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 [latex]\frac{1}{360}[/latex] of a full circular rotation
    • Full circle = [latex]360°[/latex]
    • Symbol: [latex]°[/latex] (e.g., [latex]45°[/latex])
  • Radians:
    • Based on the radius of a circle
    • One radian is the angle subtended by an arc length equal to the radius
    • Full circle = [latex]2π[/latex] radians
    • Symbol: rad (e.g., [latex]\frac{π}{2}[/latex] rad)
  • Conversion:
    • [latex]π[/latex] radians [latex]= 180°[/latex]
    • [latex]1[/latex] radian [latex]≈ 57.3°[/latex]
    • Conversion formulas:
      • Degrees to Radians: [latex]\text{Angle in Degrees} = \text{Angle in Radians }\times\frac{180}{π}[/latex]
      • Radians to Degrees: [latex]\text{Angle in Radians} = \text{Angle in Degrees }\times\frac{π}{180}[/latex]
  • Usage:
    • Degrees: common in everyday situations, navigation, construction
    • Radians: preferred in advanced mathematics, physics, engineering
  1. Express [latex]210°[/latex] using radians.
  2. Express [latex]\dfrac{11\pi}{6}[/latex] rad using degrees.


The Six Basic Trigonometric Functions

The Main Idea 

  • The Six Functions:
    • Sine ([latex]\sin[/latex]): [latex]\frac{\text{opposite}}{\text{hypotenuse}}[/latex]
    • Cosine ([latex]\cos[/latex]): [latex]\frac{\text{adjacent}}{\text{hypotenuse}}[/latex]
    • Tangent ([latex]\tan[/latex]): [latex]\frac{\text{opposite}}{\text{adjacent}}[/latex]
    • Cosecant ([latex]\csc[/latex]): [latex]\frac{\text{hypotenuse}}{\text{opposite}}[/latex] (reciprocal of sine)
    • Secant ([latex]\sec[/latex]): [latex]\frac{\text{hypotenuse}}{\text{adjacent}}[/latex] (reciprocal of cosine)
    • Cotangent ([latex]\cot[/latex]): [latex]\frac{\text{adjacent}}{\text{opposite}}[/latex] (reciprocal of tangent)
  • Mnemonic Device:
    • SOH-CAH-TOA for sine, cosine, and tangent
  • Reciprocal Relationships:
    • [latex]\sin{\theta} = \frac{1}{\csc{\theta}}[/latex]
    • [latex]\cos{\theta} = \frac{1}{\sec{\theta}}[/latex]
    • [latex]\tan{\theta} = \frac{1}{\cot{\theta}}[/latex]
  • Applications:
    • Finding unknown side lengths in right triangles
    • Solving real-world problems involving angles and distances
Using the triangle shown below ,evaluate [latex]\sin t[/latex],[latex]\cos t[/latex],[latex]\tan t[/latex],[latex]\sec t[/latex],[latex]\csc t[/latex],and [latex]\cot t[/latex].
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 [latex]20[/latex]-ft ladder against a house. If the angle between the base of the ladder and the ground is to be [latex]60^{\circ}[/latex], 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., [latex]\csc \theta =\large \frac{1}{\sin \theta}[/latex])
    • Pythagorean Identities (e.g., [latex]\sin^2 \theta +\cos^2 \theta =1[/latex])
    • 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 [latex]1+\cot^2 \theta =\csc^2 \theta[/latex].

Simplify the expression by rewriting and using identities:

[latex]\csc^2{\theta}-\cot^2{\theta}[/latex]

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 [latex]2π[/latex]
    • Tangent, cotangent: period of [latex]π[/latex]
  • 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: [latex]f(x) = A \sin{(B(x - α))} + C[/latex](similar for cosine)
    • [latex]A[/latex]: Amplitude (vertical stretch/compression)
    • [latex]B[/latex]: Frequency (horizontal stretch/compression, affects period)
    • [latex]α[/latex]: Phase shift (horizontal shift)
    • [latex]C[/latex]: Vertical shift
  • Effects of Transformations:
    • Amplitude: [latex]|A|[/latex] is the height from midline to peak/trough
    • Period: [latex]d\frac{2 \pi}{|B|}[/latex] for sine and cosine
    • Phase Shift: Right by [latex]α[/latex] if positive, left if negative
    • Vertical Shift: Up by [latex]C[/latex] if positive, down if negative

Describe and sketch the graph of [latex]f(x) = 2 \cos{(\frac{1}{2}(x + \frac{π}{3}))} - 1[/latex]