Problem 49 - Algebra 2

Question

Critical thinking Does the Law of Cosines apply to a right triangle? That is does \(c^{2}=a^{2}+b^{2}-2 a b \cos C\) remain true when \(\angle C\) is a right angle? Justify your answer.

Step-by-Step Solution

Answer

Yes, the Law of Cosines applies to a right triangle. When the included angle is 90 degrees, the formula simplifies to the Pythagorean theorem: \(c^{2}=a^{2}+b^{2}\).

Step 1: Substitute the value of angle C in the equation of Law of Cosines

As the problem stated, we have a right triangle where angle C is 90 degrees. We know that the cosine of 90 degrees is 0. So, replace \(C\) with 90 degrees in the equation \(c^{2}=a^{2}+b^{2}-2 a b \cos C\). Hence, the equation becomes \(c^{2}=a^{2}+b^{2}-2 a b \cos 90 \degrees \).

Step 2: Calculate the cosine of 90 degrees

We have to find the value of \(\cos 90 \degrees\). The value of \(\cos 90 \degrees\) is 0.

Step 3: Simplify the equation

Now, replace \(\cos 90 \degrees\) in the equation with 0. Hence, the equation becomes \(c^{2}=a^{2}+b^{2}-2 a b * 0 = a^{2}+b^{2}\).

Key Concepts

Right Triangle

In geometry, a right triangle is a type of triangle where one angle measures exactly 90 degrees. This special triangle has unique properties that make it important for various calculations in mathematics. The sides of a right triangle have specific names:

• The side opposite the right angle is known as the hypotenuse.
• The other two sides are called the adjacent side and the opposite side, depending on which non-right angle they relate to.

Right triangles are particularly useful because they allow us to apply different mathematical principles, such as the Law of Cosines and the Pythagorean theorem, to solve problems related to angles and distances.

Cosine Function

The cosine function is a fundamental trigonometric function that relates an angle of a right triangle to the length of its adjacent side and the hypotenuse. For a given angle \( \theta \) in a right triangle:

• The cosine is the ratio of the adjacent side to the hypotenuse, expressed as \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \).

When the angle is 90 degrees, as in a right triangle where the cosine function is applied, the value of \( \cos(90^{\circ}) \) is 0. This is crucial to understand because it transforms many equations, including the Law of Cosines. Knowing how to use the cosine function helps in various mathematical computations, especially in trigonometry.

Trigonometric Identities

Trigonometric identities are equations that hold true for any angle and are used to simplify complex trigonometric expressions. These identities are essential tools for solving problems involving triangles. Some of the basic trigonometric identities include:

• \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

In the context of right triangles, using the identity \( \cos(90^{\circ}) = 0 \) simplifies calculations when applying the Law of Cosines, allowing us to reduce it to the well-known Pythagorean theorem.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone of geometry, specifically for right triangles. It states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, it is represented as:\[ c^2 = a^2 + b^2 \]Where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides. This relationship is particularly crucial in validating the Law of Cosines. When in a right triangle, the Law of Cosines simplifies to the Pythagorean theorem. This simplification occurs because \( \cos(90^{\circ}) = 0 \), thus removing the cosine term and leaving the well-known relationship outlined above.