Questions and Answers
Which of the following is a Pythagorean identity?
The function cos(-θ) is equal to -cos(θ).
False
What is the formula for sin(2θ)?
2sin(θ)cos(θ)
For the equation tan(θ) = 0, the general solution is θ = ______.
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Match each type of identity with its corresponding identity:
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Who established the conservation of mass principle?
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In an open system, mass is conserved during a chemical reaction.
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What must be true about the total mass of reactants and products in a closed system?
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The principle of conservation of mass states that mass is neither __________ nor __________ in a chemical reaction.
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Match the following terms with their definitions:
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Which process is an example of the conservation of mass?
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Balancing chemical equations reflects the conservation of mass.
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What happens to atoms during a chemical change according to the conservation of mass?
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Study Notes
Trigonometric Identities
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Fundamental Identities
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Pythagorean Identities
- sin²(θ) + cos²(θ) = 1
- 1 + tan²(θ) = sec²(θ)
- 1 + cot²(θ) = csc²(θ)
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Reciprocal Identities
- sin(θ) = 1/csc(θ)
- cos(θ) = 1/sec(θ)
- tan(θ) = 1/cot(θ)
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Quotient Identities
- tan(θ) = sin(θ)/cos(θ)
- cot(θ) = cos(θ)/sin(θ)
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Pythagorean Identities
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Even-Odd Identities
- sin(-θ) = -sin(θ)
- cos(-θ) = cos(θ)
- tan(-θ) = -tan(θ)
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Co-Function Identities
- sin(π/2 - θ) = cos(θ)
- cos(π/2 - θ) = sin(θ)
- tan(π/2 - θ) = cot(θ)
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Sum and Difference Formulas
- sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)
- cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β)
- tan(α ± β) = (tan(α) ± tan(β)) / (1 ∓ tan(α)tan(β))
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Double Angle Formulas
- sin(2θ) = 2sin(θ)cos(θ)
- cos(2θ) = cos²(θ) - sin²(θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ)
- tan(2θ) = 2tan(θ) / (1 - tan²(θ))
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Half Angle Formulas
- sin(θ/2) = √((1 - cos(θ))/2)
- cos(θ/2) = √((1 + cos(θ))/2)
- tan(θ/2) = sin(θ)/(1 + cos(θ)) or (1 - cos(θ))/sin(θ)
Trigonometric Equations
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Basic Equations
- sin(θ) = 0: θ = nπ
- cos(θ) = 0: θ = (2n + 1)(π/2)
- tan(θ) = 0: θ = nπ
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Solving Trigonometric Equations
- Isolate the trigonometric function.
- Use identities to simplify.
- Find general solutions and specific solutions within the interval [0, 2π).
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Common Techniques
- Factoring: e.g., sin(θ)(1 - cos(θ)) = 0
- Using identities: Transform to a single function.
- Graphical Method: Use graphs of functions to find intersections.
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Applications
- Used to solve real-world problems in physics, engineering, and architecture.
- Essential for modeling periodic phenomena like sound waves, light waves, and circular motion.
Fundamental Identities
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Pythagorean Identities link sine and cosine functions, forming a foundation of trigonometry:
- sin²(θ) + cos²(θ) = 1
- 1 + tan²(θ) = sec²(θ)
- 1 + cot²(θ) = csc²(θ
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Reciprocal Identities show relationships between sine, cosine, and their reciprocals:
- sin(θ) = 1/csc(θ)
- cos(θ) = 1/sec(θ)
- tan(θ) = 1/cot(θ)
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Quotient Identities express tangent and cotangent as ratios of sine and cosine:
- tan(θ) = sin(θ)/cos(θ)
- cot(θ) = cos(θ)/sin(θ)
Even-Odd Identities
- Sine Function is odd: sin(-θ) = -sin(θ)
- Cosine Function is even: cos(-θ) = cos(θ)
- Tangent Function is odd: tan(-θ) = -tan(θ)
Co-Function Identities
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Complementary Angle Relationships:
- sin(π/2 - θ) = cos(θ)
- cos(π/2 - θ) = sin(θ)
- tan(π/2 - θ) = cot(θ)
Sum and Difference Formulas
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Sine Addition and Subtraction:
- sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)
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Cosine Addition and Subtraction:
- cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β)
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Tangent Addition and Subtraction:
- tan(α ± β) = (tan(α) ± tan(β)) / (1 ∓ tan(α)tan(β))
Double Angle Formulas
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Double Angle for Sine:
- sin(2θ) = 2sin(θ)cos(θ)
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Double Angle for Cosine:
- cos(2θ) can be expressed in three forms:
- cos²(θ) - sin²(θ)
- 2cos²(θ) - 1
- 1 - 2sin²(θ)
- cos(2θ) can be expressed in three forms:
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Double Angle for Tangent:
- tan(2θ) = 2tan(θ) / (1 - tan²(θ))
Half Angle Formulas
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Half Angle for Sine:
- sin(θ/2) = √((1 - cos(θ))/2)
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Half Angle for Cosine:
- cos(θ/2) = √((1 + cos(θ))/2)
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Half Angle for Tangent:
- tan(θ/2) can be calculated using two formulas:
- sin(θ)/(1 + cos(θ)
- (1 - cos(θ))/sin(θ)
- tan(θ/2) can be calculated using two formulas:
Trigonometric Equations
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Basic Solutions:
- sin(θ) = 0 yields solutions θ = nπ
- cos(θ) = 0 yields solutions θ = (2n + 1)(π/2)
- tan(θ) = 0 yields solutions θ = nπ
Solving Trigonometric Equations
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Methods:
- Isolate the trigonometric function, simplifying as necessary by applying identities.
- Determine general and specific solutions in the interval [0, 2π).
Common Techniques
- Factoring: Apply techniques such as sin(θ)(1 - cos(θ)) = 0 to find solutions.
- Using Identities: Transform expressions to a single function for easier solving.
- Graphical Method: Utilize graphs to identify intersections representing solutions.
Applications
- Trigonometric equations are crucial for solving practical problems in physics, engineering, and architecture.
- They also model periodic phenomena, including sound waves, light waves, and circular motion.
Conservation of Mass
- Definition: Mass cannot be created or destroyed during chemical reactions; total mass remains constant throughout the process.
- Closed System Importance: Conservation applies only in a closed system where no external mass is introduced or removed.
- Reactants vs. Products: Total mass of reactants entering a reaction equals total mass of products formed; essential for assessing reaction efficiency.
- Atomic Rearrangement: Atoms are rearranged during chemical changes, but their total number remains unchanged; no atoms are lost or gained.
- Balancing Equations: Balancing chemical equations reflects conservation of mass, ensuring identical counts for each element on both sides.
- Historical Significance: Antoine Lavoisier established this principle in the late 18th century through meticulous mass measurements before and after chemical reactions.
- Stoichiometry Role: Key in stoichiometry for calculating amounts of reactants needed and products yielded, vital for precise chemical manufacturing.
- Industrial Applications: Helps optimize resource use and reduce waste in manufacturing processes, contributing to environmental sustainability.
- Limitations in Open Systems: In open systems, mass changes can occur due to gas escape or environmental absorption, leading to apparent deviations from conservation.
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Practical Examples:
- Combustion of Hydrocarbon Fuels: Mass of consumed fuel and oxygen equals the mass of produced carbon dioxide and water.
- Decomposition of Water: Mass of resulting hydrogen and oxygen gases matches the mass of the original water, demonstrating mass conservation in chemical transformations.
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Description
Test your knowledge of trigonometric identities, including fundamental, reciprocal, and even-odd identities. This quiz covers essential concepts that are crucial for anyone studying trigonometry. Challenge yourself to recall these identities and solidify your understanding.
