Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity. $$\sin 2 x=2 \sin x \cos x$$

Step-by-Step Solution

Answer

The equation \(\sin 2x = 2 \sin x \cos x\) is a trigonometric identity.

Step 1: Understanding the Identity to Verify

The given equation is \( \sin 2x = 2 \sin x \cos x \). We need to verify if this equation holds for all values of \(x\). This is a well-known trigonometric identity called the Double Angle Formula for sine.

Step 2: Recall the Double Angle Sine Formula

The Double Angle Formula for sine states that: \( \sin 2x = 2 \sin x \cos x \). This formula is a standard identity in trigonometry, meaning it should hold true for any angle \(x\).

Step 3: Substitute a Value for Testing

Choose a specific value for \(x\) to test the identity. Let's try \( x = \frac{\pi}{4} \).

Step 4: Evaluate Both Sides

Evaluate the left-hand side (LHS) of the equation: \( \sin 2\left( \frac{\pi}{4} \right) = \sin \left( \frac{\pi}{2} \right) = 1 \).Evaluate the right-hand side (RHS) of the equation: \( 2 \sin \left( \frac{\pi}{4} \right) \cos \left( \frac{\pi}{4} \right) = 2 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2} = 2 \times \frac{1}{2} = 1 \).

Step 5: Conclusion from Evaluation

Since the LHS and RHS are equal (both are 1), this verifies that the formula \(\sin 2x = 2 \sin x \cos x\) holds true for \(x = \frac{\pi}{4}\). As this is a known identity, it holds for all \(x\).

Key Concepts

Double Angle Formulas

In trigonometry, the double angle formulas are crucial tools that simplify expressions involving trigonometric functions. They specifically pertain to angles that are double another, offering expressions for trigonometric functions like sine, cosine, and tangent in terms of themselves. The double angle formula for sine is expressed as \( \sin 2x = 2 \sin x \cos x \). This identity is renowned for its simplicity and utility, as it helps in reducing complex trigonometric expressions into more manageable forms.
One reason the double angle formulas are so beneficial is that they provide a straightforward way to evaluate trigonometric functions at larger angles by using smaller, more familiar angles. This makes them invaluable in calculus, physics, engineering, and other fields requiring precise trigonometric calculations. As a result, this formula, alongside others like \( \cos 2x = \cos^2 x - \sin^2 x \), serves as a fundamental building block in mathematics.
Utilizing these formulas often involves substituting specific values, simplifying expressions, or proving further trigonometric identities. Mastering these formulas elevates one's ability to tackle a wide range of mathematical problems with greater ease. They exemplify the beauty and elegance of trigonometric identities in connecting various mathematical concepts smoothly.

Sine Function

The sine function is one of the primary trigonometric functions, symbolized as \( \sin x \). It plays a crucial role in the study of triangles and circles. In a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse. More specifically, if you have a right triangle with an angle \(x\), the sine is calculated as:

• \( \sin x = \frac{\text{opposite side}}{\text{hypotenuse}} \)

Beyond triangles, sine is also essential in the cycle of a unit circle, where \( \sin x \) represents the y-coordinate of a point on the circle corresponding to an angle \(x\) from the positive x-axis.
Mathematical properties of sine include:

• It's periodic with a period of \(2\pi\), which means \( \sin(x + 2\pi) = \sin x \).
• It is an odd function, indicating \( \sin(-x) = -\sin x \).

These characteristics make sine function highly useful in modeling cyclical phenomena like sound waves and tides. Understanding how sine function behaves is essential when working with trigonometric identities and transformations. This knowledge allows you to simplify expressions and solve equations effectively.

Angle Evaluation

Evaluating angles in trigonometry often pertains to the process of simplifying expressions or calculating the function's value at specific angles. This becomes crucial in verifying trigonometric identities, like those discussed in the original exercise. To evaluate angles effectively:

• Start by choosing a simple value for the angle, often something manageable like \( \frac{\pi}{4} \), \( \frac{\pi}{6} \), or \( \frac{\pi}{3} \).
• This substitution helps check if the identity holds true for these specific angle values.
• Ensuring both sides of the equation yield the same results strengthens the validity of the identity at any angle.

Let's consider evaluating the sine function at \( x = \frac{\pi}{4} \), a commonly used angle:

• For \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

The process often helps create simple verifications and understanding. By recalculating known angles and performing these evaluations, one gains insight into how angles influence the results of trigonometric functions. This is particularly beneficial when working through mathematical proofs or when establishing the foundational knowledge needed for more advanced studies in mathematics and sciences.