Problem 8 - Trigonometry

Question

For each of the following triangles, solve for \(B\) and use the results to explain why the triangle has the given number of solutions. \(A=30^{\circ}, b=10 \mathrm{~cm}, a=5 \mathrm{~cm}\); one solution

Step-by-Step Solution

Answer

The angle \(B\) is \(90^{\circ}\), making it a right triangle, explaining the one solution.

Step 1: Understand the Given Information

We are given a triangle with an angle \(A = 30^{\circ}\) and sides \(a = 5\, \text{cm}\) and \(b = 10\, \text{cm}\). We need to determine the measure of angle \(B\) and explain why the triangle has only one solution.

Step 2: Apply the Law of Sines

The Law of Sines states \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Using the given values, substitute into the equation: \( \frac{5}{\sin 30^{\circ}} = \frac{10}{\sin B} \).

Step 3: Solve for \( \sin B \)

Calculate \( \sin 30^{\circ} = 0.5 \). Therefore, \( \frac{5}{0.5} = 10 \), which simplifies to \( 10 = \frac{10}{\sin B} \). Thus, \( \sin B = 1 \).

Step 4: Find the Measure of Angle \(B\)

If \( \sin B = 1 \), then \( B = 90^{\circ} \). In a triangle, the angle opposite the hypotenuse (the longest side) is a right angle.

Step 5: Explain the Number of Solutions

Since we found \( B = 90^{\circ} \), the triangle is a right triangle with the hypotenuse as the longest side (\(b = 10\, \text{cm} \)). There is only one such configuration possible, hence there is one solution.

Key Concepts

Law of Sines

The Law of Sines is an essential tool in trigonometry for solving triangles. It is especially helpful when dealing with non-right triangles, but it can be applied to right triangles as well in certain contexts. The Law of Sines relates the sides of a triangle to the sines of its angles. Specifically, it states that the ratio of the length of a side of a triangle to the sine of the angle opposite is constant for all three sides. Mathematically, this is shown as:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

This formula is useful for determining unknown angles or sides in a triangle, as you can see in our original exercise. By applying this law, we find that if two angles and one side are known, or two sides and a non-enclosed angle are known, the rest of the triangle can be solved. Remember, if the three sides are known, the Law of Sines can still help establish possible angle measures.

Triangle Solutions

Finding solutions for triangles involves knowing which laws and formulas to apply in different scenarios. There are two common scenarios: finding missing sides when angles are known, or finding missing angles when sides are given.

• The Law of Sines works well for both scenarios and allows you to find all the potential angles or sides of a triangle. But sometimes, you may stumble upon the so-called ambiguous case, which occurs in non-right triangles when you know two sides and a non-enclosed angle (SSA condition). This might lead to two possible triangle configurations, one, or sometimes even none.
• In the solution of our exercise, the problem provides a setup where this ambiguity doesn't occur: the triangle resolves into a single solution, a right triangle, due to angle \( B \) becoming \( 90^{\circ} \).

By understanding triangle solutions and potential ambiguities, you can better anticipate and resolve different triangular setups.

Right Triangle

A right triangle is uniquely defined by having one of its angles equal to \( 90^{\circ} \). This right angle creates special properties and facilitates the use of trigonometric ratios such as sine, cosine, and tangent. In right triangles, the side opposite the right angle is the longest and is known as the hypotenuse.New
Understanding the distinction between right and other triangles is crucial in trigonometry. Right triangles allow the use of Pythagorean Theorem:

• \( a^2 + b^2 = c^2 \)

where \( c \) is the hypotenuse. The original exercise demonstrates this when determining that angle \( B \) is \( 90^{\circ} \), thus identifying the triangle as right. With known sides and angles, the matter of additional triangles or ambiguous cases is eliminated, leaving only one solution.