Problem 184 $$ \cot A=\frac{1}{2}\left(\co... [FREE SOLUTION]

Chapter 11: Problem 184

$$ \cot A=\frac{1}{2}\left(\cot \frac{A}{2}-\tan \frac{A}{2}\right) $$

Short Answer

Expert verified

The exercise involves converting the cotangent and tangent functions into their sine and cosine counterparts, followed by simplification using trigonometric identities. The equation is verified after applying the identities.

Step by step solution

Rewrite cotangent and tangent in terms of sine and cosine

The formula for cotangent is $\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}$ and for tangent $\tan x = \frac{\sin x}{\cos x}$.\n Using these definitions, the given equation becomes: \[\frac{\cos A}{\sin A} = \frac{1}{2} \left( \frac{\cos \frac{A}{2}}{\sin \frac{A}{2}} - \frac{\sin \frac{A}{2}}{\cos \frac{A}{2}} \right)\]

Simplify

Cross multiply to simplify the equation: \[2\cos A \sin \frac{A}{2} \cos \frac{A}{2} = \cos A \cos \frac{A}{2} - \sin A \sin \frac{A}{2} \] This equation can be further simplified using the trigonometric identity $\cos(2x) = \cos^{2}(x) - \sin^{2}(x)$. Now replace $2x$ by A: \[2\cos A \sin \frac{A}{2} \cos \frac{A}{2} = \cos (2x)\]

Simplify further

The equation can be further simplified using the identity $\sin(2x) = 2\sin x\cos x$. Thus the equation becomes: \[2\cos A \sin A = \cos (2A)\] which is a known trigonometric identity, and thus verifies the given equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cotangent Function

Cotangent, often abbreviated as cot, is one of the fundamental trigonometric functions. It is the reciprocal of the tangent function and is represented mathematically as $ \cot x = \frac{1}{\tan x} $, which simplifies to $ \cot x = \frac{\cos x}{\sin x} $.
This relation shows how cotangent compares the ratio of cosine and sine. Understanding this definition is essential because it allows you to express trigonometric equations in terms of sine and cosine, making them often easier to work with in algebra.

Cotangent measures the length of the adjacent side over the opposite side in a right triangle.
It can help to switch between cotangent and its reciprocal, tangent, depending on the problem's requirements.

Mastery of cotangent is pivotal in simplifying complex trigonometric forms.

Tangent Function

The tangent function, denoted as $ \tan x $, is another core trigonometric function defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. Mathematically, it is expressed as $ \tan x = \frac{\sin x}{\cos x} $.
This ratio is significant because it relates two essential trigonometric functions, sine and cosine, combining their properties into one function.

The tangent function is periodic with a period of $ \pi $.
It has vertical asymptotes where $ \cos x = 0 $ (i.e., $ x = \frac{\pi}{2} + n\pi $, where $ n $ is an integer).

Using tangent and understanding its properties can drastically simplify solving trigonometric equations, especially when paired with cotangent expressions.

Simplifying Trigonometric Equations

Simplifying trigonometric equations often involves rewriting complex expressions using well-known trigonometric identities. In the provided problem, the equation is first rewritten using the definitions of cotangent and tangent.
Then, to simplify and solve the equation, identities like the double angle and product-to-sum formulas are used.
Here is a quick guide:

Start by replacing trigonometric functions with their sine and cosine forms.
Use known trigonometric identities to combine like terms.
Balance the equation by cross-multiplying if necessary, aiming to eliminate fractions.

Mastering these steps allows you to navigate through apparently complex trigonometric equations with ease.

Double Angle Formula

The double angle formula is a powerful technique in trigonometry for converting expressions that involve multiple angles. These formulas simplify many trigonometric equations, bending them into more manageable forms.
Key double angle identities include:

$ \cos(2x) = \cos^2(x) - \sin^2(x) $
$ \sin(2x) = 2\sin(x)\cos(x) $
$ \tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)} $

These identities emerge from the addition and subtraction formulas for sine and cosine.
In practice, they help reduce the complexity of equations by lowering the degree of the trigonometric functions involved. As seen in the exercise, substituting the double angle formula led to solving the equation by identifying it as a known identity, providing a seamless path to a solution.

91影视

$$ \cot A=\frac{1}{2}\left(\cot \frac{A}{2}-\tan \frac{A}{2}\right) $$

Short Answer

Step by step solution

Rewrite cotangent and tangent in terms of sine and cosine

Simplify

Simplify further

Key Concepts

Cotangent Function

Tangent Function

Simplifying Trigonometric Equations

Double Angle Formula

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