Problem 55 - Algebra and Trigonometry

Question

A triangle has an area of \(16 \mathrm{in}^{2},\) and two of the sides of the triangle have lengths 5 \(\mathrm{in.}\) and 7 in. Find the angle included by these two sides.

Step-by-Step Solution

Answer

The angle is approximately 66.42 degrees.

Step 1: Understanding the Problem

We are given a triangle with an area of 16 square inches and two sides measuring 5 inches and 7 inches. We need to find the angle between these two sides.

Step 2: Use the Formula for Area of a Triangle

The formula for the area of a triangle when two sides and the included angle is known is \( \text{Area} = \frac{1}{2}ab \sin(C) \), where \(a\) and \(b\) are the sides and \(C\) is the angle between them. Here, \(a = 5\), \(b = 7\), and the area is 16.

Step 3: Set Up the Equation

Plug in the known values into the formula. This gives us the equation: \( 16 = \frac{1}{2} \times 5 \times 7 \times \sin(C) \).

Step 4: Solve for \(\sin(C)\)

Rearrange the equation to solve for \( \sin(C) \). First, multiply both sides by 2 to get rid of the fraction: \( 32 = 35\sin(C) \). Then, divide both sides by 35: \( \sin(C) = \frac{32}{35} \).

Step 5: Calculate the Angle

Use the inverse sine function to calculate \( C \). This gives \( C = \sin^{-1}\left(\frac{32}{35}\right) \). Calculate this using a calculator.

Key Concepts

Area of a Triangle

One of the most useful formulas for determining the area of a triangle involves the sine function. It’s particularly handy when you know two sides of a triangle and the angle included between them. This method is often referred to as the "Area Rule." In general, the formula to find the area of such a triangle is given by:

• \( \text{Area} = \frac{1}{2}ab \sin(C) \)

Here:

• \(a\) and \(b\) are the lengths of the two sides.
• \(C\) is the angle between these two sides.

For example, if you have a triangle with sides measuring 5 inches and 7 inches and you know the area is 16 square inches, you can use this formula to find the angle between the sides. You simply plug in the values of the sides and the area to solve for \( \sin(C) \). Then you use inverse trigonometry to find the angle \( C \) itself.

Sine Function

The sine function is one of the fundamental trigonometric functions, vital in studying triangles and oscillating phenomena. It comes from comparing the ratios in a right triangle and extends to the unit circle.In the context of triangles, specifically, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse. But, it can also relate to non-right triangles using the formula for the triangle's area when two sides and the included angle are known.
Mathematically, if you have a triangle with sides \(a\) and \(b\), and want to find the area given an angle \(C\), you use:

• \( \sin(C) = \frac{2 \times \text{Area}}{a \times b} \)

This formula helps find the sine of the included angle by rearranging the given area formula. Once you have \( \sin(C) \), you can use the inverse sine (often written as \( \sin^{-1} \)) to compute the exact angle.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential for solving problems involving unknown angles in triangles. They effectively reverse the standard trigonometric functions to find angles when the function's value is known.Each trigonometric function has its inverse:

• Sine (\( \sin \)) becomes inverse sine (\( \sin^{-1} \))
• Cosine (\( \cos \)) becomes inverse cosine (\( \cos^{-1} \))
• Tangent (\( \tan \)) becomes inverse tangent (\( \tan^{-1} \))

Using the inverse sine function, denoted as \( \sin^{-1} \), you can find an angle \( C \) from the sine value. For instance, if \( \sin(C) = \frac{32}{35} \), you use a calculator to find:

• \( C = \sin^{-1}\left(\frac{32}{35}\right) \)

This calculation will provide the size of angle \( C \), enabling you to solve the triangle completely. Understanding how these inverse functions work is crucial to gaining deeper insights into trigonometry and solving angles efficiently.