Problem 70 - Precalculus

Question

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. $$ \sin \left(\frac{\pi}{3}+x\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{\sqrt{3}}{2}\cos(x) + \frac{1}{2}\sin(x) \). Graphs confirm identity.

Step 1: Identify Trigonometric Formula

The given expression is \[ \sin\left(\frac{\pi}{3} + x\right) \]which resembles the sum of angles formula for sine. This formula is:\[\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b). \]

Step 2: Apply the Sum of Angles Formula

Let's apply the sum of angles formula with \( a = \frac{\pi}{3} \) and \( b = x \).\[\sin\left(\frac{\pi}{3} + x\right) = \sin\left(\frac{\pi}{3}\right)\cos(x) + \cos\left(\frac{\pi}{3}\right)\sin(x)\]

Step 3: Evaluate Sine and Cosine at \(\frac{\pi}{3}\)

The values for \(\sin\left(\frac{\pi}{3}\right)\) and \(\cos\left(\frac{\pi}{3}\right)\) are:\[\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \quad \text{and} \quad \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\]

Step 4: Simplify the Expression

Substituting the values we calculated:\[\sin\left(\frac{\pi}{3} + x\right) = \frac{\sqrt{3}}{2}\cos(x) + \frac{1}{2}\sin(x)\]This is the simplified expression.

Step 5: Graph Both Expressions

Graph the functions\[ y_1 = \sin\left(\frac{\pi}{3} + x\right) \]and\[ y_2 = \frac{\sqrt{3}}{2}\cos(x) + \frac{1}{2}\sin(x) \] to verify they produce identical graphs, confirming the expressions are equivalent.

Key Concepts

Sum of Angles Formula

The sum of angles formula is a key component in simplifying trigonometric expressions involving angle addition. When dealing with an expression like \( \sin(a + b) \), we use this formula:

• \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \)

This formula allows us to break down the sine of the sum of two angles into individual sines and cosines. Knowing these values at specific angles helps transform an expression to make it easier to handle. In our exercise, \( a = \frac{\pi}{3} \) and \( b = x \), simplifying \( \sin\left(\frac{\pi}{3} + x\right) \) into more manageable parts that ultimately aid in understanding and graphing the expression. This method is particularly useful in calculus, physics, and engineering where angle transformations are frequent.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, used to relate angles to side lengths in right triangles. There are six main trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. The most common and perhaps most crucial are sine and cosine. These functions recur in various sciences and engineering fields due to their periodic nature, which models waves and oscillations.

• The sine function, \( \sin(x) \), represents the ratio of the opposite side to the hypotenuse in a right triangle.
• The cosine function, \( \cos(x) \), represents the adjacent side to the hypotenuse ratio.

In our example, calculating \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \) and \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \) is crucial. These values are derived from special triangles, specifically the 30-60-90 triangle. Understanding them aids in breaking down and simplifying complex expressions.

Simplifying Expressions

Simplifying trigonometric expressions can make an otherwise complex problem much easier to solve or graph. The process involves breaking down an expression using known values and identities. In simplifying \( \sin\left(\frac{\pi}{3} + x\right) \), we successfully turned it into:

• \( \frac{\sqrt{3}}{2}\cos(x) + \frac{1}{2}\sin(x) \)

This transformation uses calculated constants for the sine and cosine of \( \frac{\pi}{3} \), making the expression simple enough to graph. Simplifying expressions avoids potential round-off errors and facilitates comparing different forms of the original problem. By confirming the simplified form produces identical graphs, this validates the transformation and deepens the understanding of trigonometric identities in action.