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Questions and Answers
When simplifying the expression $ rac{1+ ext{cos}θ}{1- ext{cos}θ} - rac{1- ext{cos}θ}{1+ ext{cos}θ}$, what trigonometric function is equal to $4 ext{cot}θ ext{cosec}θ$?
What result do you obtain when you add $ rac{ ext{sin}θ}{1+ ext{cos}θ}$ and $ rac{1+ ext{cos}θ}{ ext{sin}θ}$?
If $ ext{cos}θ - ext{sin}θ = ext{√}2 ext{sin}θ$, what should be proved about $ ext{cos}θ + ext{sin}θ$?
Given $tanθ = n tanφ$ and $sinθ = m sinφ$, what does the relationship $ ext{cos}^2θ = rac{m^2 - 1}{n^2 - 1}$ imply?
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When substituting $x = a ext{cosec}θ$ and $y = b ext{cot}θ$, what identity is established by $ rac{x^2}{a^2} - rac{y^2}{b^2} = 1$?
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Study Notes
Trigonometric Equations Overview
- Trigonometric equations involve relations between angles and their corresponding sine, cosine, tangent values.
- Key identities simplify these equations, aiding in finding solutions.
Problem Equations
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Equation 6:
- ((1 + \cos θ) / (1 - \cos θ) - (1 - \cos θ) / (1 + \cos θ) = 4 \cot θ \csc θ)
- Involves subtraction of fractions with trigonometric functions and is set to equal a product of cotangent and cosecant.
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Equation 7:
- (\sin θ / (1 + \cos θ) + (1 + \cos θ) / \sin θ = 2 \csc θ)
- Combines sine and cosine in a manner that requires manipulation to reach a simple trigonometric function.
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Equation 8:
- Given ( \cos θ - \sin θ = \sqrt{2} \sin θ), prove that ( \cos θ + \sin θ = \sqrt{2} \cos θ).
- Involves proofs using basic trigonometric properties.
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Equation 9:
- Establish ( \tan θ = n \tan φ) and ( \sin θ = m \sin φ), then prove ( \cos^2 θ = (m^2 - 1) / (n^2 - 1)).
- Involves the relationship between tangent, sine, and cosine values.
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Equation 10:
- If ( x = a \csc θ) and ( y = b \cot θ) then prove ((x^2/a^2) - (y^2/b^2) = 1).
- Connects cosecant and cotangent to derive a Pythagorean identity.
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Equation 11:
- Let ( x = r \sin A \cos C), ( y = r \sin A \sin C) and ( z = r \cos A) to prove ( r^2 = x^2 + y^2 + z^2).
- Demonstrates how to express three-dimensional space using trigonometric functions.
Applications of Trigonometric Identities
- Trigonometric identities serve to simplify complex equations into more manageable forms.
- These principles extend to real-world problems, allowing for calculations involving angles, heights, distances, and other applications in physics and engineering.
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Description
This quiz covers key aspects of trigonometric equations, focusing on relationships between angles and their sine, cosine, and tangent values. It includes problem-solving involving complex equations and proofs using fundamental trigonometric identities.
