Question

In a right triangle, the tangent of one of the acute angles is 1 . How are the measures of the two legs related?

Short answer

The two legs of the triangle are equal in length.

Step-by-step solution

Understanding the problem

We have a right triangle, and we are given that the tangent of one of its acute angles is 1. We need to determine how the measures of the two legs of this triangle are related.

Recalling the Definition of Tangent

The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite leg to the length of the adjacent leg. Mathematically, it's expressed as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

Applying the Given Tangent Value

We are given that \( \tan(\theta) = 1 \). This means that \( \frac{\text{opposite}}{\text{adjacent}} = 1 \). For this equality to hold, the lengths of the opposite and adjacent legs must be equal.

Conclusion: Relationship Between the Legs

Since the tangent of the angle equals 1, the two legs of the triangle (opposite and adjacent) must be equal in length.

Key concepts

Tangent Ratio

In a right triangle, the tangent ratio is one of the fundamental ways to relate the angles and sides. It specifically deals with the lengths of the two legs adjoining the right angle. The tangent of an angle, usually denoted as \( \tan(\theta) \), is the ratio of the length of the opposite leg to the length of the adjacent leg.

This can be represented mathematically as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

In practical terms, this means:

• The opposite side is the one across from the angle \(\theta\).
• The adjacent side is next to the angle \(\theta\) and forms the right angle with the hypotenuse.

Thus, the tangent ratio provides a simple way to determine the relationship between these two sides given an angle. For instance, in this exercise where \(\tan(\theta) = 1\), it reveals that the opposite and adjacent sides are of equal length. This can only happen in an isosceles right triangle where both acute angles are 45°.

Trigonometric Functions

Trigonometric functions are essential tools for understanding the relationships between angles and sides in right triangles. These functions include sine, cosine, and tangent, among others.

Each function provides a distinct ratio:

• Sine (\( \sin \)) is the ratio of the length of the opposite side to the hypotenuse.
• Cosine (\( \cos \)) is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \)) we've already discussed as the opposite over the adjacent.

Trigonometric functions are critical in solving problems involving right triangles because they allow us to make connections between angles and side ratios. These functions are ubiquitous in mathematics, physics, engineering, and many other scientific fields.

The tangent function, in particular, stands out when working with problems involving equal legs of a triangle, as it clearly signals this relationship when its value is 1. This serves as a gateway to understanding deeper properties and solving more complex problems.

Acute Angles

Acute angles are angles that measure less than 90 degrees. They are a crucial component in right triangles, where the triangle consists of a right angle and two acute angles.

In right triangles:

• The two acute angles are complementary, meaning they add up to 90 degrees.
• Each acute angle has a unique trigonometric value that can be calculated and used in various applications.

Understanding the properties of acute angles helps in determining side lengths when an angle measure is known. For example, knowing the tangent of an acute angle helps us find the relationship between the two legs of the triangle. This is relevant in many real-world applications, such as in navigation and architecture, where precise angle measurements are crucial.

Therefore, comprehending the behavior of acute angles in conjunction with trigonometric functions offers deeper insight into geometric relationships. In this problem, where one acute angle has a tangent of 1, it helps us quickly identify that the angle is 45°, and the triangle is isosceles.