Problem 26 - Precalculus Mathematics for Calculus

Question

Use the Law of sines to solve for all possible triangles that satisfy the given conditions. $$a=100, \quad b=80, \quad \angle A=135^{\circ}$$

Step-by-Step Solution

Answer

\( \angle B \approx 27.78^{\circ} \), \( \angle C \approx 17.22^{\circ} \), and \( c \approx 44.77 \).

Step 1: Understanding the Law of Sines

The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We will use this law to find the missing sides and angles in the triangle.

Step 2: Setting up the Equation

Given the side lengths and angle \( a = 100 \), \( b = 80 \), and \( \angle A = 135^{\circ} \), we set up the equation using the Law of Sines: \( \frac{100}{\sin 135^{\circ}} = \frac{80}{\sin B} \).

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

Calculate \( \sin 135^{\circ} = \sin(180^{\circ} - 45^{\circ}) = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \). Substitute \( \sin 135^{\circ} \) in the equation: \( \frac{100}{\frac{\sqrt{2}}{2}} = \frac{80}{\sin B} \). This simplifies to \( \sin B = \frac{80 \times \frac{\sqrt{2}}{2}}{100} \).

Step 4: Calculate \( \sin B \)

Simplify and calculate \( \sin B \): \( \sin B = \frac{80 \sqrt{2}}{200} = \frac{2 \sqrt{2}}{5} \).

Step 5: Determining \( \angle B \)

Using \( \sin B = \frac{2 \sqrt{2}}{5} \), calculate \( \angle B \) using an inverse sine function. Approximate \( \angle B \) with a calculator to find \( \angle B \approx 27.78^{\circ} \).

Step 6: Determine \( \angle C \)

Use the angle sum theorem for triangles which states \( \angle A + \angle B + \angle C = 180^{\circ} \). Substitute the known angles: \( 135^{\circ} + 27.78^{\circ} + \angle C = 180^{\circ} \), so \( \angle C = 180^{\circ} - 135^{\circ} - 27.78^{\circ} \approx 17.22^{\circ} \).

Step 7: Calculate Side \( c \)

Use the Law of Sines again to calculate side \( c \): \( \frac{c}{\sin C} = \frac{a}{\sin A} \). Substitute known values: \( \frac{c}{\sin 17.22^{\circ}} = \frac{100}{\frac{\sqrt{2}}{2}} \) which simplifies to \( c \approx 44.77 \).

Key Concepts

Triangle Properties

Triangles, fundamental structures in geometry, have certain properties that are essential to understand. A triangle consists of three sides, three angles, and three vertices. These elements are interconnected through various geometric principles and trigonometric laws.
One important property of triangles is the angle sum theorem, which states that the sum of all internal angles in a triangle is always 180 degrees. This property allows us to find unknown angles when given other angle measures.
Additionally, when dealing with specific types of triangles, such as scalene, isosceles, or equilateral, recognizing their properties can help simplify problems. For example, in an isosceles triangle, two sides are equal, and the base angles are also equal.

• Scalene Triangle: All sides and angles are different.
• Isosceles Triangle: Two sides and angles are equal.
• Equilateral Triangle: All sides and angles are equal.

Understanding triangle properties helps when applying formulas and solving geometric problems efficiently.

Trigonometric Functions

Trigonometric functions are tools used to relate the angles and sides of triangles, especially right triangles. The three primary trigonometric functions are sine, cosine, and tangent, which are defined as:

• Sine (\( \sin \theta \) ): The ratio of the opposite side to the hypotenuse of a right triangle.
• Cosine (\( \cos \theta \) ): The ratio of the adjacent side to the hypotenuse of a right triangle.
• Tangent (\( \tan \theta \) ): The ratio of the opposite side to the adjacent side of a right triangle.

These functions, beyond right triangles, extend to work with any triangle through laws such as the Law of Sines and the Law of Cosines.
The Law of Sines, given by the equation \( \frac{a}{\sin A}= \frac{b}{\sin B }= \frac{c}{\sin C } \), applies to any triangle, allowing the calculation of unknown sides or angles when given some initial conditions.
Trigonometric functions are fundamental to solving complex geometric problems and are utilized in various fields such as physics, engineering, and architecture.

Inverse Sine Function

The inverse sine function, also known as arcsin, is used to find an angle when the value of its sine is known. The inverse functions are crucial for solving problems involving unknown angles in triangles.
The notation for the inverse sine function is \( \sin^ {-1}(x) \) or \( \mathrm{arcsin}(x) \). When you have a sine value, this function helps determine the corresponding angle. In the context of the Law of Sines, once you find \( sin B \) or any angle's sine value, you can use the inverse inute determine the angle \( B \).
It's important to understand the range of the inverse sine function, which is from \( -90^{\circ } \) to \( 90^{\circ } \), limiting the possible angles it can return. This becomes crucial in solving ambiguous triangle problems where more than one angle value might satisfy the given trigonometric condition.
Mastering the use of inverse sine makes solving triangles more straightforward, especially when computers and calculators can handle the computational work, leaving room for strategized problem-solving.