Angles and Their Types

Angles

• An angle is formed by two rays or lines that share a common endpoint.
• The rays are called the sides of the angle.
• The initial side is the starting position of the ray.
• The terminal side is the position of the ray after rotation.
• A standard angle has its vertex at the origin and its initial side coincides with the x-axis.
• The measure of an angle is the amount of rotation needed to move from the initial side to the terminal side.
• Angles can be measured in degrees or radians:
  • 360° = 2π radians (one full circle)
  • π radians = 180°
• Types of angles:
  • Acute angles are between 0° and 90°.
  • Right angles are equal to 90°.
  • Obtuse angles are between 90° and 180°.
  • Straight angles are equal to 180°.
  • Reflex angles are greater than 180° and less than 360°.
• Coterminal angles share the same initial and terminal sides.
• A reference angle is the acute angle formed between the terminal side and the x-axis.
• To convert from radians to degrees, multiply by $\frac{180}{\pi}$.
• To convert from degrees to radians, multiply by $\frac{\pi}{180}$.

Circular Functions

• Trigonometric functions relate the angles of a right triangle to the ratios of its sides.
• SOH CAH TOA is a mnemonic to remember the trigonometric functions:
  • Sine = Opposite / Hypotenuse
  • Cosine = Adjacent / Hypotenuse
  • Tangent = Opposite / Adjacent
• The reciprocal trigonometric functions are:
  • Cosecant (csc θ) = 1 / sin θ
  • Secant (sec θ) = 1 / cos θ
  • Cotangent (cot θ) = 1 / tan θ
• Pythagorean Identities: Fundamental relationships between trigonometric functions:
  • sin² θ + cos² θ = 1
  • 1 + tan² θ = sec² θ
  • 1 + cot² θ = csc² θ
• Quotient Identities: Express tangent and cotangent in terms of sine and cosine:
  • tan θ = sin θ / cos θ
  • cot θ = cos θ / sin θ

Evaluating Trigonometric Functions

• To evaluate trigonometric functions when given a point on the terminal side of an angle:
  • Determine the value of r (the hypotenuse) using the Pythagorean theorem: r² = x² + y²
  • Apply the SOH CAH TOA definitions to find the values of the trigonometric functions.
• For the point P(-5, 12):
  • r = √((-5)² + (12)²) = 13
  • sin θ = y/r = 12/13
  • cos θ = x/r = -5/13
  • tan θ = y/x = -12/5
  • csc θ = r/y = 13/12
  • sec θ = r/x = -13/5
  • cot θ = x/y = -5/12

Trigonometric Identities

• An identity equation is true for all possible values of the variable (domain).
• A conditional equation is true for specific values of the variable.
• A contradiction equation is never true for any value of the variable.
• Fundamental Trigonometric Identities:
  • Reciprocal Identities: connect a function with its reciprocal, like sin θ and csc θ.
  • Quotient Identities: show how to express tangent and cotangent in terms of sine and cosine.
  • Pythagorean Identities: show a relationship between the squares of sine and cosine functions.