Problem 51 - Algebra 2

Question

Use the definitions of trigonometric ratios in right \(\triangle A B C\) to verify each identity. \(\cos ^{2} A+\sin ^{2} A=1\)

Step-by-Step Solution

Answer

Yes, with the definitions of cosine and sine and the Pythagorean Theorem, we have verified that in any right triangle, \(\cos^2 A+\sin ^{2} A = 1\).

Step 1: Assign Variables

Let the length of the hypotenuse be \(c\), the length of the side adjacent to angle \(A\) be \(b\), and the length of the side opposite angle \(A\) be \(a\). This makes \(\cosA = \frac{b}{c}\) and \(\sinA = \frac{a}{c}\).

Step 2: Square Both Sides

Square both the cosine and sine ratios to get \(\cos^2A = \left(\frac{b}{c}\right)^2\) and \(\sin^2A = \left(\frac{a}{c}\right)^2\).

Step 3: Addition

Add the squared ratios together to get \(\cos^2 A+\sin ^{2} A = \left(\frac{b}{c}\right)^2 + \left(\frac{a}{c}\right)^2\). Simplifying, we get \(\cos^2A + \sin^2A = \frac{b^2}{c^2} + \frac{a^2}{c^2}\).

Step 4: Pythagorean Identity

By the Pythagorean Theorem, we know that \(a^2 + b^2 = c^2\), for any right triangle. So, substituting \(c^2\) for \(a^2 + b^2\), we get \(\cos^2A + \sin^2A = \frac{c^2}{c^2} = 1\).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry that relates the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed with the formula: \[ c^2 = a^2 + b^2 \]where:

• \( c \) is the hypotenuse,
• \( a \) and \( b \) are the other two sides of the triangle.

This theorem helps us understand the relationship between the angles and sides of right triangles, laying a foundation for trigonometry. By applying the Pythagorean Theorem to trigonometric identities, for example, you can derive identities like \( \cos^2 A + \sin^2 A = 1 \), which reflects the integrity of the triangle's angles and sides.

Trigonometric Ratios

Trigonometric ratios are based on the relationships between the angles and sides of right triangles. The primary trigonometric ratios are sine, cosine, and tangent, each associated with a specific angle in a right triangle. Here's how they are defined:

• Sine (\( \sin \)) of angle \( A \) is the ratio of the opposite side to the hypotenuse:
• \( \sin A = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{a}{c} \)
• Cosine (\( \cos \)) of angle \( A \) is the ratio of the adjacent side to the hypotenuse:
• \( \cos A = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{b}{c} \)
• Tangent (\( \tan \)) of angle \( A \) is the ratio of the opposite side to the adjacent side:
• \( \tan A = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{a}{b} \)

Understanding these ratios is essential for solving various problems involving right triangles, and they form the basis of many identities used in trigonometry.

Right Triangle

A right triangle is a type of triangle that has one angle measuring 90 degrees. This characteristic angle makes it a powerful element in mathematical investigations because it allows for the application of the Pythagorean Theorem and trigonometric ratios directly. In a right triangle:

• The side opposite the right angle is called the hypotenuse, which is always the longest side.
• The other two sides are referred to as the legs, which are perpendicular to each other.

Solving problems involving right triangles often requires calculating unknown lengths or angles using trigonometric ratios or the Pythagorean Theorem. These triangles are used in various fields, including architecture and engineering, because of their properties which simplify many calculations. Understanding right triangles is crucial as they form the basis of advanced geometric concepts and real-world applications.