General Solutions of Trigonometric Equations 2

Hey everyone, my name is Angeleen Antazo.

I apologize for not being able to do my homework last Thursday due to some technical difficulties. We continued the lesson about General Solutions of Trigonometric Equations 2.

Solving the equation over the domain of all real numbers.

Example: sin3θ= -√2/2

· - √2/2 = - 1/√2

· sin θ is O/H = - 1 /√2

· Then you have to find what angle it is by looking back to 30°, 45° or 60° and look to see which radian value it matches.

· In this case it is 45° or π/4.

· Once you’ve figured that out, you have find where sin θ is negative by using the CAST Rule.

· sin θ is negative is quadrant III and IV.

- QIII: π + π/4 = 5π/4

- QIV: 2π – π/4 = 7π/4

· θ = 5π/4 + 2Kπ , 7π/4 + 2Kπ

· But since it’s asking for 3θ you would either multiply or divide them by 3.

· 3θ= 5π/4x3 + 2Kπ/3, 7π/4x3 + 2Kπ/3

θ = 3π/4 + 2Kπ/3, 7π/ 12 + 2Kπ/3 where kϵI

Solving for θ over the interval 0≤ θ ≤ 2π

Example: sin θ/2 = 0

· First, you have to find where sinθ = 0

· In this case, sinθ = 0 in 0 , π and 2π (0°, 90°, 360°)

· sin θ/2 (you have to multiply the answers by 2)

θ = 2 x0 , 2 x π , 2 x 2π

θ = 0 , 2π, 4π

· Since the interval is 0≤ θ ≤ 2π , 4π must be rejected because it falls outside the interval given.

θ = 0 , 2π

Homework: Exercise 5, Questions 2, 9-12

Trigonometric Worksheet

Graphing Circular Functions

On the next day, which was on Friday, we learned about Graphing Circular Functions.

· Graphing on a Cartesian Plane is known as “unrolling” the Unit Circle.

· We will plot θ values on the x- axis and the trigonometric function values at θ on the

· y- axis

· One cycle is a potion of the graph from one point to another at which time the graph begins to repeat itself.

· One period is the light of one cycle in the either degrees or radians. The period for a

Sinx or cosx function is 2π/|b| . The period for a tanx function is π/|b|

· The amplitude is the distance from the middle axis to the highest or lowest point for a sin x or cos x function. A change in amplitude vertically stretches or compresses the basic shape of the curve. The amplitude for sin x or cos x function is |a| . The amplitude for a tan x function is unlimited or infinite.

· Basic equations: y= a sin bx y= a cos bx y= a tan bx

· x represents θ values and y represents the trigonometric function’s value of θ .

· use quadrantal values for x when graphing sin x or cos x.

· Use quadrantal and π/4 values for x when graphing tan x.

· The tan x function will have asymptotes at quadrantal values where tan x is undefined.

Example: y=sin x

- y=a sin b

- Period= 2π/|b| = 2π/|1| = 2π

- Amplitude: a=1

- Domain= (-∞,∞)

- Range= [-1,-1]

- x-intercept(s): x= K π where kϵI

x	y
0	sin= o
π/2	sinπ/2= 1
π	sinπ = 0
3π/2	sin3π/2= -1
2π	sin2 π= 0

Example: y=tan x

- Period= π/|b|= π

- Amplitude= Undefined

- Domain= xϵR, x≠ π/2 + k π

- Range= (-∞, ∞)

- x- intercept(s): x= k π

x	y
0	tan0= 0
π/4	tanπ/4= 1
π/2	tanπ/2=undefined
3π/4	tan3π/4= -1
π	tanπ= 0
5π/4	tan5π/4= 1
3π/2	tan3π/2=undefined
7π/4	tan7 π/4= -1
2π	tan2 π= 0