Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 37-46: Verify the identity. $$ \cos \left(\theta+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\cos \theta-\sin \theta) $$

Step-by-Step Solution

Answer

The identity is verified as correct.

Step 1: Recall the Angle Addition Formula for Cosine

The cosine of the sum of two angles \( \theta \) and \( \frac{\pi}{4} \) can be expanded using the angle addition formula for cosine. This formula is \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \).

Step 2: Apply the Angle Addition Formula

In the given identity, substitute \( a = \theta \) and \( b = \frac{\pi}{4} \). This gives us \( \cos\left(\theta + \frac{\pi}{4}\right) = \cos(\theta)\cos\left(\frac{\pi}{4}\right) - \sin(\theta)\sin\left(\frac{\pi}{4}\right) \).

Step 3: Substitute Trigonometric Values for \( \frac{\pi}{4} \)

For \( x = \frac{\pi}{4} \), we know \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) and \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \). Substitute these values into the expression.

Step 4: Simplify the Equation

Substituting the values, we get:\[ \cos\left(\theta + \frac{\pi}{4}\right) = \cos(\theta) \cdot \frac{\sqrt{2}}{2} - \sin(\theta) \cdot \frac{\sqrt{2}}{2} \]Factor out \( \frac{\sqrt{2}}{2} \) to get:\[ \cos\left(\theta + \frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} (\cos(\theta) - \sin(\theta)) \].

Step 5: Verify the Identity

The expression \( \frac{\sqrt{2}}{2} (\cos(\theta) - \sin(\theta)) \) matches the right side of the given identity, confirming that the identity is indeed correct.

Key Concepts

Angle Addition Formula

The Angle Addition Formula is a foundational concept in trigonometry. It helps to calculate the trigonometric values of the sum or difference of two angles. For the cosine function, the formula is expressed as

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

This formula is particularly useful when you need to find the cosine of a compound angle, where direct measurement is not straightforward. For instance, in the exercise, we were given the compound angle \( \theta + \frac{\pi}{4} \). Using the Angle Addition Formula allowed the expansion of this angle into manageable parts involving \( \theta \) and \( \frac{\pi}{4} \).

Using this method, you can break down complex trigonometric expressions into simpler components, making the calculations much easier. In essence, the Angle Addition Formula acts like a "cheat sheet" for deconstructing and solving trigonometric problems involving sums of angles.

Cosine Function

The cosine function is one of the primary trigonometric functions, alongside sine and tangent. It represents the horizontal coordinate of a point on the unit circle and is crucial in calculating angles and sides in triangles. The key aspect of the cosine function is how it reacts to different angle inputs.

When the cosine function is applied to an angle like \( \frac{\pi}{4} \), it results in a specific value, which in this case, is

• \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)

This value is derived from the geometry of the unit circle, where \( \frac{\pi}{4} \) (or 45 degrees) is an angle in the first quadrant. Both sine and cosine values are positive here, simplifying many calculations.

The ability to substitute these fixed trigonometric values, like \( \frac{\sqrt{2}}{2} \), into more complex expressions allows for easier manipulation and simplification of trigonometric equations. Understanding the role and behavior of the cosine function is essential for mastering various trigonometric identities and problems.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions often involves using identities and algebraic techniques to make complex equations more manageable. In the given exercise, simplification was achieved by applying the Angle Addition Formula and known trigonometric values.

Once the formula \( \cos(a+ b) = \cos(a)\cos(b) - \sin(a)\sin(b) \) is applied, we substituted the trigonometric values for \( \cos\left(\frac{\pi}{4}\right) \) and \( \sin\left(\frac{\pi}{4}\right) \), both of which are \( \frac{\sqrt{2}}{2} \).

• This substitution allows us to combine and factor out common terms like \( \frac{\sqrt{2}}{2} \), resulting in a simpler expression: \( \frac{\sqrt{2}}{2} (\cos(\theta) - \sin(\theta)) \).

By factoring out common terms, the original trigonometric identity is confirmed.Understanding simplification helps in verifying identities and solving equations much more effectively, making it a crucial skill in trigonometry. These steps not only confirm identities but also aid in practical applications like solving problems in physics and engineering related to waves and rotational motion.