Problem 18 - Algebra and Trigonometry

Question

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. $$ a=30, \quad c=40, \quad \angle A=37^{\circ} $$

Step-by-Step Solution

Answer

There are two possible triangles: (37°, 53.1°, 89.9°) and (37°, 126.9°, 16.1°).

Step 1: Apply the Law of Sines

The Law of Sines states \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). First, use it to find \( \sin C \): \( \frac{30}{\sin 37^\circ} = \frac{40}{\sin C} \).

Step 2: Solve for \( \sin C \)

Rearrange the equation \( \frac{30}{\sin 37^\circ} = \frac{40}{\sin C} \) to solve for \( \sin C \): \[ \sin C = \frac{40 \cdot \sin 37^\circ}{30} \].

Step 3: Calculate \( \sin C \)

Find \( \sin 37^\circ \) using a calculator: \( \sin 37^\circ \approx 0.6018 \). Then, calculate \( \sin C \): \( \sin C = \frac{40 \times 0.6018}{30} \approx 0.8024 \).

Step 4: Determine possible angles for \( C \)

Since \( \sin C = 0.8024 \), \( C \) could be \( \angle C = \sin^{-1}(0.8024) \) which is approximately \( 53.1^\circ \), or the supplementary angle \( C = 126.9^\circ \) due to the sine rule.

Step 5: Calculate angle \( B \) for each possibility

For \( C = 53.1^\circ \),\( B = 180^\circ - (37^\circ + 53.1^\circ) = 89.9^\circ \). For \( C = 126.9^\circ \), \( B = 180^\circ - (37^\circ + 126.9^\circ) = 16.1^\circ \).

Step 6: Check the possibility of the two triangles

For each case, ensure the sum of angles equals \( 180^\circ \). Both \( (37^\circ, 53.1^\circ, 89.9^\circ) \) and \( (37^\circ, 126.9^\circ, 16.1^\circ) \) sum to \( 180^\circ \), confirming both triangles are possible.

Key Concepts

Solving Triangles

Solving triangles is a fundamental task in trigonometry, often involving finding unknown angles and sides of a triangle using given information. In this exercise, we focus on utilizing the Law of Sines to solve for all potential triangles with certain given measurements. When you have a triangle with a known side, an angle opposite that side, and another side, the Law of Sines becomes incredibly helpful. Here's a step-by-step for solving triangles:

• First, identify what you know: sides, angles, or any other clues.
• Apply relevant trigonometric rules—here, the Law of Sines helps to link sides to their opposite angles.
• Use the given angles to find missing angles when possible. Remember, all angles in a triangle sum to 180°.
• Rearrange and solve the equations for the unknown values, checking for both potential solutions since a sine function can have an angle and its supplement as solutions.
You'll need to check which angles and sides create viable triangles, such that all angles sum to 180° and all sides maintain the triangle's integrity.

Angle Calculation

Calculating angles is an essential part of solving triangles, especially when given some side lengths and certain angles. In trigonometry, this usually involves using the relationships that govern the properties of triangles, like the Law of Sines. Here's how you can calculate an angle:First, understand that the Law of Sines is articulated as:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]For example, to find an angle given two sides and an angle (like finding \( \angle C \) in our exercise), use the known values to solve for \( \sin C \). With \( a = 30, \angle A = 37^\circ, \text{ and } c = 40 \), apply:\[ \frac{30}{\sin 37^\circ} = \frac{40}{\sin C} \]Solve for \( \sin C \) first, then calculate \( C = \sin^{-1}(\sin C) \). Keep in mind that if \( \sin C \) resolves to a positive number less than or equal to 1, \( C \) could also be \( 180^\circ - C \) due to the sine's even function property.

Trigonometric Identities

Trigonometric identities are formulas that relate the angles and sides of a triangle. They are crucial for solving various trigonometric problems, such as this example with the Law of Sines. The key is understanding these identities to manipulate and solve for the unknowns in a triangle problem.Important Trigonometric Identities include:

• Law of Sines: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
• Sine Rule: The sine of an angle can have two possible angles, \( \theta \) and \( 180^\circ - \theta \).
• Other basic identities like \( \sin(\theta) = \sin(180^\circ - \theta) \), illustrating reflection properties.
These identities allow us to solve triangles by finding missing side lengths or angles, often under the SSA (side-side-angle) setup. Applying these identities accurately helps verify whether a single or multiple triangle solutions exist for a given problem scenario.