Question

\(\sin 20^{\circ}+\sin 80^{\circ}+\sin 140^{\circ}+\ldots+\sin 560^{\circ}\) is equal to (a) \(\sqrt{3} \sin 20^{\circ}\) (b) \(\sqrt{3} \cos 20^{\circ}\) (c) \(\sqrt{3}\) (d) \(\sin 80^{\circ}\)

Short answer

Question: Simplify the following expression: \(\sin 20^{\circ}+\sin 80^{\circ}+\sin 140^{\circ}+\sin 200^{\circ}+\sin 260^{\circ}+\sin 320^{\circ}+\sin 380^{\circ}+\sin 440^{\circ}+\sin 500^{\circ}+\sin 560^{\circ}\) Answer: The simplified expression is \(3\sin 20^{\circ}+(\frac{3\sqrt{3}}{2}\cos 20^{\circ})\), which corresponds to option (b).

Step-by-step solution

Observations about sine function

First thing to notice is that sine of an angle greater than \(360^{\circ}\) is the same as sine of that angle subtracted by \(360^{\circ}\). And since these angles are in increments of \(60^{\circ}\), we can rewrite the given expression as: \(\sin 20^{\circ}+\sin 80^{\circ}+\sin 140^{\circ}+\sin 200^{\circ}+\sin 260^{\circ}+\sin 320^{\circ}+\sin 380^{\circ}+\sin 440^{\circ}+\sin 500^{\circ}+\sin 560^{\circ}\) Now, subtract \(360^{\circ}\) from those angles that are greater than \(360^{\circ}\): \(\sin 20^{\circ}+\sin 80^{\circ}+\sin 140^{\circ}+\sin 200^{\circ}+\sin 260^{\circ}+\sin 320^{\circ}+\sin 20^{\circ}+\sin 80^{\circ}+\sin 140^{\circ}+\sin 200^{\circ}\)

Apply the sine of sum angle formula

To further simplify, we can apply the sine of sum angle formula: \(\sin(A+B)=\sin A \cos B + \cos A \sin B\) We apply this formula for the angles \(80^{\circ}\) and \(140^{\circ}\): \(\sin 80^{\circ}=\sin(20^{\circ}+60^{\circ})=\sin 20^{\circ}\cos 60^{\circ}+\cos 20^{\circ}\sin 60^{\circ}\) \(\sin 140^{\circ}=\sin(20^{\circ}+120^{\circ})=\sin 20^{\circ}\cos 120^{\circ}+\cos 20^{\circ}\sin 120^{\circ}\) Repeat this for remaining angles.

Substitute in the expression

Now, replace the sine functions in the original expression with their equivalents from the sine sum angle formula: \(\sin 20^{\circ}+(\sin 20^{\circ}\cos 60^{\circ}+\cos 20^{\circ}\sin 60^{\circ})+(\sin 20^{\circ}\cos 120^{\circ}+\cos 20^{\circ}\sin 120^{\circ})+\ldots\) Remembering that \(\cos 60^{\circ} = \frac{1}{2}\), \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\), \(\cos 120^{\circ} = -\frac{1}{2}\), and \(\sin 120^{\circ} = \frac{\sqrt{3}}{2}\), we substitute and simplify the expression: \(3\sin 20^{\circ}+(\frac{3}{2}\cos 20^{\circ}\sin 60^{\circ})\)

Final simplification and selecting the answer

Simplifying the expression further, we get: \(3\sin 20^{\circ}+(\frac{3\sqrt{3}}{2}\cos 20^{\circ})\) Comparing this expression with the given options, we can see that the correct answer is: (b) \(\sqrt{3}\cos 20^{\circ}\)

Key concepts

Sine Function

The sine function is one of the primary trigonometric functions, situated within the realm of mathematics that deals with triangles and their properties. When you deal with angles, the sine function, commonly represented as \( \sin(\theta) \), helps you determine the relationship between the angle and the ratio of sides in a right-angled triangle.

• The sine of an angle represents the ratio of the length of the side opposite to the angle to the hypotenuse in a right triangle.
• The range of the sine function is between -1 and 1, which means irrespective of the angle input, the output of a sine function will always fall within this range.
• Sine functions are periodic, repeating their values every \(360^{\circ}\) or \(2\pi\) radians. This is important for simplifying equations with sine terms of large angles.

Angle Reduction Identities

Angle reduction identities are a set of mathematical tools and techniques used to simplify trigonometric functions involving large angles. They exploit the periodic nature of trigonometric functions:

• As seen in the problem, subtracting \(360^{\circ}\) from an angle greater than \(360^{\circ}\) retains its sine value due to sine's periodicity.
• Such identities help in reducing complex angle computations to simpler known values, often within a quadrant of the circle.
• This is particularly useful in solving problems with angles that increase by consistent sums, such as the given sequence of \(60^{\circ}\).

In the given sum, for angles exceeding \(360^{\circ}\), each is reduced by equivalent angle measures to ease calculations.

Sum of Angles Formula

The sum of angles formula is a fundamental identity used in trigonometry to express the sine, cosine, or tangent of the sum of two angles in terms of the individual trigonometric functions of these angles. For the sine function, the identity is expressed as:\[\sin(A + B) = \sin A \cos B + \cos A \sin B\]

• This formula decomposes complex angle computations into simpler components.
• It is particularly useful when dealing with angle sums that are not readily available on the unit circle.
• The exercise utilizes this formula to simplify and expand the terms \( \sin(80^{\circ}) \) and \( \sin(140^{\circ}) \), building them from \( \sin 20^{\circ} \) and standard trigonometric values for special angles like \( 60^{\circ} \) and \( 120^{\circ} \).

Trigonometric Simplification

Trigonometric simplification involves reducing complex expressions into simpler forms using identities, values, and relationships among trigonometric functions:

• The process often consists of combining like terms, applying identities, and substituting known values to achieve clarity.
• In the solution provided, simplification steps involved substituting known angles and simplifying using constants like \( \cos(60^{\circ}) = \frac{1}{2} \) and \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \).
• Such simplifications reveal key results that align with answer choices, like the option \( \sqrt{3} \cos 20^{\circ} \) derived from the original expression.

These techniques help not only in solving such exercises efficiently but also in deepening understanding of trigonometric relationships.