Problem 8 - Algebra and Trigonometry

Question

True or False A special case of the Law of Cosines is the Pythagorean Theorem.

Step-by-Step Solution

Answer

True. The Pythagorean Theorem is a special case of the Law of Cosines for a right-angled triangle.

Step 1: Understand the Law of Cosines

The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and an angle \theta opposite side c, the following relationship holds: \[ c^2 = a^2 + b^2 - 2ab \, \cos(\theta) \]

Step 2: Examine the Pythagorean Theorem

The Pythagorean Theorem applies specifically to right-angled triangles, and it states: \[ c^2 = a^2 + b^2 \] Here, c is the hypotenuse, and a and b are the other two sides.

Step 3: Connect the two equations

Observe that in the Pythagorean Theorem, the angle opposite the hypotenuse (c) is 90 degrees. Recall from trigonometry that \( \cos(90^\circ) = 0 \).

Step 4: Simplify the Law of Cosines for a right-angled triangle

When \( \theta = 90^\circ \), the Law of Cosines simplifies as follows: \[ c^2 = a^2 + b^2 - 2ab \, \cos(90^\circ) \]Since \( \cos(90^\circ) = 0 \), this equation reduces to \[ c^2 = a^2 + b^2 \].

Step 5: Conclusion

The simplified form of the Law of Cosines when the angle is 90 degrees is the same as the Pythagorean Theorem. Therefore, the Pythagorean Theorem is indeed a special case of the Law of Cosines.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry that applies to right-angled triangles. It states that in a right-angled 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. Mathematically, this can be written as: \[ c^2 = a^2 + b^2 \] Here:

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

It is crucial to recognize that this theorem only holds true for right-angled triangles. In a broader context, we can see this as a specific case of the Law of Cosines when the angle is 90 degrees.

Right-Angled Triangles

A right-angled triangle consists of one angle that is exactly 90 degrees. The side opposite this right angle is known as the hypotenuse, which is always the longest side of the triangle. The other two sides are referred to as the legs.
In addition to the Pythagorean Theorem, several trigonometric identities are used extensively in the study of right-angled triangles. These include sine, cosine, and tangent, each of which relates the angles of the triangle to the lengths of its sides. For example:

• \( \text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
• \( \text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

Understanding these relationships helps in extending the concept of right-angled triangles to more complex problems, including applications of the Law of Cosines.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. These identities help in simplifying complex trigonometric expressions and solving trigonometric equations.
One of the most essential trigonometric identities used in conjunction with the Law of Cosines is the cosine of 90 degrees: \( \text{cos}(90^\text{o}) = 0 \). When used in the Law of Cosines formula for a right-angled triangle, this identity simplifies the equation:
\[ c^2 = a^2 + b^2 - 2ab \text{cos}(90^\text{o}) \] Since \( \text{cos}(90^\text{o}) = 0 \), the equation reduces to: \[ c^2 = a^2 + b^2 \] This is precisely the Pythagorean Theorem, confirming that the theorem is a special case of the Law of Cosines. Trigonometric identities like these form the backbone of trigonometry, linking together various principles and theorems.