trigonometry

• Trigonometry


  A branch of mathematics that focuses on relationships between the sides and angles of triangles

  
• Trigonometry is an intricate piece of other branches of mathematics such as, Geometry, Algebra, and Calculus
• Topics covered in this tutorial

  • Understand how angles are measured
  • Use trig functions to find information about right triangles
  • Use definitions and fundamental Identities of trig functions
  • Understand key features of graphs of trig functions

  
• Degrees


  A circle is comprised of 360°, which is called one revolution
• Radians


  1 revolution measured in radians is 2π, where π is the constant approximately 3.14
• Converting between degrees and radians

  1. 1° = π/180 radians

  2. 1 radian = 180/π degrees

  
• Unit Circle

  • A circle that is centered at the origin and always has a radius of 1
  • The equation of the Unit Circle is x² + y² = 1

  
• Trigonometric ratios

  Sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot)

  
• Sine (sin θ)


  Opposite side / Hypotenuse
• Cosine (cos θ)


  Adjacent side / Hypotenuse
• Tangent (tan θ)


  Opposite side / Adjacent side
• Cosecant (csc θ)


  Reciprocal of sine
• Secant (sec θ)


  Reciprocal of cosine
• Cotangent (cot θ)


  Reciprocal of tangent
• Finding trigonometric ratios given a right triangle

  Use Pythagorean Theorem to find the hypotenuse

  2. Plug the side lengths into the trigonometric ratio formulas

  
• Example 5


  • Find the six trig ratios for the given right triangle

  
• Trigonometric ratios


  The six trigonometric ratios are: sine, cosine, tangent, cosecant, secant, and cotangent

  
• Finding trigonometric ratios


  1. Use Pythagorean Theorem to find the hypotenuse

  2. Use the definitions of the trigonometric ratios to find the other sides

  
• As temperature increases


  The rate of reaction increases
• Total utility is the sum of marginal utilities
• Pythagorean Theorem


  r = √(x² + y²), where r is the hypotenuse and x, y are the legs of a right triangle
• Finding trigonometric ratios using Pythagorean Theorem


  1. Use Pythagorean Theorem to find hypotenuse

  2. Plug hypotenuse and given sides into definitions to find other ratios

  
• Special right triangles


  30-60-90 and 45-45-90 triangles have known side length ratios

  
• Finding trigonometric ratios using special right triangles

  1. Identify if the angle is 30°, 45°, or 60°

  2. Use the known side length ratios to calculate the other sides and trigonometric ratios

  
• Using a calculator to find trigonometric ratios

  Use inverse trig functions (sin^-1, cos^-1, tan^-1) to find angle measures given a ratio

  
• Finding missing side lengths using trigonometric ratios

  1. Identify the known ratio and side length

  2. Rearrange the ratio equation to solve for the missing side

  
• Finding angle measures using inverse trigonometric functions

  1. Identify the known trigonometric ratio

  2. Use the inverse function button on the calculator to find the angle

  
• Fundamental trigonometric identities


  • Reciprocal identities
  • Quotient identities
  • Pythagorean identities
  • Negative angle identities
  • Complementary angle theorem
• Using fundamental trigonometric identities

  1. Identify the relevant identity

  2. Substitute known values and simplify
• Sum and difference formulas

  • Sine and cosine
  • Tangent
• Using sum and difference formulas

  1. Identify the relevant formula

  2. Substitute known angle measures

  3. Simplify the expression
• Double and half angle formulas

  • Double angle formulas
  • Half angle formulas

  
• Using double and half angle formulas

  1. Identify the relevant formula

  2. Substitute known angle measure

  3. Simplify the expression

  
• Product to sum and sum to product formulas

  • Product to sum
  • Sum to product
• Using product to sum and sum to product formulas

  1. Identify the relevant formula

  2. Substitute known angle measures

  3. Simplify the expression
• Law of Sines


  Relates the sides and angles of a triangle

  
• Law of Cosines


  Relates the sides and angles of a triangle

  
• Product to Sum Formulas

  Formulas to convert products of trigonometric functions to sums
• Sum to Product Formulas

  Formulas to convert sums of trigonometric functions to products
• Law of Sines


  Relationship between the sides and angles of an oblique triangle