Problem 31 - Algebra and Trigonometry

Question

Prove the identity. $$ \cos \left(x+\frac{\pi}{6}\right)+\sin \left(x-\frac{\pi}{3}\right)=0 $$

Step-by-Step Solution

Answer

The expression simplifies to 0, proving the identity.

Step 1: Use Angle Addition Formulas

Apply the angle addition formulas for both cosine and sine.\[\cos \left(x + \frac{\pi}{6}\right) = \cos x \cos \frac{\pi}{6} - \sin x \sin \frac{\pi}{6}\\sin \left(x - \frac{\pi}{3}\right) = \sin x \cos \frac{\pi}{3} - \cos x \sin \frac{\pi}{3}\]

Step 2: Substitute Known Values

Substitute the values for \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\), \(\sin \frac{\pi}{6} = \frac{1}{2}\), \(\cos \frac{\pi}{3} = \frac{1}{2}\), and \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\) into the expressions.\[\cos \left(x + \frac{\pi}{6}\right) = \cos x \cdot \frac{\sqrt{3}}{2} - \sin x \cdot \frac{1}{2}\\sin \left(x - \frac{\pi}{3}\right) = \sin x \cdot \frac{1}{2} - \cos x \cdot \frac{\sqrt{3}}{2}\]

Step 3: Combine Expressions

Combine the two expressions from Step 2. \[\begin{align*}\cos \left(x + \frac{\pi}{6}\right) + \sin \left(x - \frac{\pi}{3}\right) = & \left(\cos x \cdot \frac{\sqrt{3}}{2} - \sin x \cdot \frac{1}{2}\right) \& + \left(\sin x \cdot \frac{1}{2} - \cos x \cdot \frac{\sqrt{3}}{2}\right)\end{align*}\]

Step 4: Simplify to Prove the Identity

Simplify the expression by combining like terms. Notice that the terms involving \(\cos x\) and \(\sin x\) cancel each other.\[\frac{\sqrt{3}}{2} \cos x - \frac{1}{2} \sin x + \frac{1}{2} \sin x - \frac{\sqrt{3}}{2} \cos x = 0\]Since all terms cancel each other out, the equation simplifies to 0, proving the identity.

Key Concepts

Angle Addition Formulas

Angle addition formulas are an essential tool in trigonometry. They allow us to calculate the cosine or sine of the sum or difference of two angles. This is particularly useful when dealing with composite angles that are not standard angles for which we easily know the sine and cosine values.

The angle addition formulas for cosine and sine are given as follows:

• For cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
• For sine: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \)

These identities are powerful because they allow complex trigonometric expressions to be broken down into simpler parts. In our exercise, we use these formulas to solve the equation \( \cos \left(x+\frac{\pi}{6}\right)+\sin \left(x-\frac{\pi}{3}\right)=0 \). By expanding using the angle addition formulas, we simplify the expression and can directly substitute known trigonometric values.

Cosine Function

The cosine function is a fundamental part of trigonometry. It describes the ratio of the adjacent side to the hypotenuse in a right triangle for a given angle, as well as its behavior on the unit circle. The cosine function has periodic properties and is essential in understanding wave-like phenomena.

Some key points to remember about cosine are:

• It ranges from -1 to 1.
• The cosine of 0 degrees (or 0 radians) is 1.
• The function is even, meaning \( \cos(-x) = \cos(x) \).
• It has a period of \( 2\pi \), completing one full cycle every \( 2\pi \) radians.

In the context of our problem, knowing specific cosine values for common angles such as \( \frac{\pi}{6} \) and \( \frac{\pi}{3} \) is critical. For example, \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \) and \( \cos \frac{\pi}{3} = \frac{1}{2} \). These values allow us to simplify expressions quickly, making them invaluable when proving trigonometric identities.

Sine Function

The sine function is another critical function in trigonometry that represents the ratio of the opposite side to the hypotenuse in a right triangle. Like the cosine function, it also describes wave-like behaviors and is periodic.

Important aspects of the sine function include:

• It also ranges from -1 to 1.
• Sine of 0 degrees (or 0 radians) is 0.
• It is an odd function, meaning \( \sin(-x) = -\sin(x) \).
• Sine has a period of \( 2\pi \), similar to the cosine function.

In our specific trigonometric equation, knowing the sine of commonly used angles makes it easier to prove identities. For instance, \( \sin \frac{\pi}{6} = \frac{1}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \). These values were substituted during our solution process to simplify and prove the given identity. Understanding these fundamental elements of the sine function is key to making trigonometry more manageable.