Problem 22 - 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. $$\boldsymbol{\beta}, \boldsymbol{a} ; \boldsymbol{c}$$

Step-by-Step Solution

Answer

\( b = a \cdot \cot(\beta) \).

Step 1: Introduction: Understand Triangle Properties

Triangle \(ABC\) is a right triangle with \(\gamma = 90^\circ\). In a right triangle, we use the Pythagorean theorem and trigonometric identities. For right triangles \(\sin\), \(\cos\), and \(\tan\) of angles are useful: \(\sin(\beta) = \frac{a}{c}\), \(\cos(\beta) = \frac{b}{c}\). Our given parts are angle \(\beta\), side \(a\), and hypotenuse \(c\), and we want to express \(b\) in terms of \(\beta, a,\) and \(c\).

Step 1: Use Trigonometric Identity for \( \cos \beta \)

Given \(\beta\) and \(c\), use the trigonometric identity for cosine: \[ \cos(\beta) = \frac{b}{c} \]. Solve for \(b\): \[ b = c \cdot \cos(\beta) \].

Step 2: Express \( c \) in Terms of \( a \) and Trigonometric Functions

Since \(\sin(\beta) = \frac{a}{c}\), rearrange to express \(c\) in terms of \(a\): \[ c = \frac{a}{\sin(\beta)} \].

Step 3: Combine the Expressions

Substitute \(c\) from Step 2 into the expression found in Step 1: \[ b = \left(\frac{a}{\sin(\beta)}\right) \cdot \cos(\beta) \]. Simplify the expression using the identity \( \cot(\beta) = \frac{\cos(\beta)}{\sin(\beta)}\): \[ b = a \cdot \cot(\beta) \].

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one of its angles measuring exactly 90 degrees. This angle is known as the right angle. The side opposite this angle is called the hypotenuse, and it is usually the longest side of the triangle. The other two sides are known as the legs of the triangle. A right triangle has some interesting properties:

• The sum of the angles in a triangle is always 180 degrees. Therefore, the other two angles sum up to 90 degrees.
• The longest side, or hypotenuse, is always opposite the right angle.
• Right triangles make trigonometric functions remarkably useful.

In geometry, right triangles are foundational because they allow us to use trigonometric ratios like sine, cosine, and tangent to find unknown side lengths and angles. Each trigonometric function expresses a ratio of two sides of the right triangle related to an angle of interest.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle specifically used with right triangles. It relates the lengths of the three sides of a right triangle: If you label the hypotenuse as \(c\) and the other two sides as \(a\) and \(b\), the theorem can be expressed as:\[ a^2 + b^2 = c^2 \]This theorem provides an essential formula for calculating the length of any one side given the lengths of the other two. It's especially useful in many areas of mathematics and physics because:

• It helps find the distance between two points in a plane.
• It supports geometrical constructions and proofs.
• It serves as the basis for trigonometric functions by defining consistent relationships between side lengths.

In trigonometry, the Pythagorean theorem ensures that the trigonometric ratios (sine, cosine, tangent) hold steady relationships, enabling the solution of other complex problems using trigonometric identities.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any angle. In a right triangle setup, they are particularly powerful tools for relating the angles with the ratio of the triangle's sides:- **Sine** (\(\sin\)) of an angle \(\beta\) is the ratio of the opposite side to the hypotenuse: \(\sin(\beta) = \frac{a}{c}\).

• This allows you to solve for unknown side lengths or angles when only partial information is given.

- **Cosine** (\(\cos\)) of an angle \(\beta\) is the ratio of the adjacent side to the hypotenuse: \(\cos(\beta) = \frac{b}{c}\).

• Similar to sine, this identity helps determine unknown geometric features of the triangle efficiently.

- **Tangent** (\(\tan\)) of an angle \(\beta\) is the ratio of the opposite side to the adjacent side: \(\tan(\beta) = \frac{a}{b}\).Trigonometric identities simplify the process of discovering unknown measures within the triangle, leveraging the consistent relationships these identities highlight. They also provide ways to transform expressions involving these angles, aiding in more complex calculations such as those involving angles or adding multiple trigonometric expressions into single formulas. These identities are critical when solving real-world problems like angle construction, wave analysis, and other scientific computations.