Problem 27 - Trigonometry

Question

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

Step-by-Step Solution

Answer

For a right triangle with \( A = 90^{\circ} \), \( a^2 = b^2 + c^2 \) by the Law of Cosines.

Step 1: Understanding the Law of Cosines

The Law of Cosines states that for any triangle with sides of lengths \( a \), \( b \), \( c \), and an angle \( A \) opposite side \( a \), it holds that \( a^2 = b^2 + c^2 - 2bc \cos(A) \).

Step 2: Applying to Right Triangle

Since angle \( A \) is \( 90^{\circ} \), the triangle is a right triangle and \( \cos(90^{\circ}) = 0 \). We substitute \( A = 90^{\circ} \) into the Law of Cosines formula.

Step 3: Simplifying the Formula

Substitute \( A = 90^{\circ} \) into the Law of Cosines formula: \[ a^2 = b^2 + c^2 - 2bc \cos(90^{\circ}) \] Since \( \cos(90^{\circ}) = 0 \), the formula simplifies to \[ a^2 = b^2 + c^2 \].

Step 4: Concluding the Proof

Through substitution and simplification, we have shown that if \( A = 90^{\circ} \), then \( a^2 = b^2 + c^2 \), which is the well-known Pythagorean Theorem for right triangles.

Key Concepts

Law of Cosines

The Law of Cosines is a powerful tool in trigonometry. It is used to find unknown sides or angles in any triangle, not just right triangles. The formula is given by:

• For a triangle with sides of lengths \( a \), \( b \), and \( c \), and an angle \( A \) opposite side \( a \), the Law is stated as: \[ a^2 = b^2 + c^2 - 2bc \cos(A) \]
• This formula helps to relate the lengths of the sides of a triangle to one of its angles.

The Law of Cosines is like a generalized version of the Pythagorean Theorem. When the angle \( A \) is \( 90^{\circ} \), this formula simplifies directly into the Pythagorean Theorem. This makes it versatile for solving various triangle problems, both in academic contexts and real-world applications.

Right Triangle

A right triangle is a specific type of triangle where one of the angles is exactly \( 90^{\circ} \). This special property makes calculations simpler and is the basis of the Pythagorean Theorem.

• In a right triangle, the side opposite the right angle is called the hypotenuse, traditionally denoted as \( c \).
• The other two sides, called the legs, are labeled as \( a \) and \( b \).

The defining feature is the relationship between the sides, described by the formula:\[ a^2 + b^2 = c^2 \]This relationship is the essence of the Pythagorean Theorem. It is not only central in geometry but also widely used in various branches of science and engineering to calculate distances and design structures related to right angles.

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable included. These identities are fundamental in simplifying expressions and solving equations that involve trigonometric functions.

• One important identity for right angles is that \( \cos(90^{\circ}) = 0 \).
• Another cornerstone identity is \( \sin^2(A) + \cos^2(A) = 1 \) for any angle \( A \).

When applying the Law of Cosines to a right triangle, the fact that \( \cos(90^{\circ}) = 0 \) simplifies the equation significantly. This leads directly to the Pythagorean Theorem. Understanding and using these identities make solving trigonometric problems much more manageable, providing clear and elegant solutions in both theoretical and practical situations.