Problem 28 - Trigonometry

Question

$$ \text { Solve each of the following triangles. } $$ $$ \text { Use the law of cosines to show that, if } a^{2}=b^{2}+c^{2} \text {, then } A=90^{\circ} \text {. } $$

Step-by-Step Solution

Answer

If \( a^2 = b^2 + c^2 \), then angle \( A \) is \( 90^{\circ} \) by the Law of Cosines.

Step 1: Understand the Law of Cosines

The Law of Cosines is given by the formula \( c^2 = a^2 + b^2 - 2ab \cos(C) \), where \(c\) is the side opposite angle \(C\), and \(a\) and \(b\) are the other two sides of the triangle.

Step 2: Substitute Given Condition

The problem states \( a^2 = b^2 + c^2 \). Substitute this into the Law of Cosines formula with \( a \) corresponding to \( c \) in the law: \[ a^2 = a^2 - 2ab \cos(A) \].

Step 3: Simplify Equation

Rearrange the expression obtained in Step 2: \[ 0 = - 2bc \cos(A) \].

Step 4: Solve for \( \cos(A) \)

For the equation \( -2bc \cos(A) = 0 \) to hold true, either \( b = 0 \), \( c = 0 \), or \( \cos(A) = 0 \). Since we are dealing with a proper triangle, \( b \) and \( c \) cannot be zero. Therefore, \( \cos(A) = 0 \).

Step 5: Determine \( A \) from \( \cos(A) \)

Since \( \cos(A) = 0 \), the angle \( A \) must be \( 90^{\circ} \) because the cosine of \( 90^{\circ} \) is \( 0 \). This confirms that if \( a^2 = b^2 + c^2 \), then \( A = 90^{\circ} \).

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 and is typically denoted by a small square in diagrams. Right triangles are unique because they follow specific mathematical rules, including the Pythagorean theorem, which only applies to them.

In the context of the Law of Cosines, a right triangle plays a special role. When the angle opposite the side you're considering is 90 degrees, the law simplifies to the Pythagorean theorem. Here's why:

• In a right triangle, the side opposite the right angle is the longest and is called the hypotenuse.
• The other two sides are often referred to as the adjacent and opposite sides, relative to an angle.
• Determining the exact measurements of these three sides and angles requires and complements the use of basic trigonometric rules.

Understanding right triangles is crucial for applying principles of trigonometry and the Law of Cosines correctly when solving triangles.

Angle Measurement

Angle measurement is essential when solving any triangle, as it defines the triangle's overall shape. In geometry, angles are measured in degrees, with a full rotation equating to 360 degrees.

For triangles, the sum of the three interior angles always equals 180 degrees. Specifically:

• If one angle is 90 degrees, as in right triangles, the two remaining angles together add up to 90 degrees.
• The size of each angle directly affects the calculation of sides and the use of trigonometric identities like the Law of Sines or Cosines.

When calculating angles using the Law of Cosines in a right triangle setting, identifying a right angle, where the cosine value equals zero, is critical. It simplifies calculations, confirming the triangle's status as right-angled by demonstrating that the principal angle opposite the longest side is 90 degrees.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, specifically dealing with right triangles. It establishes a relationship between the sides of a right triangle. The theorem is succinctly described by the equation:\[ a^2 + b^2 = c^2 \]Here, \(c\) represents the hypotenuse, or the side opposite the right angle, while \(a\) and \(b\) are the triangle's other two sides.

This fundamental geometrical rule helps to:

• Verify whether a triangle is a right triangle: If \( a^2 + b^2 = c^2 \), then the triangle must be right-angled.
• Calculate the length of one side if the other two are known, simplifying many practical applications.
• Support the transition from specific triangle rules, like the Law of Cosines, into broader trigonometric analysis.

Understanding the Pythagorean theorem makes transitioning to related concepts, like the Law of Cosines, more intuitive, as it primarily extends the theorem's principles to non-right triangles.