefore, \(\sin^2 90^\circ = (1)^2 = 1\).
Sum = \(7 + 1 = 8\).
Final Result of the Sine Squares Sum
The value of the series \(\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 84^\circ + \sin^2 90^\circ\) is 8.
Revision Table: Key Concepts for Sine Sum
| Concept | Identity/Value | Application |
|---|---|---|
| Pythagorean Identity | \(\sin^2 \theta + \cos^2 \theta = 1\) | Fundamental relation between sine and cosine squares. |
| Complementary Angle Identity | \(\cos \theta = \sin (90^\circ - \theta)\) | Relates sine and cosine of complementary angles. |
| Sum of Complementary Sine Squares | \(\sin^2 \theta + \sin^2 (90^\circ - \theta) = 1\) | Used to group terms in the series. |
| Value of \(\sin 90^\circ\) | \(\sin 90^\circ = 1\) | Used to evaluate the final term in the series. |
Additional Information: Related Trigonometry Concepts
This problem demonstrates a common technique for summing trigonometric series, especially when the angles form an arithmetic progression. Here are some related concepts:
- Cosine Series: A similar technique can be applied to sums of cosine squares, like \(\cos^2 \theta + \cos^2 (90^\circ - \theta) = \cos^2 \theta + \sin^2 \theta = 1\).
- Series with Different Intervals: The same pairing method works for any arithmetic progression of angles where terms can be paired to sum to \(90^\circ\), as long as the angle \(45^\circ\) (if present) or \(90^\circ\) (if present) is handled separately.
- Angles Beyond 90°: For sums involving angles beyond \(90^\circ\), reduction formulas like \(\sin (180^\circ - \theta) = \sin \theta\) or \(\sin (90^\circ + \theta) = \cos \theta\) might be used in conjunction with square properties (\(\sin^2 (-\theta) = \sin^2 \theta\), \(\sin^2 (180^\circ \pm \theta) = \sin^2 \theta\)).
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