Problem 41 - Algebra 2

Question

Give a counterexample to the statement that \(\sin (\alpha+\beta)=\) \(\sin \alpha+\sin \beta\) is an identity.

Step-by-Step Solution

Answer

The statement is false as shown by \( \alpha = \frac{\pi}{3} \), \( \beta = \frac{\pi}{6} \), where \( \sin(\alpha+\beta) \neq \sin\alpha + \sin\beta \).

Step 1: Introduction to Counterexample

To show that a mathematical statement is not an identity, we can provide a counterexample, which is a specific case where the statement does not hold true.

Step 2: Choose Specific Angles

Let's choose two angles: \( \alpha = \frac{\pi}{3} \) and \( \beta = \frac{\pi}{6} \). These angles are convenient because they correspond to well-known trigonometric values.

Step 3: Evaluate \( \sin(\alpha + \beta) \)

Calculate \( \sin(\alpha + \beta) = \sin\left(\frac{\pi}{3} + \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{2}\right) \). We know \( \sin\left(\frac{\pi}{2}\right) = 1 \).

Step 4: Evaluate \( \sin \alpha + \sin \beta \)

Now calculate \( \sin \alpha + \sin \beta = \sin\left(\frac{\pi}{3}\right) + \sin\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} + \frac{1}{2} = \frac{\sqrt{3} + 1}{2} \).

Step 5: Compare and Conclude

Compare the two results: \( \sin(\alpha + \beta) = 1 \) and \( \sin \alpha + \sin \beta = \frac{\sqrt{3} + 1}{2} \). Since \( 1 eq \frac{\sqrt{3} + 1}{2} \), this shows that the statement \( \sin(\alpha + \beta) = \sin \alpha + \sin \beta \) is not a valid identity for all angle values.

Key Concepts

Trigonometric Identity

A trigonometric identity is a mathematical equation involving trigonometric functions that hold true for all values of the variable for which the equation is defined. These identities are essential tools in mathematics because they allow us to transform and simplify trigonometric expressions. Some well-known examples include

• \(\sin^2\theta + \cos^2\theta = 1\) which is known as the Pythagorean identity.
• \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), which relates three basic trigonometric functions.

A common misconception is that an expression is true for all angle measures when it may only hold in specific instances. In the example of \(\sin (\alpha + \beta)\), if someone claims it’s the same as \(\sin \alpha + \sin \beta\) for any angles \(\alpha\) and \(\beta\), we must provide a counterexample to show this isn’t an identity. Understanding the proper identities helps solve complex problems by converting an expression into a simpler form.

Sine Addition Formula

The Sine Addition Formula is a fundamental identity in trigonometry. It allows you to find the sine of a sum of two angles, expressed as: \[\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta.\]This formula is crucial because it breaks down the sine of a sum into terms involving the sines and cosines of the individual angles. Contrary to the incorrect assertion in our exercise, this is the true relationship defining how sine behaves with angle addition. For example, using specific values like \(\alpha = \frac{\pi}{3}\) and \(\beta = \frac{\pi}{6}\), you can verify this formula by substitution:

• \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\) and \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\).
• \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\) and \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\).

Substitute these into the formula:\[\sin(\alpha + \beta) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{1}{2} \cdot \frac{1}{2} = \frac{3}{4} + \frac{1}{4} = 1.\]This agrees with \(\sin\left(\frac{\pi}{2}\right) = 1\), proving the formula's accuracy.

Angle Sum Identities

Angle sum identities in trigonometry include the sine, cosine, and tangent addition formulas, which express trigonometric functions of sums of angles. They are invaluable when working with problems involving multiple angles. Here is a brief overview:- **Sine Addition:** As mentioned, this is given by: \[\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta.\]- **Cosine Addition:** This identity states: \[\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta.\]- **Tangent Addition:** For tangent, the identity is: \[\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}.\]These identities enable us to calculate the trigonometric functions of angles that are sums of other angles, facilitating more advanced trigonometric problem-solving. They are used extensively in various branches of mathematics, including calculus and complex analysis, and are also vital in physics and engineering for dealing with wave patterns and oscillations. Understanding these identities allows us to decompose trigonometric expressions, evaluate complex angles, and simplify calculations in practical applications.