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Questions and Answers
Calculate the values of sin(60°) and cos(60°) and explain their significance in a right triangle.
Calculate the values of sin(60°) and cos(60°) and explain their significance in a right triangle.
sin(60°) = √3/2 and cos(60°) = 1/2; these values represent the ratios of the opposite and adjacent sides to the hypotenuse in a right triangle with a 60° angle.
Explain the relationship between sine and cosine in terms of the Pythagorean identity.
Explain the relationship between sine and cosine in terms of the Pythagorean identity.
The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1, illustrating how the sine and cosine of an angle are geometrically related to the sides of a right triangle.
What is the period of the sine and cosine functions, and why is this concept important in trigonometry?
What is the period of the sine and cosine functions, and why is this concept important in trigonometry?
The period of the sine and cosine functions is 2π, meaning they repeat every 2π radians, which is important for analyzing oscillatory behavior in various applications.
Describe the significance of the unit circle in understanding trigonometric functions.
Describe the significance of the unit circle in understanding trigonometric functions.
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What are the reciprocal functions of sine, cosine, and tangent, and how are they defined?
What are the reciprocal functions of sine, cosine, and tangent, and how are they defined?
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Study Notes
Trigonometry
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Definition: A branch of mathematics dealing with the relationships between the angles and sides of triangles, especially right triangles.
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Basic Functions:
- Sine (sin): Opposite side / Hypotenuse
- Cosine (cos): Adjacent side / Hypotenuse
- Tangent (tan): Opposite side / Adjacent side
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Reciprocal Functions:
- Cosecant (csc): 1/sin
- Secant (sec): 1/cos
- Cotangent (cot): 1/tan
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Key Angles:
- 0°, 30°, 45°, 60°, 90°
- Corresponding sine values:
- sin(0°) = 0
- sin(30°) = 1/2
- sin(45°) = √2/2
- sin(60°) = √3/2
- sin(90°) = 1
- Corresponding cosine values:
- cos(0°) = 1
- cos(30°) = √3/2
- cos(45°) = √2/2
- cos(60°) = 1/2
- cos(90°) = 0
- Corresponding tangent values:
- tan(0°) = 0
- tan(30°) = 1/√3
- tan(45°) = 1
- tan(60°) = √3
- tan(90°) = undefined
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Pythagorean Identity:
- sin²(θ) + cos²(θ) = 1
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Angle 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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Unit Circle:
- A circle with a radius of 1, centered at the origin (0,0) in the Cartesian plane.
- Angles measured in radians, where:
- 360° = 2π radians
- π/2, π, 3π/2 are key angles.
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Applications:
- Used in physics, engineering, computer graphics, and any field involving periodic phenomena.
- Important for solving triangles (law of sines and law of cosines).
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Graphs:
- Sine and cosine functions oscillate between -1 and 1.
- Period of sine and cosine is 2π.
- Tangent function has a period of π and asymptotes at odd multiples of π/2.
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Important Properties:
- Periodicity: Functions repeat at regular intervals.
- Symmetry: Sine is odd (sin(-x) = -sin(x)), cosine is even (cos(-x) = cos(x)), tangent is odd (tan(-x) = -tan(x)).
Trigonometry Overview
- A mathematical discipline focused on the relationships between angles and sides of triangles, predominantly right triangles.
Basic Trigonometric Functions
- Sine (sin): Ratio of the opposite side to the hypotenuse.
- Cosine (cos): Ratio of the adjacent side to the hypotenuse.
- Tangent (tan): Ratio of the opposite side to the adjacent side.
Reciprocal Trigonometric Functions
- Cosecant (csc): The reciprocal of sine (1/sin).
- Secant (sec): The reciprocal of cosine (1/cos).
- Cotangent (cot): The reciprocal of tangent (1/tan).
Key Angles and Their Values
- Notable angles: 0°, 30°, 45°, 60°, 90°.
- Sine values:
- sin(0°) = 0
- sin(30°) = 1/2
- sin(45°) = √2/2
- sin(60°) = √3/2
- sin(90°) = 1
- Cosine values:
- cos(0°) = 1
- cos(30°) = √3/2
- cos(45°) = √2/2
- cos(60°) = 1/2
- cos(90°) = 0
- Tangent values:
- tan(0°) = 0
- tan(30°) = 1/√3
- tan(45°) = 1
- tan(60°) = √3
- tan(90°) = undefined
Pythagorean Identity
- The fundamental relationship described by the equation sin²(θ) + cos²(θ) = 1.
Angle Sum and Difference Formulas
- Sine addition and subtraction: sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β).
- Cosine addition and subtraction: cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β).
- Tangent addition and subtraction: tan(α ± β) = (tan(α) ± tan(β)) / (1 ∓ tan(α)tan(β)).
Unit Circle
- Defined as a circle with radius 1, centered at the origin (0, 0) in the Cartesian plane.
- Angle measurements: 360° corresponds to 2π radians.
Applications of Trigonometry
- Extensively utilized in fields like physics, engineering, computer graphics, and other areas dealing with periodic behavior.
- Essential in solving triangles through methods such as the law of sines and the law of cosines.
Graphs and Their Properties
- Sine and cosine functions oscillate within the range of -1 to 1, with a period of 2π.
- The tangent function has a period of π and is characterized by asymptotes at odd multiples of π/2.
Important Properties of Functions
- Periodicity: Trigonometric functions exhibit repetition at consistent intervals.
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Symmetry:
- Sine is an odd function: sin(-x) = -sin(x).
- Cosine is an even function: cos(-x) = cos(x).
- Tangent is also an odd function: tan(-x) = -tan(x).
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Description
Test your knowledge on the fundamental concepts of trigonometry, including definitions, basic functions, and key angles. This quiz covers essential relationships between angles and sides of triangles, along with reciprocal functions and identities.
