Question

A right triangle has a hypotenuse of length 8 inches. If one angle is \(35^{\circ},\) find the length of each leg.

Short answer

The lengths of the legs are approximately 4.59 inches and 6.55 inches.

Step-by-step solution

- Identify the Right Triangle Components

In a right triangle, we have a hypotenuse of 8 inches and an angle of 35 degrees. We need to find the lengths of the other two sides, which are the legs of the triangle.

- Use Trigonometric Ratios

To find the lengths of the legs, use the sine and cosine functions. For the angle of 35 degrees, let the opposite leg be denoted as 'a' and the adjacent leg as 'b'.

- Calculate Opposite Leg Using Sine

Use the equation for the sine: \[\text{sin}(35^{\circ}) = \frac{a}{8}\] Rearrange to solve for 'a': \[a = 8 \cdot \text{sin}(35^{\circ})\] Calculate: \[a \approx 8 \cdot 0.5736 \approx 4.5888 \] Thus, 'a' is approximately 4.59 inches.

- Calculate Adjacent Leg Using Cosine

Use the equation for the cosine: \[\text{cos}(35^{\circ}) = \frac{b}{8}\] Rearrange to solve for 'b': \[b = 8 \cdot \text{cos}(35^{\circ})\] Calculate: \[b \approx 8 \cdot 0.8192 \approx 6.5536 \] Thus, 'b' is approximately 6.55 inches.

Key concepts

Right Triangle

A right triangle is a type of triangle that has one angle equal to 90 degrees. This 90-degree angle is called the right angle. The side opposite this right angle is known as the hypotenuse, which is the longest side of the triangle. The other two sides are referred to as the legs of the triangle.
In our exercise, we are given a right triangle with a hypotenuse of 8 inches and an angle of 35 degrees. We need to determine the lengths of the two legs using this information.
Remember, the properties of a right triangle make it easier to apply trigonometric functions to find unknown lengths or angles.

Sine Function

The sine function is an essential trigonometric ratio used in right triangles. It is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The formula for sine is:
\(\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)
For our exercise, we are given the angle \(35^\text{circ}\) and the hypotenuse (8 inches). To find the length of the side opposite this angle (denoted as ‘a’), we use the sine function:

\(\text{sin}(35^{\text{circ}}) = \frac{a}{8}\)
By rearranging and solving for ‘a’, we get:

\(a = 8 \times \text{sin}(35^{\text{circ}})\)
\(a \approx 8 \times 0.5736 \approx 4.59\)
So, the length of the opposite side is approximately 4.59 inches.

Cosine Function

The cosine function is another critical trigonometric ratio used to solve right triangles. It is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. The formula for cosine is:
\(\text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
In our exercise, to find the length of the side adjacent to the 35-degree angle (denoted as ‘b’), we use the cosine function:

\(\text{cos}(35^{\text{circ}}) = \frac{b}{8}\)
By rearranging and solving for ‘b’, we have:

\(b = 8 \times \text{cos}(35^{\text{circ}})\)
\(b \approx 8 \times 0.8192 \approx 6.55\)
Thus, the length of the adjacent side is approximately 6.55 inches.

Angle Measurement

In trigonometry, angle measurement is crucial because it helps in calculating the side lengths of triangles using trigonometric functions. Angles can be measured in degrees or radians, but for most beginner problems, degrees are used.
In our exercise, we work with an angle of 35 degrees. It's important to understand that this angle, along with the right angle (90 degrees), helps in defining the trigonometric ratios.
Knowing one angle besides the right angle enables us to uniquely determine the other angles and potentially all sides of the triangle using trigonometric functions such as sine and cosine.
This principle is particularly useful when dealing with right triangles, as we demonstrated by finding the lengths of the legs using known angles and the hypotenuse.