Problem 61 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Law of cosines to prove each identity. (a) \(\frac{1}{2} b c(1+\cos A)=\left(\frac{a+b+c}{2}\right)\left(\frac{-a+b+c}{2}\right)\). (b) \(\frac{1}{2} b c(1-\cos A)=\left(\frac{a-b+c}{2}\right)\left(\frac{a+b-c}{2}\right)\).

Step-by-Step Solution

Answer

Using the Law of Cosines and simple algebraic manipulation, both identities (a) and (b) have been proven true.

Step 1 - Understand the Law of Cosines

The Law of Cosines states that in any triangle with sides of lengths a, b, and c, and the angle \(A\) opposite side a, the following relationship holds true: \(a^2 = b^2 + c^2 - 2bc \cos A \).

Step 2 - Express CosineA in terms of a, b, c

By manipulating the law of cosines, we can express \( \cos A = \frac{b^2 + c^2 - a^2}{2bc}\)

Step 3 - Prove Identity (a)

We will substitute the expression for \(\cos A\) into the left-hand side of identity (a) \(\frac{1}{2} b c(1+\cosA)\). It becomes \(\frac{1}{2} b c + \frac{1}{4}(b^2+c^2 - a^2)\). With simple algebraic manipulation, it can be shown that this equals to the right-hand side: \(\left(\frac{a+b+c}{2}\right)\left(\frac{-a+b+c}{2}\right)\).

Step 4 - Prove Identity (b)

We substitute the expression for \(\cos A\) into the left-hand side of identity (b) \(\frac{1}{2} b c(1-\cos A)\). It becomes \(\frac{1}{2} b c - \frac{1}{4}(b^2+c^2 - a^2)\). With simple algebraic manipulation, it can be proven that this equals the right-hand side: \(\left(\frac{a-b+c}{2}\right)\left(\frac{a+b-c}{2}\right)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are special equations involving trigonometric functions that hold true for all possible values of the variables involved. They are vital tools in trigonometry and can simplify complex mathematical problems.

• Sine, Cosine, and Tangent: These are the basic trigonometric functions. Cosine (cos) specifically relates the adjacent side to the hypotenuse in a right triangle.


• Law of Cosines Identity: For any triangle, the Law of Cosines is expressed as \(a^2 = b^2 + c^2 - 2bc \cos A\). This relates the lengths of the sides of a triangle to the cosine of one of its angles.


• Transforming Identities: Through algebraic manipulation, trigonometric identities can be transformed to express one function in terms of others, like transforming \(\cos A\) into \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\).

Understanding and proving identities is a crucial skill as it allows one to confirm the correctness of equations involved in geometrical proofs or transformations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions to make them easier to understand or solve.

• Equation Transformation: By moving terms from one side of an equation to another, we can transform complex expressions into simpler forms.


• Substitution: In trigonometry, substitution is often used to replace terms with equivalent expressions to facilitate problem-solving, such as substituting the expression for \(\cos A\) from the Law of Cosines into other identities.


• Simplification: This is the process of reducing an expression to its simplest form, which is essential for proving identities or solving equations like those in the exercise.

Mastering algebraic manipulation is crucial in trigonometry as it lays the foundation for proving more complex identities and simplifying equations.

Triangle Properties

Triangles are fundamental geometrical shapes with unique properties that are extensively used in trigonometry.

• Sides and Angles: A triangle consists of three sides (a, b, and c) and three angles (A, B, and C). The sum of the angles in a triangle is always 180 degrees.


• Types of Triangles: Triangles can be classified based on their sides (equilateral, isosceles, scalene) or angles (acute, obtuse, right).


• Law of Cosines Application: This law is particularly useful in non-right triangles for finding an unknown side when two sides and an angle are known. It is an extension of the Pythagorean theorem.

Understanding the properties and classifications of triangles is essential as it allows for the correct application of trigonometric laws, such as the Law of Cosines, to solve various geometrical problems.