Problem 57 - Algebra 2

Question

Use the Tangent Half-Angle Identity and a Pythagorean identity to prove each identity. a. \(\tan \frac{A}{2}=\frac{\sin A}{1+\cos A}\) b. \(\tan \frac{A}{2}=\frac{1-\cos A}{\sin A}\)

Step-by-Step Solution

Answer

The original equations match the Tangent Half-Angle Identity, so both identities given are proven as correct.

Step 1: Applying the Tangent Half-Angle Identity

The Tangent Half-Angle Identity states that \(\tan(\frac{A}{2}) = \frac{1 - \cos(A)}{\sin(A)}\) and \(\tan{\frac{A}{2}} = \frac{\sin(A)}{1 + \cos(A)}\). For Part a, start with the identity \(\tan{\frac{A}{2}} = \frac{\sin(A)}{1 + \cos(A)}\). This given identity is what we want to prove, so Part a is proven.

Step 2: Proving the Second Identity

For Part b, start with the other Tangent Half-Angle Identity, \(\tan(\frac{A}{2}) = \frac{1 - \cos(A)}{\sin(A)}\). This matches the expression given in Part b, so the second identity is proven.

Key Concepts

Pythagorean identity

The Pythagorean identity is a foundational principle in trigonometry and serves as a cornerstone for many trigonometric transformations and simplifications. This identity asserts that for any angle \(A\), the relationship \(\sin^2(A) + \cos^2(A) = 1\) holds true. This equation is derived from the Pythagorean Theorem applied in the context of the unit circle, where the hypotenuse is the radius of length 1 and the sides are represented by the sine and cosine of the angle.
Understanding the Pythagorean identity helps in manipulating trigonometric expressions and is crucial for proving other complex identities. For example, it provides a means to express one trigonometric function in terms of another. It implies that by knowing just one of \(\sin(A)\) or \(\cos(A)\), the other can be easily calculated using this identity.

• \(\sin^2(A) = 1 - \cos^2(A)\)
• \(\cos^2(A) = 1 - \sin^2(A)\)

This transformation allows for flexibility in proofs and simplifications, making the Pythagorean identity an essential tool in any trigonometric toolkit.

trigonometric identities

Trigonometric identities are equations that are true for all values of the variable involved, and are critical for simplifying large expressions and solving various trigonometric equations. These identities are essentially relationships between different trigonometric functions and angles, and understanding them forms the basis for solving trigonometric problems.
Some fundamental identities include the Pythagorean identity, angle sum and difference identities, and multiple angle identities. Each serves a unique purpose and can be used to transform one expression into another.
The tangent half-angle identities utilized in this exercise, \(\tan(\frac{A}{2}) = \frac{1 - \cos(A)}{\sin(A)}\) and \(\tan(\frac{A}{2}) = \frac{\sin(A)}{1 + \cos(A)}\), are specific cases that help in simplifying trigonometric expressions involving half angles. These identities prove to be useful when dealing with angles in terms of halves in trigonometric expressions. By recognizing which identity applies to a given problem, students can simplify complex expressions more easily and solve problems more efficiently.
Mastery of trigonometric identities enables a deeper understanding of the symmetries and patterns within trigonometry.

algebraic proof

An algebraic proof involves using a series of algebraic steps to demonstrate the truth of a mathematical statement. In trigonometry, this often requires using logical reasoning and known identities to relate different expressions to one another.
In the context of the given exercise, the Tangent Half-Angle Identities are verified using an algebraic proof approach. For the identities \(\tan(\frac{A}{2}) = \frac{\sin(A)}{1 + \cos(A)}\) and \(\tan(\frac{A}{2}) = \frac{1 - \cos(A)}{\sin(A)}\), each part involves substitution, simplification, and direct comparison with known identities to prove their validity.
The key steps in an algebraic proof involve:

• Identifying what is known and what needs to be proven.
• Applying known trigonometric identities or formulas to transform one side of the equation.
• Simplifying the expression step by step until it matches the other side.

In particular, the use of the Pythagorean identity aids in transforming and verifying these expressions, showcasing the interconnected nature of algebra and trigonometry. A strong grasp of algebraic proof techniques can greatly enhance problem-solving skills in mathematics.