Problem 4 - Algebra 2 and Trigonometry

Question

In \(\triangle A B C,\) if \(a=24, \mathrm{m} \angle A=\frac{\pi}{6},\) and \(\mathrm{m} \angle B=\frac{\pi}{2},\) find the exact value of \(b\) in simplest form.

Step-by-Step Solution

Answer

The value of \( b \) is 48.

Step 1: Identify Known Values

We know that in the triangle \( \triangle ABC \), \( a = 24 \), \( \angle A = \frac{\pi}{6} \), and \( \angle B = \frac{\pi}{2} \). Since \( \angle B \) is a right angle, triangle \( \triangle ABC \) is a right triangle.

Step 2: Use Angle Sum Property

Use the angle sum property of triangles which states that the sum of the angles in a triangle is \( \pi \). Therefore, \( \angle A + \angle B + \angle C = \pi \). This means \( \frac{\pi}{6} + \frac{\pi}{2} + \angle C = \pi \).

Step 3: Solve for \( \angle C \)

Substitute the known values: \( \frac{\pi}{6} + \frac{\pi}{2} + \angle C = \pi \). Convert \( \frac{\pi}{2} \) to \( \frac{3\pi}{6} \) for a common denominator, giving \( \frac{\pi}{6} + \frac{3\pi}{6} + \angle C = \pi \). Simplifying gives \( \frac{4\pi}{6} + \angle C = \pi \). So, \( \angle C = \pi - \frac{4\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3} \).

Step 4: Apply Sine Rule

In a right triangle, \( a = 24 \) is opposite \( \angle C \) and adjacent to \( \angle A \). Therefore, we use the sine rule for \( b \), the side opposite \( \angle A \): \( \frac{b}{\sin B} = \frac{a}{\sin A} \). Since \( \sin B = 1 \) (because \( B = \frac{\pi}{2} \)), the formula becomes \( b = \frac{a}{\sin A} \).

Step 5: Calculate \( b \)

Substitute the known values: \( \sin A = \sin(\frac{\pi}{6}) = \frac{1}{2} \). Thus, \( b = \frac{24}{\frac{1}{2}} = 24 \times 2 = 48 \).

Key Concepts

Right Triangle

A **right triangle** is a special type of triangle where one of the angles is exactly 90 degrees or \( \frac{\pi}{2} \) radians. This angle is known as the right angle. In such triangles, there are only two distinct types of components to understand:

• The hypotenuse: the side opposite the right angle, and the longest side of the triangle.
• The two legs: the sides adjacent to the right angle.

Right triangles have unique properties, including the Pythagorean theorem, which is not the focus of this exercise but important to know. In this particular problem, \( \triangle ABC \) is identified as a right triangle because \( \angle B = \frac{\pi}{2} \). With this knowledge, specific trigonometric rules can be applied to find unknown side lengths and angles.

Sine Rule

The **Sine Rule**, also known as the Law of Sines, is a helpful tool in finding unknown sides or angles in any triangle, be it right or not. However, in right triangles, this rule simplifies under certain conditions. Here’s the formula:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]where \( a, b, c \) are the side lengths opposite angles \( A, B, C \), respectively. In our exercise, it simplifies since \( \sin B = 1 \) because \( B \) is \( 90 \) degrees, which makes calculations much easier. For right triangles when seeking for a side like \( b \), the relationship is straightforward since \( \sin B = 1 \), applying the formula: \[ b = \frac{a}{\sin A} \]This is very useful when direct angle-side relationships are known, as it allows us to easily solve for missing sides and verify triangle consistencies.

Angle Sum Property

The **angle sum property** is a fundamental characteristic of triangles, stating the sum of the internal angles in any triangle is always equal to \( \pi \) radians (or 180 degrees). In our case, using this property helps in deducing unknown angles when the other angles in the triangle are known. For \( \triangle ABC \), the relationship can be expressed as:

• \( \angle A + \angle B + \angle C = \pi \)

Given the exercise specifics, we had \( \angle A = \frac{\pi}{6} \) and \( \angle B = \frac{\pi}{2} \). Therefore, to find \( \angle C \), simply substitute these known values and solve:\[ \angle C = \pi - \left( \frac{\pi}{6} + \frac{\pi}{2} \right) = \frac{\pi}{3} \]Using these steps simplifies working with triangle angles, ensuring the calculations abide by the total \( \pi \) radians in sum.

Exact Values

**Exact values** in trigonometry refer to exact trigonometric ratios known for specific angles. These are vital in solving problems without relying on approximations or numeric calculators. Some commonly used exact values include angles like \( \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{4} \). For \( \frac{\pi}{6} \), we have the trigonometric identity:

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

In the given problem, we leverage **exact values** to efficiently find \( b \) by substituting directly while using the sine rule:\[ b = \frac{24}{\frac{1}{2}} = 48 \]Utilizing these known values means we can remain in the domain of precision, which is critical when solving algebraic expressions involving trigonometry.