Problem 111 - Algebra and Trigonometry

Question

Exercises \(110-112\) will help you prepare for the material covered in the next section. Use the appropriate values from Exercise 110 to answer each of the following. a. Is \(\sin \left(2 \cdot 30^{\circ}\right),\) or \(\sin 60^{\circ},\) equal to \(2 \sin 30^{\circ} ?\) b. Is \(\sin \left(2 \cdot 30^{\circ}\right),\) or \(\sin 60^{\circ},\) equal to \(2 \sin 30^{\circ} \cos 30^{\circ} ?\)

Step-by-Step Solution

Answer

a. No, \(\sin 2\cdot 30^\circ\) or \(\sin 60^\circ\) is not equal to \(2 \sin 30^\circ\).\n b. Yes, \(\sin 2\cdot 30^\circ\) is equal to \(2 \sin 30^\circ \cos 30^\circ\), which is an application of the double-angle identity, but not \(\sin 60^\circ\).

Step 1: Initial Calculation

Use the given values from the exercise to substitute in the given sin functions. That means you will calculate the value for: \(\sin 2\cdot 30^\circ\), \(\sin 60^\circ\), and \(2 \sin 30^\circ\).

Step 2: Comparison

Compare the values calculated in Step 1. This will confirm if \(\sin 2\cdot 30^\circ\) or \(\sin 60^\circ\) is equal to \(2 \sin 30^\circ\).

Step 3: Application of Double-Angle Identity

Now, apply the double-angle identity \(\sin{2\theta} = 2\sin{\theta}\cos{\theta}\). Calculate the right side of the equation using given values: \(2\sin 30^\circ\cos 30^\circ\)

Step 4: Comparison

Compare the values calculated in Step 3 with \(\sin 2\cdot 30^\circ\) and \(\sin 60^\circ\). This will provide the answer for part (b) of the exercise.

Key Concepts

Understanding the Double-Angle Identity

The double-angle identity is a useful tool in trigonometry that helps simplify expressions involving angles that are twice the size of another angle. Specifically, the identity for sine states that \( \sin{2\theta} = 2\sin{\theta}\cos{\theta} \). This equation allows us to find the sine of twice an angle by using the sine and cosine of the original angle.

• This identity helps simplify and solve trigonometric equations by breaking down more complex angles into simpler, known quantities.
• It is particularly helpful when evaluating angles that are not standard or when finding identities for angle measure transformations.

Once understood, it becomes a powerful tool for solving various trigonometric problems, such as those involving transformations or those needing specific simplifications.

Exploring the Sine Function

The sine function is one of the most fundamental functions in trigonometry. It deals with the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.

• Sine of an angle \( \theta \) within a right triangle is defined as \( \sin{\theta} = \frac{\text{opposite side}}{\text{hypotenuse}} \).
• It is periodic and oscillates between -1 and 1, making it perfect for modeling wave-like phenomena.

In the exercise, when calculating \( \sin 60^{\circ} \), it is commonly recognized that \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \). Knowing these fundamental sine values can greatly aid in solving more complex equations and understanding further trigonometric functions.

The Role of Angle Measurement

Angle measurement is vital in trigonometry and defines how we perceive and calculate the properties of angles. It is often seen in degrees or radians, which are the two standard units.

• Common angles used include 30°, 45°, 60°, 90°, and their radian equivalents.
• Understanding these standard angles and their trigonometric values is crucial for performing quick calculations and comparisons.

In this exercise, for instance, comparing \( \sin 60^{\circ} \) involves familiarity with these angle measures to confirm identities like the double-angle theorem. Knowing angle measures is essential for both basic exercises and advanced trigonometric applications.

Applying Mathematical Comparison in Trigonometry

Mathematical comparison in trigonometry involves comparing trigonometric values or identities to confirm relationships or solutions. This technique helps determine accuracy and consistency in solving trigonometric problems.

• It involves directly computing values such as \( \sin 60^{\circ} \), \( 2 \sin 30^{\circ} \), and \( 2 \sin 30^{\circ} \cos 30^{\circ} \) to confirm if they are equal.
• Comparisons help understand whether a trigonometric identity, like the double-angle identity, holds true for specific angles.

In the given exercise, by performing such comparisons, we can clearly evaluate whether the expressions given are valid or whether there are discrepancies in understanding or calculations. This reinforces strong problem-solving skills and precise trigonometric comprehension.