12-05: Trigonometric Equations

Periodic function

a function where the y values repeat over regular x value intervals

(x,y) → (1/k x

p, ay + q)

min = q

|a|

Calculator

sin⁻¹( ), cos⁻¹( ), tan⁻¹( )

Rearrange

isolate for trig ratios

Factor

exponents, same ratio repeats

Identities

multiple ratios

In all cases

must consider CAST rule

periodic function

a function where the y values repeat over regular x value intervals

period

the duration of one cycle

negative a value

r in x

a value

vertical stretch: |a| > 1; vertical compression: 0 < |a| < 1 --> |a| is the amplitude, |a| = (max - min)/2

k value

horizontal stretch: |k| >1; horizontal compression: 0 < |k| < 1 --> period = 2π/k --> k = 2π/period

p value

horizontal shift --> + left, - right --> moves starting point

q value

vertical shift --> + up, - down --> q = average, q= (max + min)/2

graphing using properties

• y-scale:

avg = q

max = q + |a|

min = q - |a|

• x-scale:

period = 2π / k

QI (quarter intervals) = Period/4

avg (Sin), max (Cos), avg, min, (and so on)

solving trig equations

1- Rearrange - isolate for trig ratios

2 - Factor - exponents, same ratio repeats

3 - Identities - multiple ratios

In all cases: must consider CAST rule

AROC

slope of secant of curve

IROC

slope of tangent line ( + & - 0.001)