Trigonometric Functions: Fresh Take

  • Convert angle measures between degrees and radians
  • Interpret and apply basic trigonometric definitions and identities
  • Analyze trigonometric functions by identifying graphs and periods, and describing shifts in sine or cosine graphs.

Degrees versus Radians

The Main Idea 

  • Degrees:
    • One degree is 1360 of a full circular rotation
    • Full circle = 360°
    • Symbol: ° (e.g., 45°)
  • Radians:
    • Based on the radius of a circle
    • One radian is the angle subtended by an arc length equal to the radius
    • Full circle = 2π radians
    • Symbol: rad (e.g., π2 rad)
  • Conversion:
    • π radians =180°
    • 1 radian 57.3°
    • Conversion formulas:
      • Degrees to Radians: Angle in Degrees=Angle in Radians ×180π
      • Radians to Degrees: Angle in Radians=Angle in Degrees ×π180
  • Usage:
    • Degrees: common in everyday situations, navigation, construction
    • Radians: preferred in advanced mathematics, physics, engineering
  1. Express 210° using radians.
  2. Express 11π6 rad using degrees.


The Six Basic Trigonometric Functions

The Main Idea 

  • The Six Functions:
    • Sine (sin): oppositehypotenuse
    • Cosine (cos): adjacenthypotenuse
    • Tangent (tan): oppositeadjacent
    • Cosecant (csc): hypotenuseopposite (reciprocal of sine)
    • Secant (sec): hypotenuseadjacent (reciprocal of cosine)
    • Cotangent (cot): adjacentopposite (reciprocal of tangent)
  • Mnemonic Device:
    • SOH-CAH-TOA for sine, cosine, and tangent
  • Reciprocal Relationships:
    • sinθ=1cscθ
    • cosθ=1secθ
    • tanθ=1cotθ
  • Applications:
    • Finding unknown side lengths in right triangles
    • Solving real-world problems involving angles and distances
Using the triangle shown below ,evaluate sint,cost,tant,sect,csct,and cott.
Right triangle with sides 33, 56, and 65. Angle t is also labeled which is opposite to the side labeled 33.

A house painter wants to lean a 20-ft ladder against a house. If the angle between the base of the ladder and the ground is to be 60, how far from the house should she place the base of the ladder?

Trigonometric Identities

The Main Idea 

  • Trigonometric identities are equations involving trigonometric functions that are true for all values where they are defined.
  • Types of Identities:
    • Reciprocal Identities (e.g., cscθ=1sinθ)
    • Pythagorean Identities (e.g., sin2θ+cos2θ=1)
    • Addition and Subtraction Formulas
    • Double-Angle Formulas
  • Importance:
    • Crucial for solving trigonometric equations
    • Used in proving other mathematical statements
    • Frequently applied in calculus and higher mathematics
  • Verification Methods:
    • Manipulate one side of the equation to match the other
    • Use known identities to make substitutions
    • Convert all terms to sines and cosines if needed

Prove the trigonometric identity 1+cot2θ=csc2θ.

Simplify the expression by rewriting and using identities:

csc2θcot2θ

Graphs and Periods of the Trigonometric Functions

The Main Idea 

  • Periodicity:
    • Trigonometric functions repeat their values at regular intervals
    • Sine, cosine, secant, cosecant: period of 2π
    • Tangent, cotangent: period of π
  • Basic Graphs:
    • Sine and cosine: smooth wave patterns
    • Tangent and cotangent: repeating vertical asymptotes
    • Secant and cosecant: repeating reciprocal patterns of cosine and sine
  • Transformations: General form: f(x)=Asin(B(xα))+C(similar for cosine)
    • A: Amplitude (vertical stretch/compression)
    • B: Frequency (horizontal stretch/compression, affects period)
    • α: Phase shift (horizontal shift)
    • C: Vertical shift
  • Effects of Transformations:
    • Amplitude: |A| is the height from midline to peak/trough
    • Period: d2π|B| for sine and cosine
    • Phase Shift: Right by α if positive, left if negative
    • Vertical Shift: Up by C if positive, down if negative

Describe and sketch the graph of f(x)=2cos(12(x+π3))1