Problem 31 - Precalculus Mathematics for Calculus

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 identity is proven as both terms cancel out, resulting in 0.

Step 1: Apply Angle Sum Formula for Cosine

First, we'll use the angle sum formula for cosine. The formula is \( \cos(a+b) = \cos a \cos b - \sin a \sin b \). Here, \( a = x \) and \( b = \frac{\pi}{6} \). So:\[\cos \left(x + \frac{\pi}{6}\right) = \cos x \cos \frac{\pi}{6} - \sin x \sin \frac{\pi}{6}\]Knowing that \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \) and \( \sin \frac{\pi}{6} = \frac{1}{2} \), we substitute these values:\[\cos \left(x + \frac{\pi}{6}\right) = \cos x \cdot \frac{\sqrt{3}}{2} - \sin x \cdot \frac{1}{2}\]

Step 2: Apply Angle Difference Formula for Sine

Next, we apply the angle difference formula for sine. The formula is \( \sin(a-b) = \sin a \cos b - \cos a \sin b \). Here, \( a = x \) and \( b = \frac{\pi}{3} \). So:\[\sin \left(x - \frac{\pi}{3}\right) = \sin x \cos \frac{\pi}{3} - \cos x \sin \frac{\pi}{3}\]Knowing that \( \cos \frac{\pi}{3} = \frac{1}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), substitute these values:\[\sin \left(x - \frac{\pi}{3}\right) = \sin x \cdot \frac{1}{2} - \cos x \cdot \frac{\sqrt{3}}{2}\]

Step 3: Combine Results and Simplify

Now, let's add the results from Steps 1 and 2.\[\cos \left(x + \frac{\pi}{6}\right) + \sin \left(x - \frac{\pi}{3}\right) = \left(\cos x \cdot \frac{\sqrt{3}}{2} - \sin x \cdot \frac{1}{2}\right) + \left(\sin x \cdot \frac{1}{2} - \cos x \cdot \frac{\sqrt{3}}{2}\right)\]Notice how the terms cancel out:\[\cos x \cdot \frac{\sqrt{3}}{2} - \cos x \cdot \frac{\sqrt{3}}{2} + \sin x \cdot \frac{1}{2} - \sin x \cdot \frac{1}{2} = 0\]Thus, the left side simplifies to zero, proving the identity.

Key Concepts

Angle Sum Formula

The angle sum formula is a crucial identity in trigonometry that allows us to simplify expressions involving the sum of two angles. For cosine, the angle sum formula is given by:\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \].
In the problem, our objective was to prove the identity involving \( \cos(x + \frac{\pi}{6}) \). By using the angle sum formula for cosine, we can break down the trigonometric functions into more manageable parts.
This involves substituting known values of sine and cosine for specific angles like \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \) and \( \sin \frac{\pi}{6} = \frac{1}{2} \).
By plugging in these values and simplifying, we are able to express \( \cos(x + \frac{\pi}{6}) \) in terms of \( \cos x \) and \( \sin x \), which is the first step in proving the trigonometric identity. This property is extremely powerful when it comes to working with complex trigonometric equations.

Angle Difference Formula

Similar to the angle sum formula, the angle difference formula is another fundamental identity in trigonometry. It helps us simplify expressions involving the difference between two angles. The formula for sine when dealing with angle differences is:\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \].
In the given exercise, we apply this formula to \( \sin(x - \frac{\pi}{3}) \). By substituting the known values for the cosine and sine of \( \frac{\pi}{3} \) - specifically, \( \cos \frac{\pi}{3} = \frac{1}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), we can express \( \sin(x - \frac{\pi}{3}) \) in terms of \( \sin x \) and \( \cos x \).
Understanding these formulas not only helps in solving complex trigonometric identities but also enhances our ability to transform and simplify complicated expressions for easier computation.

Sine and Cosine Values

Knowing the sine and cosine values of common angles is essential when working with trigonometric identities. These values can often be found in trigonometry tables or derived from the unit circle.
For example, in this problem we frequently use:

• \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \)
• \( \sin \frac{\pi}{6} = \frac{1}{2} \)
• \( \cos \frac{\pi}{3} = \frac{1}{2} \)
• \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \)

These values stem from the geometric properties of the unit circle, where each point on the circle is defined by coordinates \((\cos \theta, \sin \theta \)).
Accurate recollection of sine and cosine values for these common angles greatly simplifies the calculation process when working through trigonometric problems. They are the building blocks that allow us to use complex identities like the angle sum and difference formulas effectively.