Questions and Answers
What is the definition of sine in relation to a right triangle?
Which identity represents the relationship between sine and cosine for a given angle?
What is the cosine of 0°?
How can the tangent function be expressed in terms of sine and cosine?
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Which of the following is a reciprocal function of sine?
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What is the period of the tangent function?
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What is the sum formula for sine?
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In what format are angles commonly measured in trigonometry?
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Study Notes
Trigonometry
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Definition: Branch of mathematics dealing with the relationships between the angles and sides of triangles, especially right triangles.
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Key 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 (tan = sin/cos).
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Reciprocal Functions:
- Cosecant (csc): Reciprocal of sine (csc = 1/sin).
- Secant (sec): Reciprocal of cosine (sec = 1/cos).
- Cotangent (cot): Reciprocal of tangent (cot = 1/tan).
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Pythagorean Identity:
- sin²(θ) + cos²(θ) = 1
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Angle Measures:
- Degrees: Full circle = 360 degrees.
- Radians: Full circle = 2π radians (π radians = 180 degrees).
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Unit Circle:
- Circle of radius 1 centered at the origin (0,0).
- Helps in defining trigonometric functions for all angles.
- Coordinates of key angles:
- (1, 0) at 0° (0 rad)
- (0, 1) at 90° (π/2 rad)
- (-1, 0) at 180° (π rad)
- (0, -1) at 270° (3π/2 rad)
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Trigonometric Identities:
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Angle Sum/Difference:
- sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)
- cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β)
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Double Angle:
- sin(2θ) = 2sin(θ)cos(θ)
- cos(2θ) = cos²(θ) - sin²(θ)
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Angle Sum/Difference:
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Applications:
- Solving triangles in geometry.
- Modeling periodic phenomena (e.g., waves, oscillations).
- Navigation and physics problems.
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Graphs:
- Sine and cosine functions have a range of [-1, 1] and a period of 2π.
- Tangent function has a vertical asymptote and a period of π.
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Inverse Trigonometric Functions:
- Used to find angles when the values of trigonometric functions are known.
- Notation:
- sin⁻¹(x), cos⁻¹(x), tan⁻¹(x).
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Common Angles:
- 0°, 30°, 45°, 60°, 90° (and their radian equivalents).
- Known sine, cosine, and tangent values for easy reference.
Definition of Trigonometry
- Trigonometry focuses on the relationships between the angles and sides of triangles, primarily right triangles.
Key Functions
- Sine (sin): Opposite side divided by the hypotenuse.
- Cosine (cos): Adjacent side divided by the hypotenuse.
- Tangent (tan): Opposite side divided by the adjacent side, expressed as tan = sin/cos.
Reciprocal Functions
- Cosecant (csc): Inverse of sine, calculated as csc = 1/sin.
- Secant (sec): Inverse of cosine, calculated as sec = 1/cos.
- Cotangent (cot): Inverse of tangent, calculated as cot = 1/tan.
Pythagorean Identity
- Fundamental relationship: sin²(θ) + cos²(θ) = 1, applicable for any angle θ.
Angle Measures
- Degrees: Circle is divided into 360 degrees.
- Radians: Circle is divided into 2π radians, with π radians equivalent to 180 degrees.
Unit Circle
- Represents a circle with a radius of 1, centered at the origin (0,0).
- Essential for defining trigonometric functions across all angles, with key angle coordinates:
- 0° (0 rad) corresponds to (1, 0).
- 90° (π/2 rad) corresponds to (0, 1).
- 180° (π rad) corresponds to (-1, 0).
- 270° (3π/2 rad) corresponds to (0, -1).
Trigonometric Identities
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Angle Sum/Difference identities express sine and cosine of summed or subtracted angles:
- sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)
- cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β)
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Double Angle identities for sine and cosine:
- sin(2θ) = 2sin(θ)cos(θ)
- cos(2θ) = cos²(θ) - sin²(θ)
Applications of Trigonometry
- Crucial for solving triangles in geometric contexts.
- Useful in modeling periodic phenomena such as waves and oscillations.
- Relevant in navigation and various physics problems.
Graphs of Trigonometric Functions
- Sine and cosine functions have a range from -1 to 1, with a periodicity of 2π.
- The tangent function exhibits vertical asymptotes and has a periodicity of π.
Inverse Trigonometric Functions
- Employed to determine angles when certain trigonometric function values are known.
- Notation includes sin⁻¹(x), cos⁻¹(x), and tan⁻¹(x).
Common Angles
- Basic angles include 0°, 30°, 45°, 60°, and 90°, along with their corresponding radian measures.
- Sine, cosine, and tangent values for these angles are essential for quick reference.
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Description
Test your knowledge of the fundamentals of trigonometry, including key functions and identities. This quiz covers essential concepts such as sine, cosine, tangent, and the unit circle. Perfect for students looking to reinforce their understanding of this important mathematical branch.
