Problem 11 - Technical Mathematics with Calculus

Question

Solve triangle \(A B C\). $$A=45.6^{\circ} \quad a=7.83 \quad c=10.4$$

Step-by-Step Solution

Answer

Using the Law of Sines to find angle B and then the triangle's angle sum to find angle C, followed by Law of Sines again gives: angle B is approximately 33.7 degrees, angle C is approximately 100.7 degrees, and side b is approximately 6.30 units.

Step 1: Use the Law of Sines to find angle B

To find angle B, use the Law of Sines: \[\frac{a}{\sin A} = \frac{c}{\sin C}\]. Since angle A and side a are opposite to each other and side c is known, we can write \[\sin B = \frac{a \cdot \sin A}{c}\] and then find angle B by taking the inverse sine.

Step 2: Calculate the value of angle B

Substitute the known values into the equation from Step 1: \[\sin B = \frac{7.83 \cdot \sin 45.6^\circ}{10.4}\], and then use a calculator to find angle B.

Step 3: Find angle C using the angle sum of a triangle

Use the fact that the sum of angles in a triangle is 180 degrees to find angle C: \[ C = 180^\circ - A - B \].

Step 4: Use the Law of Sines to find side b

Now that we have angle C, use the Law of Sines again to find side b: \[\frac{a}{\sin A} = \frac{b}{\sin B}\]. Rearrange to solve for b: \[b = \frac{a \cdot \sin B}{\sin A}\] and substitute the known values to find the length of side b.

Key Concepts

Solving Triangles

When we refer to 'solving a triangle', we are talking about finding the unknown angles and sides of a triangle when some of these values are already given. In trigonometry, several methods are used to complete this task, with the Law of Sines being one of the foremost techniques for solving non-right triangles.

The Law of Sines is applicable in situations where we know either two angles and a side (AAS or ASA) or two sides and an angle opposite one of them (SSA). The process involves setting up ratios of the sides to the sine of their opposite angles. Once you find one missing value, you can typically use other trigonometric laws or the angle sum property of triangles to find the remaining unknowns.

Trigonometric Functions

The trigonometric functions, including sine, cosine, and tangent, are fundamental to many areas of mathematics, especially in the study of triangles. These functions relate the angles of a triangle to the ratios of its sides.

For example, the sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. These functions allow us to solve triangles because they provide a connection between angles and side lengths. The Law of Sines makes use of the sine function specifically to compare the ratio of each side's length to the sine of its opposite angle, which can be used to solve for unknown sides or angles.

Inverse Sine

Inverse sine, also known as arcsine (denoted as \(\sin^{-1}\)), is a function that reverses the action of the sine function. If you have the sine of an angle and you wish to find the angle itself, the inverse sine function is used. This is especially important when solving for angles after setting up the Law of Sines.

For example, when you have calculated the numerical value of the sine of an angle, the inverse sine function on a calculator can help you find the actual angle measurement in degrees or radians. However, it's important to remember that sine can have the same value for two different angles within a full circle (0 to 360 degrees), so knowing the context of your triangle is vital to determine the correct angle.

Angle Sum of Triangle

One of the fundamental properties of triangles is that the sum of their internal angles will always equal 180 degrees. This angle sum property is particularly valuable when two angles are known, and the third is being sought. It provides a simple but effective rule to complete the puzzle of missing angle values in a triangle.

By deducting the sum of the known angles from 180 degrees, you can find the measurement of the missing angle. This step is often used after applying the Law of Sines to find one or two angles as it naturally complements the process, leading to a complete solution for the rest of the triangle's measurements.