Problem 17 - Precalculus: Functions and Graphs

Question

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) express the third part in terms of the first two. $$\alpha, c ; \quad b$$

Step-by-Step Solution

Answer

\(b = c \cdot \cos(\alpha)\)

Step 1: Identify the Known Elements of the Right Triangle

In triangle \(ABC\), \(\gamma = 90^\circ\) indicates that this is a right triangle with \(C\) being the right angle. The known parts in this problem are angle \(\alpha\) and hypotenuse \(c\), and we are asked to express side \(b\).

Step 2: Understand Trigonometric Relationships

In a right triangle, the relationship between an acute angle \(\alpha\), its opposite side \(a\), the hypotenuse \(c\), and the adjacent side \(b\) can be determined through trigonometric ratios. Specifically, \(\cos(\alpha) = \frac{b}{c}\).

Step 3: Rearrange the Trigonometric Formula to Find \(b\)

From the cosine definition, \(\cos(\alpha) = \frac{b}{c}\) can be rearranged to solve for the side \(b\): \[ b = c \cdot \cos(\alpha) \]

Step 4: Conclusion

The side \(b\) in the right triangle can be expressed in terms of the hypotenuse \(c\) and angle \(\alpha\) as \(b = c \cdot \cos(\alpha)\).

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle equal to 90 degrees. This specific angle is known as the right angle. In every right triangle, the side opposite the right angle is the longest side, termed the hypotenuse. The other two sides that form the right angle are known as the legs or catheti of the triangle.
Right triangles are fundamental in trigonometry because they allow us to define and explore trigonometric functions using their consistent angle-side relationships.

• The hypotenuse is always opposite to the right angle and is the longest side of the triangle.
• The two legs of the triangle can be referred to as either opposite or adjacent, depending on the angle in question.

Understanding the properties of right triangles is the foundation for solving many trigonometric problems, such as determining side lengths or angles using known values.

Trigonometric Ratios

Trigonometric ratios are mathematical tools that relate the angles and sides of a right triangle. These ratios include sine, cosine, and tangent, each corresponding to a specific ratio between sides of the triangle. Knowing at least one angle besides the right angle and one side of the triangle allows you to find the other sides using these ratios.
The primary trigonometric ratios are:

• Sine ( \( \sin \theta \) ): This ratio compares the length of the side opposite the angle to the length of the hypotenuse, or \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).
• Cosine ( \( \cos \theta \) ): This ratio compares the length of the adjacent side to the hypotenuse, or \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• Tangent ( \( \tan \theta \) ): This ratio compares the length of the opposite side to the adjacent side, or \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).

These ratios are pivotal for solving problems involving right triangles by providing a method to determine unknown quantities when certain values are known.

Cosine Function

The cosine function is one of the main trigonometric functions and is particularly useful for finding unknown side lengths of right triangles when an angle and one side are known. The function is defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle.
Understanding the cosine function involves:

• Recognizing that \( \cos(\alpha) \) gives you the ratio \( \frac{b}{c} \) , where:
  • \( b \) is the length of the side adjacent to angle \( \alpha \)
  • \( c \) is the hypotenuse
  In the context of the exercise, using the cosine function allows us to find the adjacent side, \( b \), when \( \alpha \) and \( c \) are given.
• Applying the formula: Rearranging the cosine formula \( \cos(\alpha) = \frac{b}{c} \) to solve for \( b \) gives \[ b = c \cdot \cos(\alpha) \] . This step is crucial in determining the side length of \( b \) in the triangle.

The cosine function is essential in various applications beyond geometry, but its foundation in right triangles helps in easily solving many practical problems.