Problem 353 - Precalculus

Question

For the following exercises, use the given information to find the lengths of the other two sides of the right triangle. $$ \cos B=\frac{3}{5}, a=6 $$

Step-by-Step Solution

Answer

The lengths of the other two sides are 6 and 8.

Step 1: Understand the Known Values

In the given right triangle, we know that \( \cos B = \frac{3}{5} \) and one side \( a = 6 \). Here, \( a \) is the side adjacent to angle \( B \), and \( \cos B = \frac{\text{Adjacent}}{\text{Hypotenuse}} \). This implies that the hypotenuse \( c = \frac{5}{3} \times 6 = 10 \).

Step 2: Use the Pythagorean Theorem

To find the other side, use the Pythagorean theorem: \( a^2 + b^2 = c^2 \). Plug in the known values: \( 6^2 + b^2 = 10^2 \).

Step 3: Simplify the Equation

Calculate \( 6^2 = 36 \) and \( 10^2 = 100 \). Substitute these into the equation to get \( 36 + b^2 = 100 \).

Step 4: Solve for \( b \)

Rearrange the equation to solve for \( b^2 \): \( b^2 = 100 - 36 = 64 \). Taking the square root of both sides, we find \( b = \sqrt{64} = 8 \).

Step 5: Verify the Results

Verify the result using the Pythagorean theorem. Check if \( 6^2 + 8^2 = 10^2 \). Calculating yields \( 36 + 64 = 100 \), which confirms our calculations are correct.

Key Concepts

Cosine

The concept of cosine is one of the fundamental parts of trigonometry. Cosine relates the angles and side lengths in triangles, especially in right triangles. Cosine of an angle is calculated using the ratio of the length of the adjacent side to the hypotenuse in a right triangle.
In mathematical terms:

• Cosine is represented by \( \cos \theta \).
• The formula is: \( \cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \).

In our exercise, we see \( \cos B = \frac{3}{5} \), which informs us that the adjacent side over the hypotenuse is 3 to 5 in ratio.
This knowledge is crucial for solving problems involving right triangles, as it helps relate angles to specific side lengths.

Right Triangle

A right triangle is a type of triangle that features a special angle - specifically one angle is always 90 degrees. This distinctive characteristic shapes the entire triangle's properties, making it an interesting subject in geometry.
Key features of right triangles include:

• One angle is always 90 degrees.
• The side opposite the right angle is known as the hypotenuse, and is the longest side.
• The other two sides are the adjacent and opposite sides, relative to a given angle.

In trigonometry, the angles within a right triangle help define the relationships and ratios between its side lengths, such as cosine for angle B in our original exercise. Understanding right triangles is essential as they form the foundation for several trigonometric functions and theorems.

Pythagorean Theorem

The Pythagorean Theorem is a cornerstone of geometry and trigonometry dealing particularly with right triangles. It states that in every right triangle, the square of the hypotenuse's length is always equal to the sum of the squares of the other two side lengths.
The theorem's equation is:

• \( a^2 + b^2 = c^2 \)

where \( a \) and \( b \) are the lengths of the two smaller sides, and \( c \) is the hypotenuse.
In our calculation steps, we used this theorem to determine one of the other sides when given the hypotenuse and one side (adjacent). By substituting into the equation, solving further allows the calculation of the remaining side's length, ensuring a complete understanding of the triangle's dimensions. This theorem is vital in verifying side lengths and solving unknowns in right triangles.