Find the exact value of [tex]$\operatorname { s i n } 1^{\circ}+ \operatorname { s i n } 2^{\circ}+ \operatorname { s i n } 3^{\circ}+\cdots+ \operatorname { s i n } 358^{\circ}+ \operatorname { s i n } 359^{\circ}$[/tex].
Answer (2)
The exact value of the sum sin 1 ∘ + sin 2 ∘ + ⋯ + sin 35 9 ∘ is 0. This result comes from pairing terms using the property sin ( 36 0 ∘ − x ) = − sin ( x ) , leading to complete cancellation. Therefore, the entire sum equates to 0.
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Rewrite the sum using sine function.
Pair terms using the property sin ( 36 0 ∘ − x ) = − sin x .
Simplify each pair to 0.
Conclude that the entire sum is 0 .
Explanation
Understanding the Problem We are asked to find the exact value of the sum of the sines of angles from 1 degree to 359 degrees: sin 1 ∘ + sin 2 ∘ + sin 3 ∘ + ⋯ + sin 35 8 ∘ + sin 35 9 ∘ This can be written more compactly using summation notation as: k = 1 ∑ 359 sin k ∘
Using Trigonometric Identities We can use the property that sin ( 18 0 ∘ − x ) = sin x . Also, we know that sin ( 36 0 ∘ − x ) = − sin x . We can pair terms in the sum using this property.
Pairing and Simplifying Terms Let's rewrite the sum and pair the terms: S = sin 1 ∘ + sin 2 ∘ + ⋯ + sin 17 9 ∘ + sin 18 0 ∘ + sin 18 1 ∘ + ⋯ + sin 35 9 ∘ We can rewrite the terms from 181 to 359 as 180 + 1 to 360 − 1 :
S = sin 1 ∘ + sin 2 ∘ + ⋯ + sin 17 9 ∘ + sin 18 0 ∘ + sin ( 18 0 ∘ + 1 ∘ ) + ⋯ + sin ( 36 0 ∘ − 1 ∘ ) Now, we pair the terms: S = ( sin 1 ∘ + sin 35 9 ∘ ) + ( sin 2 ∘ + sin 35 8 ∘ ) + ⋯ + ( sin 17 9 ∘ + sin 18 1 ∘ ) + sin 18 0 ∘ Using the identity sin ( 36 0 ∘ − x ) = − sin x , we have: sin 1 ∘ + sin 35 9 ∘ = sin 1 ∘ − sin 1 ∘ = 0 sin 2 ∘ + sin 35 8 ∘ = sin 2 ∘ − sin 2 ∘ = 0 And so on, until: sin 17 9 ∘ + sin 18 1 ∘ = sin 17 9 ∘ + sin ( 18 0 ∘ + 1 ∘ ) = sin 17 9 ∘ − sin 1 ∘ However, we can also use sin ( 18 0 ∘ + x ) = − sin x , so sin 18 1 ∘ = − sin 1 ∘ . Since sin 17 9 ∘ = sin ( 18 0 ∘ − 1 ∘ ) = sin 1 ∘ , we have sin 17 9 ∘ + sin 18 1 ∘ = sin 1 ∘ − sin 1 ∘ = 0 .
Finally, sin 18 0 ∘ = 0 .
Final Result Therefore, the entire sum is: S = 0 + 0 + ⋯ + 0 + 0 = 0
Examples
Understanding trigonometric sums can be useful in signal processing. For example, when analyzing sound waves or electromagnetic waves, we often encounter sums of sinusoidal functions. Knowing how to simplify these sums can help us understand the overall behavior of the wave and extract useful information from it. This is also applicable in fields like image processing and data compression, where trigonometric functions are used to represent and manipulate data.
