6.6 Trig Equations & Inequalities in Quadratic Form

– Quadratics can be solved by factoring or using the quadratic formula– Now “x” will be a trig function we must deal with & solve for at the end– Note: To solve, it should have the same trig functions

Find exact solutions if possible, and if not, round to nearest hundredth in radians.

Ex 1) Solve 2

12

7 116 6 2

2sin sin 1 0

(2sin 1)(sin 1) 0

2sin 1 0 sin 1 0

sin sin 1

,

x x

x x

x x

x x

x x

but… there were no restrictions so we have to consider all answers

7 112 6 62 , 2 , 2 ;n n n n ¢

Ex 2) Solve

2

2

2

2

2

12

7 11 36 6 2

7 3 116 2 6

2cos 3sin 3 0 , where 0 2

2(1 sin ) 3sin 3 0

2 2sin 3sin 3 0

2sin 3sin 1 0

2sin 3sin 1 0

(2sin 1)(sin 1) 0

2sin 1 0 sin 1 0

sin sin 1

,

, ,

x x x

x x

x x

x x

x x

x x

x x

x x

x x

Ex 3) Solve

2

2

12

2 tan 2cot 3 , where 0 2

2 tan 2cot 3 0

2 tan 2 3tan 0

2 tan 3tan 2 0

(2 tan 1)(tan 2) 0

2 tan 1 0 tan 2 0

tan tan 2

tan ( )

x x x

x x

x x

x x

x x

x

x

x

x

x

(not famous angles)

x = 0.46or 0.46 + π = 3.60

x = –1.11–1.11 + 2πor –1.11 + π

= 5.17= 2.03

0.46, 2.03, 3.60, 5.17

Ex 4) Solve 2

2

sin sin 1 0 , for all values of

1 1 4(1)(1) 1 1 4 1 3sin

2(1) 2 2

x x x

x

can’t factor it, so use quadratic formulaa = 1, b = 1, c = 1

only imaginary answers so

No Solution

Ex 5) Use a graphing calculator to find the solution of on [0, 2π). Sketch the graph obtained by

your calculator

Set equation ≥ 0

22sin cos 1x x

Let’s put the calc in Degree mode (then we’ll change our answer to rads)1) Press Mode 2) Arrow down to 3rd line (Radian / Degree)3) Arrow right to put cursor over Degree & hit enter

Put in graphing calculator by4) Press Y = button5) On line Y1, enter: 2sin(x)2 – cos(x) – 1 6) Press Zoom button7) Press 7: ZTrig for Zoom Trig

make sure to put in the )

22sin cos 1x x

22sin cos 1 0x x

Ex 5) cont… 22sin cos 1 0x x

We want the x-values from 0 to 2π for when the graph is above the x-axis

(that’s when it’s ≥ 0)

Now press the 2nd Trace button to get CALCPress 2: zeroUse right arrow to move cursor over to left of hereUse right arrow to move cursor over to right of there & hit ENTERHit ENTER again

Do the same thing for here

So the graph is above the x-axis between 60° and 300°Change to radians!!

5

3 3x

Sketch the graph & label!

3

5

3

Calc says X = 60 Y = 0

Calc says X = 300 Y = 0

& Hit ENTER

Homework

#605 Pg 321 #1–13 odd, 17, 19, 23, 25, 27, 30, 39Sketch the graph you get from your calc for #17, 19, 39

Even Answers:#30: 7 11

6 63.34 2 , 6.08 2 , 2 , 2 ,n n n n n ¢