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Questions and Answers
What is the property of a right triangle?
What is the property of a right triangle?
An oblique triangle has no angles measuring 90°.
An oblique triangle has no angles measuring 90°.
True
What is the angle sum in a triangle?
What is the angle sum in a triangle?
180°
The __ theorem states that a² + b² = c² for right triangles.
The __ theorem states that a² + b² = c² for right triangles.
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Match the angle type with its definition:
Match the angle type with its definition:
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What is the period for periodic functions like sine, cosine, and tangent?
What is the period for periodic functions like sine, cosine, and tangent?
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Define amplitude in the context of waves.
Define amplitude in the context of waves.
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Which identity states that sin²(x) + cos²(x) = 1?
Which identity states that sin²(x) + cos²(x) = 1?
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What is the formula for sin(2x)?
What is the formula for sin(2x)?
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Give an example of an application area where trigonometry is used.
Give an example of an application area where trigonometry is used.
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Study Notes
Triangles
- Definition: A triangle is a polygon with three sides and three angles.
- Types:
- Right triangle: one angle is 90° (π/2 radians)
- Oblique triangle: no angle is 90°
- Properties:
- Angle sum: 180° (π radians)
- Side lengths: a, b, c (opposite angles A, B, C)
- Pythagorean theorem: a² + b² = c² (for right triangles)
Angles
- Measurement units: degrees (°), radians (rad)
- Conversions:
- 1 rad = 180/π °
- 1 ° = π/180 rad
- Angle types:
- Acute: 0° < θ < 90°
- Right: θ = 90°
- Obtuse: 90° < θ < 180°
- Straight: θ = 180°
Waves
- Periodic functions:
- Sine (sin), cosine (cos), and tangent (tan)
- Period: 2π (360°)
- Wave properties:
- Amplitude: maximum displacement from mean position
- Frequency: number of cycles per unit time
- Wavelength: distance between successive peaks/troughs
- Graphs:
- Sinusoidal curves: sin(x), cos(x), tan(x)
Identities
- Fundamental identities:
- Pythagorean identity: sin²(x) + cos²(x) = 1
- Sum and difference formulas:
- sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
- cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
- Double-angle and half-angle formulas:
- sin(2x) = 2sin(x)cos(x)
- cos(2x) = cos²(x) - sin²(x)
Applications
- Physics and engineering:
- Motion, forces, and energies
- Simple harmonic motion
- Navigation and geography:
- Triangulation for location and distance
- Latitude and longitude calculations
- Computer graphics and game development:
- 3D transformations and projections
- Animation and simulation
Triangles
- A triangle has three sides and three angles.
- There are two main types of triangles: right triangles with one 90° angle and oblique triangles with no 90° angle.
- The angle sum of a triangle is always 180° or π radians.
- The side lengths of a triangle are denoted by a, b, and c, opposite angles A, B, and C respectively.
- The Pythagorean theorem is a² + b² = c², applicable to right triangles.
Angles
- Angles can be measured in degrees or radians, where 1 radian is equivalent to 180/π degrees.
- One degree is equivalent to π/180 radians.
- Angles can be classified into four types: acute (0° < θ < 90°), right (θ = 90°), obtuse (90° < θ < 180°), and straight (θ = 180°).
Waves
- Periodic functions include sine, cosine, and tangent functions with a period of 2π or 360°.
- Wave properties include amplitude (maximum displacement from the mean position), frequency (number of cycles per unit time), and wavelength (distance between successive peaks or troughs).
- Graphs of sine, cosine, and tangent functions are sinusoidal curves.
Identities
- The Pythagorean identity is sin²(x) + cos²(x) = 1.
- The sum and difference formulas are sin(a+b) = sin(a)cos(b) + cos(a)sin(b) and cos(a+b) = cos(a)cos(b) - sin(a)sin(b) respectively.
- Double-angle formulas include sin(2x) = 2sin(x)cos(x) and cos(2x) = cos²(x) - sin²(x).
- Half-angle formulas exist but are not specified.
Applications
- Triangles are used in physics and engineering to calculate motion, forces, and energies, as well as in simple harmonic motion.
- Triangulation is used in navigation and geography to determine location and distance, and to calculate latitude and longitude.
- Triangles are used in computer graphics and game development for 3D transformations and projections, animation, and simulation.
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Description
Explore the fundamentals of triangles, including types of triangles, properties, and the Pythagorean theorem. Learn about different types of angles, measurement units, and conversions.
