Problem 28 - Algebra and Trigonometry

Question

A right triangle has a hypotenuse of length 10 centimeters. If one angle is \(40^{\circ},\) find the length of each leg.

Step-by-Step Solution

Answer

Adjacent leg: 7.66 cm; Opposite leg: 6.43 cm.

Step 1 - Identify the Right Triangle Components

In a right triangle, the hypotenuse is the side opposite the right angle and the two legs are the other two sides. The given hypotenuse is 10 cm, and one angle is 40 degrees.

Step 2 - Identify the Adjacent and Opposite Legs

The given angle of 40 degrees is not the right angle. Therefore, identify the legs as opposite and adjacent to this angle. In this case, the adjacent leg is next to the 40-degree angle and the opposite leg is opposite to the 40-degree angle.

Step 3 - Use the Cosine Function to Find the Adjacent Leg

The cosine of an angle in a right triangle is the ratio of the length of the adjacent leg to the length of the hypotenuse. Using the cosine function: \[ \text{cos}(40^{\text{°}}) = \frac{\text{Adjacent Leg}}{10} \]Rearrange to solve for the adjacent leg: \[ \text{Adjacent Leg} = 10 \times \text{cos}(40^{\text{°}}) \]Calculate this value to get the adjacent leg length.

Step 4 - Calculate the Adjacent Leg

Using a calculator, find \[ \text{Adjacent Leg} = 10 \times \text{cos}(40^{\text{°}}) \approx 10 \times 0.766 \approx 7.66 \text{ cm} \]

Step 5 - Use the Sine Function to Find the Opposite Leg

The sine of an angle in a right triangle is the ratio of the length of the opposite leg to the length of the hypotenuse. Using the sine function: \[ \text{sin}(40^{\text{°}}) = \frac{\text{Opposite Leg}}{10} \]Rearrange to solve for the opposite leg: \[ \text{Opposite Leg} = 10 \times \text{sin}(40^{\text{°}}) \]Calculate this value to get the opposite leg length.

Step 6 - Calculate the Opposite Leg

Using a calculator, find \[ \text{Opposite Leg} = 10 \times \text{sin}(40^{\text{°}}) \approx 10 \times 0.643 \approx 6.43 \text{ cm} \]

Key Concepts

hypotenuse

The hypotenuse is the longest side of a right triangle. It is opposite the right angle, making it a critical component in right triangle trigonometry. If you know the length of the hypotenuse and one of the other angles (other than the right angle), you can use trigonometric functions to find the lengths of the other two sides. In this exercise, our hypotenuse is 10 cm.
Knowing the hypotenuse helps set up all other calculations. It serves as the base length for finding the adjacent and opposite sides using sine and cosine functions.

cosine function

The cosine function relates the angle in a right triangle to the lengths of the adjacent side and the hypotenuse. It is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
The formula to calculate the adjacent side using the cosine function is:
\[ \cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \]
Rearranging this formula to solve for the adjacent side yields:
\[ \text{Adjacent Side} = \text{Hypotenuse} \times \cos(\theta) \]
For our exercise:
The angle \(\theta\) is 40 degrees and the hypotenuse is 10 cm:
\[ \text{Adjacent Side} = 10 \times \cos(40^{\circ}) \]
Using a calculator, \( \cos(40^{\circ}) \approx 0.766 \).
So, \( 10 \times 0.766 \approx 7.66 \text{ cm}\) for the adjacent side.

sine function

The sine function is essential in right triangle calculations. It relates the angle to the lengths of the opposite side and the hypotenuse. The equation for sine is:
\[ \sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \]
To find the opposite side, rearrange the formula:
\[ \text{Opposite Side} = \text{Hypotenuse} \times \sin(\theta) \]
In our example:
The angle \(\theta\) is 40 degrees, and the hypotenuse is 10 cm:
\[ \text{Opposite Side} = 10 \times \sin(40^{\circ}) \]
Using a calculator, \( \sin(40^{\circ}) \approx 0.643 \).
Thus, \( 10 \times 0.643 \approx 6.43 \text{ cm}\). This value represents the length of the opposite side.

triangle calculations

When solving a right triangle, you often use both the sine and cosine functions to find missing side lengths.
1. **Identify the sides and angles:** Start with what you know, like the hypotenuse and an angle.
2. **Use trigonometric functions:** Use sine for the opposite side and cosine for the adjacent side.
3. **Calculate accurately:** Use a calculator for precise values, especially for trigonometric functions.
For our problem:
- Hypotenuse = 10 cm
- Angle = 40 degrees
First, use the cosine function to find the adjacent side:
\[ \text{Adjacent Side} = \text{Hypotenuse} \times \cos(40^{\circ}) = 10 \times 0.766 = 7.66 \text{ cm} \]
Next, use the sine function to find the opposite side:
\[ \text{Opposite Side} = \text{Hypotenuse} \times \sin(40^{\circ}) = 10 \times 0.643 = 6.43 \text{ cm} \]
These steps will lead you to a complete understanding of the triangle's dimensions.