12-04: Trigonometry

Radian Measure

Radian: measure of an angle formed by rotating the radius of the circle through an arc length equal to the radius

It is a unit of measurement - rads for short

 

Radian measure of angle Ď´ is defined as:

 

If you complete 1 full revolution then:

 

  • 180Âş = Ď€ rads
  • you can use fractions of semi circles to find other values - what you do to one side you must do to the other

Official Conversion Between Degrees and Radians

 

  • Exact: leave Ď€ and fractions
  • Approximate: decimal value

Angular Velocity

  • The angular velocity of a rotating object us the rate at which the central angle changes with respect to time
  • It is a rate of change → about how much the angle changes
  • RPM: Revolutions per minute → revolutions divided by minutes
    • “Per” means division

  Use Factor Label Method Process

e.g. The hard disk of a personal computer rotates at 7200 RPM. Determine the angular velocity in degrees per second.

 

Special Triangles

 

  • Special angles have a denominator of 4, 6, or 3 in radians

Unit Circle: Radius of 1

 Unit circle shows the ratios of the angles: cos is the x value and sin is the y value

 Proof of the Unit Circle - How to find Sin and Cos of an angle

To Use The Unit Circle To Evaluate Trig Ratios For Special Angles

CosĎ´x values
SinĎ´y values
TanĎ´SinĎ´/CosĎ´ = y/x

Graph for Non-special Angles (Benchmarks)

 

  • Can be used with Ď€/2, Ď€, 3Ď€/2, and 2Ď€
  • x value is cos, y value is sin

e.g. with special triangles

 

e.g. with non-special angles - note that this only can be used with the benchmarks that are labelled below

 

Equivalent Trigonometric Expressions

Equivalent expressions: expressions that yield the same value for all values of the variable

Rule of “co”

Sine

Secant

Tangent

Steps for Determining Equivalent Trig Expressions Using Cofunction Identities

  1. Is the angle in Q1 or Q2
  2. Find the CO related angle
  3. Rule of CO for ratio (CAST rule in Q2, so only sine/cosecant are positive)

 

Compound Angle Formulas

Compound angle expression: Trig expression that depends on 2 or more angles

cos(x-y)= cosxcosy + sinxsiny
cos(x+y)= cosxcosy - sinxsiny
sin(x-y)= sinxcosy - cosxsiny
sin(x+y) = sinxcosy + cosxcosy

Trig Identities

Trig identities: both LS and RS should be equal

“Prove” tells you to do trig identities

  • Split LS and RS
  • No rearranging
  • No skipping steps
  • Replace everything with sin and cos
  • Simplify each side

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