Trigonometric equations are a fundamental part of mathematics with applications in various fields such as physics, engineering, and architecture. In this blog post, we will focus on solving a specific trigonometric equation that involves the operations of cosine, sine, and cotangent functions. We will break down the equation step by step, utilizing trigonometric identities and properties along the way to ultimately find the value of the variable involved.
Given Trigonometric Equation:
( \frac{{\cos A – \sin A + 1}}{{\cos A + \sin A – 1}} = \cos A + \cot A )
Before we dive into solving this equation, let’s revisit some key trigonometric identities that will be useful during the problem-solving process.
Trigonometric Identities:
- Pythagorean Identities:
- ( \sin^2 A + \cos^2 A = 1 )
- ( 1 + \tan^2 A = \sec^2 A )
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( 1 + \cot^2 A = \csc^2 A )
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Reciprocal Identities:
- ( \csc A = \frac{1}{\sin A} )
- ( \sec A = \frac{1}{\cos A} )
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( \cot A = \frac{1}{\tan A} )
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Double Angle Identities:
- ( \sin 2A = 2 \sin A \cos A )
- ( \cos 2A = \cos^2 A – \sin^2 A )
- ( \tan 2A = \frac{2\tan A}{1 – \tan^2 A} )
Solving the Equation:
Let’s start by simplifying the left side of the equation:
( \frac{{\cos A – \sin A + 1}}{{\cos A + \sin A – 1}} )
We can simplify this expression by rationalizing the denominator. Multiplying both the numerator and denominator by ( \cos A + \sin A + 1 ), we get:
( \frac{{(\cos A – \sin A + 1)(\cos A + \sin A + 1)}}{{(\cos A + \sin A – 1)(\cos A + \sin A + 1)}} )
Expanding the numerator and denominator, we have:
( \frac{{\cos^2 A + \cos A\sin A + \cos A – \cos A\sin A – \sin^2 A – \sin A + \cos A – \sin A + 1}}{{\cos^2 A + \cos A\sin A + \cos A\sin A + \sin^2 A – \cos A + \sin A – \cos A – \sin A – 1}} )
This simplifies to:
( \frac{{\cos^2 A – \sin^2 A + 2\cos A}}{{2\cos A}} )
Using the Pythagorean identity, ( \cos^2 A – \sin^2 A = \cos 2A ), the expression becomes:
( \frac{{\cos 2A + 2\cos A}}{{2\cos A}} )
Further simplifying:
( \frac{{\cos 2A}}{{2}} + 1 )
Now, focusing on the right side of the equation:
( \cos A + \cot A )
As ( \cot A = \frac{1}{\tan A} ) and ( \cot A = \frac{\cos A}{\sin A} ), we can rewrite this expression as:
( \cos A + \frac{\cos A}{\sin A} )
Combining these terms, we get:
( \frac{\sin A \cos A + \cos A}{\sin A} )
which simplifies to:
( \frac{\cos A(\sin A + 1)}{\sin A} )
Applying the Pythagorean identity, ( \sin A + 1 = 2 \sin(\frac{A}{2})\cos(\frac{A}{2}) ), the expression becomes:
( \frac{2\cos A \sin(\frac{A}{2})\cos(\frac{A}{2})}{\sin A} )
Simplifying further:
( \frac{2\cos(\frac{A}{2})}{1} )
Therefore, the right side of the equation simplifies to:
( 2\cos(\frac{A}{2}) )
Equating the Left and Right Sides:
Now, we equate the simplified left side (( \frac{{\cos 2A}}{{2}} + 1 )) and right side (( 2\cos(\frac{A}{2}) )) of the equation:
( \frac{{\cos 2A}}{2} + 1 = 2\cos(\frac{A}{2}) )
Multiplying by 2 to clear the fraction, we get:
( \cos 2A + 2 = 4\cos(\frac{A}{2}) )
Using double angle identity for cosine, ( \cos 2A = 2\cos^2 A – 1 ), the equation simplifies to:
( 2\cos^2 A – 1 + 2 = 4\cos(\frac{A}{2}) )
( 2\cos^2 A + 1 = 4\cos(\frac{A}{2}) )
Now, we have a quadratic equation in terms of cosine. Let ( x = \cos A ), so the equation becomes:
( 2x^2 + 1 = 4\cos(\frac{A}{2}) )
Solving for ( x ), we find:
( x = \frac{4\cos(\frac{A}{2}) – 1}{2} )
Therefore, the solution to the trigonometric equation ( \frac{{\cos A – \sin A + 1}}{{\cos A + \sin A – 1}} = \cos A + \cot A ) is ( \frac{4\cos(\frac{A}{2}) – 1}{2} ).
FAQs:
- What are some key trigonometric identities to remember when solving equations involving cosine and sine functions?
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Some fundamental trigonometric identities include Pythagorean identities, reciprocal identities, and double angle identities which are often useful in simplifying trigonometric expressions.
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How can we rationalize the denominator in a trigonometric expression?
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To rationalize the denominator, one can multiply both the numerator and denominator by an appropriate expression to eliminate any radicals or complex terms in the denominator.
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Why is it important to simplify trigonometric equations using identities and properties?
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Simplifying expressions using trigonometric identities can help in making the calculations more manageable and in identifying relationships between different trigonometric functions.
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How can double angle identities be applied in solving trigonometric equations?
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Double angle identities allow for expressing trigonometric functions in terms of double the angle, which can be helpful in simplifying expressions and solving equations involving trigonometric functions.
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What does the solution ( \frac{4\cos(\frac{A}{2}) – 1}{2} ) imply in the context of the given trigonometric equation?
- The solution represents the value of the variable ( A ) that satisfies the original trigonometric equation, showcasing the relationship between the cosine function at different angles within the equation.
