Problem 29 - Calculus

Question

Rewrite \(\cos \theta+\sin \theta\) as \(\sqrt{2} \sin (\theta+\phi)\) by choosing the correct "phase angle" \(\phi\). (Make the equation correct at \(\theta=0 .\) Square both sides to check.)

Step-by-Step Solution

Answer

The correct phase angle \(\phi\) is \(\frac{\pi}{4}\).

Step 1: Rewrite Trigonometric Expression

Start with the given expression: \[ \cos \theta + \sin \theta = \sqrt{2} \sin (\theta + \phi) \]To find \(\phi\), use the sine addition formula: \[ \sin (\theta + \phi) = \sin \theta \cos \phi + \cos \theta \sin \phi \]Thus, the equation becomes:\[ \cos \theta + \sin \theta = \sqrt{2} (\sin \theta \cos \phi + \cos \theta \sin \phi) \]

Step 2: Match Coefficients

Equate the coefficients of \(\cos \theta\) and \(\sin \theta\) from both sides:1. \(1 = \sqrt{2} \sin \phi\)2. \(1 = \sqrt{2} \cos \phi\)These equations can be rewritten as:\[ \sin \phi = \frac{1}{\sqrt{2}} \]\[ \cos \phi = \frac{1}{\sqrt{2}} \]

Step 3: Find \(\phi\)

The values \( \sin \phi = \frac{1}{\sqrt{2}} \) and \( \cos \phi = \frac{1}{\sqrt{2}} \) imply \(\phi = \frac{\pi}{4}\). This is because \(\sin\) and \(\cos\) of \(\frac{\pi}{4}\) are both \(\frac{1}{\sqrt{2}}\).

Step 4: Verification at \(\theta = 0\)

Check if the equation holds when \(\theta = 0\):\[ \cos 0 + \sin 0 = \sqrt{2} \sin (0 + \frac{\pi}{4}) \]This simplifies to:\[ 1 = \sqrt{2} \cdot \frac{1}{\sqrt{2}} \]\[ 1 = 1 \]The equation holds true.

Step 5: Squaring Both Sides for Additional Verification

Square both sides to further verify:\[ (\cos \theta + \sin \theta)^2 = (\sqrt{2} \sin (\theta + \frac{\pi}{4}))^2 \]The left side becomes:\[ \cos^2 \theta + 2\cos \theta \sin \theta + \sin^2 \theta = 1 + 2\cos \theta \sin \theta \]And the right side is:\[ 2\sin^2 (\theta + \frac{\pi}{4}) = 2(\frac{1}{2} + \frac{1}{2} \sin 2\theta) = 1 + \sin 2\theta \]Both simplify to the same result, confirming the correctness.

Key Concepts

Trigonometric Identities

Trigonometric identities form the backbone of trigonometry. They are equations involving trigonometric functions that are true for every value of the occurring variables. Two of the most fundamental identities are the Pythagorean identity and co-function identities.
The Pythagorean identity states that for any angle \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is useful in simplifying expressions and proving other identities.
Co-function identities show the relationship between functions of complementary angles, for example, \( \sin (\frac{\pi}{2} - \theta) = \cos \theta \) and \( \cos (\frac{\pi}{2} - \theta) = \sin \theta \).
When working with trigonometric identities, you can use them to simplify expressions or solve equations. Substituting equivalent expressions using identities often makes complex problems more manageable.

Angle Addition Formulas

The angle addition formulas are fundamental tools in trigonometry that allow us to find the trigonometric functions of sums or differences of angles. They can be particularly useful, for instance, in verifying or transforming expressions. For sine and cosine, the formulas are:

• \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
• \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)

In our original exercise, these formulas were applied to express \( \sin(\theta + \phi) \) in terms of \( \sin \theta \) and \( \cos \theta \) to facilitate finding a suitable \( \phi \).
By breaking down the sum of angles into these components, we can adjust the coefficients to match expressions on both sides of trigonometric equations. This strategy not only helps solve problems but also deepens our understanding of how angles interact within these functions.

Verification

Verification is a crucial step in mathematics that ensures the correctness of solutions or equivalencies. It often involves proving that an expression holds true by substituting values or demonstrating equivalence through algebraic manipulation.
In trigonometry particularly, verification can involve checking identities for specific values like \( \theta = 0 \) or using methods like squaring both sides, as seen in our exercise. After expressing \( \cos \theta + \sin \theta \) using the correct phase angle \( \phi \), verification at \( \theta = 0 \) was crucial to confirm the relationship holds across all values. Further confirmation was achieved by squaring both sides of the equation and ensuring they equated under simple trigonometric transformations. This approach can reveal any potential errors in previous steps, strengthening the validity of the equation or identity.
Consistently ensuring that your computations hold under verification is a powerful strategy in mathematics to achieve reliable results, especially when dealing with identities or complex transformation of functions.