Problem 74 - Precalculus

Question

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. $$ \cos \left(\frac{5 \pi}{4}+x\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{\sqrt{2}}{2}(\sin(x) - \cos(x)) \). Graphs should be the same.

Step 1: Use the Angle Sum Identity for Cosine

The angle sum identity for cosine states that \( \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) \). Here, we need to identify \( a = \frac{5\pi}{4} \) and \( b = x \). We will substitute these values into the formula.

Step 2: Calculate \( \cos(a) \) and \( \sin(a) \)

First, calculate \( \cos(\frac{5\pi}{4}) \) and \( \sin(\frac{5\pi}{4}) \). Since \( \frac{5\pi}{4} \) is in the third quadrant of the unit circle and it is equivalent to adding \( \pi \) to \( \frac{\pi}{4} \), the cosine and sine values will be negative for both. We have \( \cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} \) and \( \sin(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} \).

Step 3: Substitute into the Angle sum Formula

Substitute the values of \( \cos(a) \) and \( \sin(a) \) into the angle sum identity: \[ \cos \left(\frac{5\pi}{4} + x\right) = \left(-\frac{\sqrt{2}}{2}\right) \cos(x) - \left(-\frac{\sqrt{2}}{2}\right) \sin(x) \]. This simplifies to: \[ \cos \left(\frac{5\pi}{4} + x\right) = -\frac{\sqrt{2}}{2}(\cos(x) - \sin(x)) \].

Step 4: Simplify the Expression

The expression simplifies to: \[ \cos \left(\frac{5\pi}{4} + x\right) = \frac{\sqrt{2}}{2}(\sin(x) - \cos(x)) \]. The negative sign in front of the fraction has been absorbed into changing the difference of sine and cosine.

Step 5: Graph the Functions

Graph the original function \( \cos \left(\frac{5\pi}{4} + x\right) \) and the simplified function \( \frac{\sqrt{2}}{2}(\sin(x) - \cos(x)) \). Both functions should yield identical graphs given the trigonometric identity used, reaffirming that they represent the same expression.

Key Concepts

Angle Sum Identity

The Angle Sum Identity for cosine is a fundamental tool in trigonometry. It allows you to find the cosine of a sum of two angles using a specific formula. The formula is given by:\[ \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) \]In other words, it breaks down the complex expression \( \cos(a + b) \) into simpler parts, using the cosine and sine of the individual angles \( a \) and \( b \). This is particularly useful when dealing with angles that are not standard on the unit circle. With this identity, you can easily perform calculations for more complicated angles like those that involve multiples of \( \pi \) or radians.

Here's how it works in practical terms. Suppose we're working with the problem \( \cos \left(\frac{5\pi}{4} + x\right) \). Rather than grappling with this sum directly, you split it into two separate, more manageable parts where \( a = \frac{5\pi}{4} \) and \( b = x \). By applying the identity, the calculation becomes straightforward, and you break down the complexity by working with \( \cos(a) \) and \( \sin(a) \), which simplifies the process significantly.

Cosine Function

The cosine function is one of the core trigonometric functions alongside sine and tangent. It’s especially useful for determining the horizontal component of an angle on the unit circle.

• The cosine of an angle \(\theta\) is defined as the x-coordinate of the point on the unit circle corresponding to that angle.
• It is an even function, meaning \( \cos(-x) = \cos(x) \).
• The function is periodic with a period of \( 2\pi \), which means that \( \cos(\theta) = \cos(\theta + 2\pi n) \) for any integer \( n \).

In the exercise you encountered, you are using the cosine function to evaluate \( \cos \left(\frac{5\pi}{4} + x\right) \). The value of \( \cos \left(\frac{5\pi}{4}\right) \) is particularly notable because it lies in the third quadrant of the unit circle. Here, both sine and cosine are negative, leading to \( \cos \left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \), which factors into the simplified form of the expression.

Graphing Functions

Graphing trigonometric functions is an excellent way to visually verify and understand these concepts. By graphing, you can see how transformations affect the trigonometric functions.

• To graph the original expression \( \cos \left(\frac{5\pi}{4} + x\right) \), start by understanding it as a cosine wave shifted horizontally.
• Likewise, graph its equivalent, simplified expression \( \frac{\sqrt{2}}{2}(\sin(x) - \cos(x)) \). Despite being an algebraically transformed expression, it should yield the same waveform.

Both graphs will overlap perfectly. This reflects the identity property: even though algebraic manipulation might make them appear different, graphically they remain identical. Consistency in graphs confirms that simplification does not alter the inherent properties of the expression, showcasing the power and utility of trigonometric identities.

Graphing helps not only with visual verification but also deepens comprehension of function behaviors, periodicity, and transformations. It transforms abstract formulas into concrete patterns.