Question

Geometry The hypotenuse of a right triangle has a length of 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 trigonometric functions to use

Since we are dealing with a right triangle and given one angle and the hypotenuse, use the sine and cosine functions which relate angles to the lengths of the sides in a right triangle.

Set up the sine function for the given angle

The sine of an angle in a right triangle is the length of the opposite side divided by the hypotenuse. Here, the opposite side to the angle is one leg, the hypotenuse is 8. Therefore, \( \text{sin}(35^{\circ}) = \frac{opposite}{8} \).

Solve for the opposite side using sine

Solve for the opposite side by multiplying both sides by the hypotenuse (8): \( \text{opposite} = 8 \cdot \text{sin}(35^{\circ}) = 8 \cdot 0.5736 \), where \( \text{sin}(35^{\circ}) = 0.5736 \). Therefore, the opposite side is approximately 4.59 inches.

Set up the cosine function for the given angle

The cosine of an angle in a right triangle is the length of the adjacent side divided by the hypotenuse. Here, the adjacent side to the angle is the other leg. Therefore, \( \text{cos}(35^{\circ}) = \frac{adjacent}{8} \).

Solve for the adjacent side using cosine

Solve for the adjacent side by multiplying both sides by the hypotenuse (8): \( \text{adjacent} = 8 \cdot \text{cos}(35^{\circ}) = 8 \cdot 0.8192 \), where \( \text{cos}(35^{\circ}) = 0.8192 \). Therefore, the adjacent side is approximately 6.55 inches.

Key concepts

trigonometric functions

In the world of geometry, trigonometric functions play a vital role in solving right triangles. These functions are mathematical tools that help us relate the angles to the lengths of the sides in a triangle. The most commonly used trigonometric functions in right triangles are sine (sin), cosine (cos), and tangent (tan). Each of these functions links an angle in the triangle to a specific ratio of the triangle's sides.

To be more specific, in a right triangle:

• Sine (sin) relates the angle to the ratio of the length of the opposite side to the hypotenuse.
• Cosine (cos) relates the angle to the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (tan) relates the angle to the ratio of the length of the opposite side to the adjacent side.

Understanding how these functions work is fundamental in trigonometry, as they can help solve various problems involving right triangles.

sine and cosine

In our given exercise, we specifically used the sine and cosine functions.

The sine function is defined as the ratio of the length of the side opposite the given angle to the hypotenuse. It can be written as:
\( \text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
For example, if we have a right triangle where one angle is 35°, and the hypotenuse is 8 inches, the sine function can help us find the length of the opposite side.

Similarly, the cosine function is defined as the ratio of the length of the adjacent side to the hypotenuse. It can be written as:
\( \text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
In our example, with a 35° angle and an 8-inch hypotenuse, the cosine function helps us find the length of the adjacent side.

These functions are pivotal for determining unknown sides when we have one angle and the hypotenuse.

solving right triangles

Solving a right triangle means finding the lengths of all sides and the measure of all angles. With one angle and the hypotenuse known, the sine and cosine functions are particularly useful.

Here's a step-by-step method for solving our specific right triangle problem:

• First, identify which trigonometric functions to use. Since we know one angle (35°) and the length of the hypotenuse (8 inches), we use sine and cosine.
• Set up the sine function for the known angle: \( \text{sin}(35^{\text{circ}}) = \frac{\text{opposite}}{8} \)
• Solve for the opposite side by multiplying both sides by the hypotenuse: \( \text{opposite} = 8 \times \text{sin}(35^{\text{circ}}) = 8 \times 0.5736 \)
• The opposite side is approximately 4.59 inches.
• Next, set up the cosine function for the known angle: \( \text{cos}(35^{\text{circ}}) = \frac{\text{adjacent}}{8} \)
• Solve for the adjacent side by multiplying both sides by the hypotenuse: \( \text{adjacent} = 8 \times \text{cos}(35^{\text{circ}}) = 8 \times 0.8192 \)
• The adjacent side is approximately 6.55 inches.

By carefully applying these steps, we can find the lengths of each leg of the right triangle, thus solving the problem completely.