10 Questions
1 Views

# Geometry: Triangles and Angles

Created by
@AmenableCoconutTree

One angle is 90°

True

180°

### The __ theorem states that a² + b² = c² for right triangles.

<p>Pythagorean</p> Signup and view all the answers

### Match the angle type with its definition:

<p>Acute = 0° &lt; θ &lt; 90° Right = θ = 90° Obtuse = 90° &lt; θ &lt; 180° Straight = θ = 180°</p> Signup and view all the answers

### What is the period for periodic functions like sine, cosine, and tangent?

<p>2π</p> Signup and view all the answers

### Define amplitude in the context of waves.

<p>maximum displacement from mean position</p> Signup and view all the answers

### Which identity states that sin²(x) + cos²(x) = 1?

<p>Pythagorean identity</p> Signup and view all the answers

### What is the formula for sin(2x)?

<p>2sin(x)cos(x)</p> Signup and view all the answers

### Give an example of an application area where trigonometry is used.

<p>Physics and engineering</p> Signup and view all the answers

## 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

• 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
• 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.

## Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

## 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.

## More Quizzes Like This

Use Quizgecko on...
Browser
Information:
Success:
Error: