Problem 57 - Algebra and Trigonometry

Question

Find the exact value of $$ \cos 0^{\circ}+\cos 1^{\circ}+\cos 2^{\circ}+\cos 3^{\circ}+\cdots+\cos 179^{\circ}+\cos 180^{\circ} $$

Step-by-Step Solution

Answer

0

Step 1: Identify symmetry

The \(\cos\) function is symmetrical such that \(\cos(90 + x) = \cos(90 - x)\). Split the series into pairs according to this property, recognizing that we will have an unpaired \(\cos 90^{\circ}\) at the middle. We have \(\cos(90 - x) + \(\cos(90 + x)\) for \(x\) ranging from 1 to 89 degrees.

Step 2: Simplify pairs

Using the symmetry property, we see that each pair sums up to \(2 \cos 90^{\circ} = 0.\)

Step 3: Account for unpaired term

Finally, we add the unpaired term \(\cos 90^{\circ}\) which is also equals to 0.

Step 4: Sum all terms

The sum of all these terms is \(2 \cos 90^{\circ}*89 + \cos 90^{\circ} = 0*89 + 0 = 0\).

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry, reflecting a specific relationship between an angle in a right-angled triangle and the lengths of its sides. Specifically, for an angle \( \theta \), \( \cos(\theta) \) is defined as the ratio of the length of the adjacent side to the hypotenuse.

• The cosine function varies between -1 and 1.
• It is an even function, meaning \( \cos(-\theta) = \cos(\theta) \).
• Cosine is periodic with a period of \(360^{\circ}\) or \(2\pi\) radians.

The cosine of some specific angles such as \(0^{\circ}\), \(90^{\circ}\), \(180^{\circ}\), etc., are often used in exercises because they simplify calculations.Understanding how the cosine function behaves over its domain helps in solving series and trigonometric equations, much like in the problem you see above.

Symmetry in Trigonometry

Symmetry plays a key role in trigonometry, where certain functions exhibit symmetry properties that make calculations more efficient. For the cosine function, this symmetry is particularly useful.The cosine function is symmetrical around the vertical axis. This means for any angle \(x\), the value \(\cos(90 + x)\) will be the same as \(\cos(90 - x)\). We call this even symmetry.Moreover, in the context of the exercise, symmetry allowed for a simplification: by observing that each angle from \(1^{\circ}\) to \(89^{\circ}\) can be paired with its complementary angle \(179^{\circ}\) minus the original angle. Thus, using the property \(\cos(90 + x) = \cos(90 - x)\) simplifies many cosine series problems.

Symmetry can be observed visually when plotting the cosine function, seeing the mirroring effect across the y-axis emphasizes the periodic nature of the cosine wave.

Sum of Angles

The sum of angles is a significant concept when dealing with trigonometric series. It refers to adding angles together and often accompanies rules or identities simplifying calculations.In trigonometry, the sum of angles formulas, such as for sine \( \sin(a + b) \) and cosine \( \cos(a + b) \), are essential tools. They allow us to express trigonometric functions of sums of angles in terms of individual trigonometric functions.

• \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \)
• This formula helps break down complex expressions and solve equations.

In the context of the exercise, even though it involves a series of angles, the symmetry and periodic properties simplified the task, eliminating the need for complex sum formulas.

Trigonometric Series

A trigonometric series is a sum of terms that are trigonometric functions. These series can often be simplified through properties like symmetry and periodicity, which can cancel terms out as seen above.The specific problem gave a series of cosine functions from \(0^{\circ}\) to \(180^{\circ}\). By recognizing the periodic and symmetrical properties of cosine, the series became manageable.

• The series was simplified using the knowledge that pairs of terms, such as \(\cos(89^{\circ})\) and \(\cos(91^{\circ})\), summed to zero.
• Each pair reduced the series to the middle term, eliminating the need for manual calculation of all terms.

Understanding trigonometric series and their properties makes it easier to manipulate and solve angles-related problems, showing the elegance and simplicity embedded in such trigonometric challenges.By learning how to handle trigonometric series, you expand your toolset for effectively tackling complex mathematical problems!