Problem 3 - Complex numbers from A to...Z

Question

$$ \begin{aligned} &\text { Let } x, y, z \text { be real numbers such that }\\\ &\sin x+\sin y+\sin z=0 \quad \text { and } \quad \cos x+\cos y+\cos z=0 \end{aligned} $$ Prove that $$ \sin 2 x+\sin 2 y+\sin 2 z=0 \quad \text { and } \quad \cos 2 x+\cos 2 y+\cos 2 z=0 $$

Step-by-Step Solution

Answer

Question: Prove that if the sum of sin(x), sin(y), and sin(z) and the sum of cos(x), cos(y), and cos(z) both equal to zero, then the sum of sin(2x), sin(2y), and sin(2z) and the sum of cos(2x), cos(2y), and cos(2z) both equal to zero. Answer: By simplifying and manipulating the given expressions using trigonometric identities and the provided constraints, we proved that sin(2x) + sin(2y) + sin(2z) = 0 and cos(2x) + cos(2y) + cos(2z) = 0.

Step 1: Double Angle Identity for Sine

Rather than examining the expression sin(2x) + sin(2y) + sin(2z) directly, use the identity sin(2A) = 2sin(A)cos(A) for simplification. Thus, sin(2x) + sin(2y) + sin(2z) can be written as: 2sin(x)cos(x) + 2sin(y)cos(y) + 2sin(z)cos(z).

Step 2: Factorization

Factor out the 2 to get: 2 [ sin(x)cos(x) + sin(y)cos(y) + sin(z)cos(z) ].

Step 3: Sine-Cosine Summation

Now, since it's given that the sum of sin(x), sin(y), and sin(z) and the sum of cos(x), cos(y), and cos(z) both equal to zero, it can be said that whatever individual values sin(x), cos(x), sin(y), cos(y), sin(z), or cos(z) take, their collective sum must be zero. Therefore, sin(x)cos(x) + sin(y)cos(y) + sin(z)cos(z) = 0.

Step 4: The Sine Double Angle Sum

Substituting the above result into the expression from Step 2, we get: 2 [ 0 ] = 0. Hence, sin(2x) + sin(2y) + sin(2z) = 0.

Step 5: Double Angle Identity for Cosine

Similarly for proving the second part of the exercise, use the double angle identity for cosine: cos(2A)= cos²(A) - sin²(A). Substitute this identity to rewrite cos(2x) + cos(2y) + cos(2z) as: cos²(x) - sin²(x) + cos²(y) - sin²(y) + cos²(z) - sin²(z).

Step 6: Reordering

Reorder the terms to group together the squares of sins and cosines: [cos²(x) + cos²(y) + cos²(z)] - [sin²(x) + sin²(y) + sin²(z)].

Step 7: Sine-Cosine Summation

Given that the sum of the squares of the sines and the sum of the squares of the cosines both equal to one, it can be said that [cos²(x) + cos²(y) + cos²(z)] = [sin²(x) + sin²(y) + sin²(z)] = 1.

Step 8: The Cosine Double Angle Sum

Substituting the above results into the expression from Step 6, we get: 1 - 1 = 0. Hence, cos(2x) + cos(2y) + cos(2z) = 0. By working through these eight steps, the expressions given in the exercise have been proven to be correct.

Key Concepts

Double Angle Identities

The double angle identities in trigonometry are fundamental tools that allow us to simplify and solve a wide range of problems. These identities express trigonometric functions of double angles, like 2A, in terms of the trigonometric functions of A.

For the sine double angle identity, the formula is given by:
\[\[\begin{align*}\sin(2A) = 2\sin(A)\cos(A)\end{align*}\]\]
When dealing with problems involving sums of trigonometric functions, double angle identities can often simplify the terms into a more workable form. In our original problem, the double angle identity for sine was crucial in proving that the sum of the sine of the double angles equals zero. By substituting the identity, the solution transforms the complex original equation into a straightforward expression that's easy to analyze and solve.

Similarly, the double angle identity for cosine is:\[\[\begin{align*}\cos(2A) = \cos^2(A) - \sin^2(A)\end{align*}\]\]
Applying this to the sum of double angles in cosine, it assisted in establishing that the combination of these cosine terms also equates to zero, as seen in the exercise.

Trigonometric Equations

Trigonometric equations involve functions such as sine, cosine, and tangent, and solving them often requires a combination of algebraic and trigonometric techniques. Key strategies in solving trigonometric equations include using identities, factoring, and sometimes utilizing the unit circle.

To tackle these types of equations, it's crucial to remember basic trigonometric identities, including Pythagorean identities, co-function identities, and sum and difference formulas. In the provided exercise, using the double angle identities is an elegant way to transform the equations and find a solution.

Furthermore, trigonometric equations can sometimes have multiple solutions because trigonometric functions are periodic. However, in our case, the double angle identity helps narrow down the specific conditions that satisfy the given summations equaling zero, thus proving the result without the need to consider a multitude of potential solutions.

Sums of Trigonometric Functions

The sums of trigonometric functions, like summing sines or cosines with different arguments, are common in trigonometric problems. When we add these functions, patterns may emerge, depending on their arguments. For instance, these functions may cancel each other out if they are in specific relationships, like being opposite angles.

Understanding how to combine trigonometric functions algebraically and graphically can be invaluable. Algebraically, we look for identities or algebraic manipulations that can simplify the expression. Graphically, we might consider the wave patterns and how they might overlap to add to zero.

In our exercise, the condition given for sums of sines and cosines to equal zero reflects a certain symmetry or cancellation in the system described by the equations. We're not asked to find the values of the individual functions; rather, we demonstrate that under the initial conditions, the expression involving the sums of these functions simplifies to zero. This is an important concept because it shows how trigonometric functions can relate to one another and decrease the complexity of a problem.