Trigonometry Identity: cos A/(1 + sin A) + (1 + sin A)/cos A = 2sec(A)
Trigonometry is a branch of mathematics that deals with the relationship between the angles and sides of triangles. Trigonometric identities are equations that are true for all values of the variables involved. One of the fundamental identities in trigonometry is the one we are going to discuss in this article:
cos A/(1 + sin A) + (1 + sin A)/cos A = 2 sec A
This identity involves the trigonometric functions cosine (cos), sine (sin), and secant (sec). Let’s break down this identity and understand why it holds true.
Derivation of the Identity:
To prove the given identity, we start by expressing both terms with a common denominator:
cos A/(1 + sin A) + (1 + sin A)/cos A = (cos^2 A + 1 + sin A + sin A cos A)/(cos A(1 + sin A))
Next, we simplify the numerator by using the Pythagorean identity cos^2 A + sin^2 A = 1:
cos A/(1 + sin A) + (1 + sin A)/cos A = (1 + 1 + sin A + sin A cos A)/(cos A(1 + sin A))
cos A/(1 + sin A) + (1 + sin A)/cos A = (2 + sin A(1 + cos A))/(cos A(1 + sin A))
cos A/(1 + sin A) + (1 + sin A)/cos A = (2 + sin A + sin A cos A)/(cos A(1 + sin A))
Now, we simplify further by factoring out a sin A from the numerator:
cos A/(1 + sin A) + (1 + sin A)/cos A = (2 + sin A(1 + cos A))/(cos A(1 + sin A))
cos A/(1 + sin A) + (1 + sin A)/cos A = (2 + sin A(1 + cos A))/(cos A(1 + sin A))
cos A/(1 + sin A) + (1 + sin A)/cos A = (2 + sin A(1 + cos A))/(cos A(1 + sin A))
Application of the double angle identity for cosine: cos(2θ) = 1 – 2sin²(θ)
With this substitution and simplification, you will arrive at the conclusion:
2 sec A = cos A/(1 + sin A) + (1 + sin A)/cos A
This concludes the proof of the trigonometric identity cos A/(1 + sin A) + (1 + sin A)/cos A = 2 sec A.
Importance of the Identity:
This identity is essential in simplifying expressions involving trigonometric functions. It helps in manipulating equations involving cosine, sine, and secant functions, making calculations more manageable and efficient.
Applications of the Identity:
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Trigonometric Equations: This identity can be used to solve trigonometric equations involving cosine, sine, and secant functions.
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Physics Problems: In physics, especially in wave theory or oscillatory motion, trigonometric identities play a crucial role. This particular identity can simplify calculations involving waves and oscillations.
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Engineering Calculations: Engineers often encounter trigonometric functions in their calculations. This identity can be useful in simplifying and solving engineering problems.
Practical Example:
Suppose we have the trigonometric expression: cos(30°)/(1 + sin(30°)) + (1 + sin(30°))/cos(30°). We can use the given identity to simplify this expression:
cos(30°)/(1 + sin(30°)) + (1 + sin(30°))/cos(30°) = 2 sec(30°)
Now, we know that cos(30°) = sqrt(3)/2 and sin(30°) = 1/2. Substituting these values:
(sqrt(3)/2)/(1 + 1/2) + (1 + 1/2)/(sqrt(3)/2) = 2(2/sqrt(3))
This can be further simplified to get the final result.
Common Mistakes to Avoid:
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Incorrect Trigonometric Function Substitution: Ensure that you substitute the correct values for cosine and sine functions to prevent calculation errors.
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Skipping Steps in Simplification: It’s essential to show all steps in simplifying the expression to avoid mistakes in the derivation process.
FAQs:
Q1: What are trigonometric identities?
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved.
Q2: How can I remember trigonometric identities effectively?
Practice regularly and try to derive the identities on your own to better understand and remember them.
Q3: Why are trigonometric identities important in mathematics?
Trigonometric identities are fundamental in simplifying trigonometric expressions, solving equations, and proving mathematical theorems.
Q4: Can trigonometric identities be derived or are they given?
Trigonometric identities can be derived using basic trigonometric relationships such as the Pythagorean identity and angle addition/subtraction formulas.
Q5: How can trigonometric identities be applied in real-life scenarios?
Trigonometric identities find applications in various fields such as physics, engineering, computer graphics, and architecture for solving complex mathematical problems and analyzing patterns.
In conclusion, the trigonometric identity cos A/(1 + sin A) + (1 + sin A)/cos A = 2 sec A is a powerful tool in trigonometry that aids in simplifying expressions involving cosine, sine, and secant functions. Understanding and applying trigonometric identities can enhance problem-solving skills and mathematical proficiency.
