Trig Alternative Assessment

Verify

100

VERIFY: csc(θ)=(1/2)csc(θ)sec(θ)

csc(2θ)=(1/2)csc(θ)sec(θ)

> (1/sin(2θ))=(1/2)(1/sin(θ)cos(θ))

>> (1/sin(2θ))=(1/2sin(θ)cos(θ))

>>> (1/sin(2θ))=(1/sin(2θ))

VERIFIED!!!

200

True or False?

(cscθ+cotθ)(cscθ-cotθ)=1

SHOW YOUR WORK

True!

(cscθ+cotθ)(cscθ-cotθ)=1

> csc²θ+cscθcotθ-cscθcotθ-cot²θ=1

>> csc²θ-cot²θ=1

>>> 1=1

300

VERIFY: tan(x/2)=sin+((1-sin²x)/sinx)-cotx

(sinx/sinx)sinx+((1-sin²x)/sinx)-cotx

sin²x/sinx+((sin²x+cos²x-sin²x)/sinx)-cosx/sinx

(sin²x+cos²x-cosx)/sinx

(1-cosx)/sinx=tan(x/2)

VERIFIED!!

400

                              tanx(3-tan²x)

VERIFY: tan(3x)=   ____________

                               (1-3tan²x)

{solving  tan(3x) }

=(tan2x+tanx)/[1-(tan(2x)tan(x))]

>={(2tanx)/(1-tan²x)+(tanx)}/{1-(tanx)(2tanx/1-tan²x)}

>={(2tanx)/(1-tan²x)+(tanx(1-tan²x)/(1-tan²x))}/{(1-tan²x/1-tan²x)-((tanx2tanx)/1-tan²x)}

>=(2tanx+tanx-tan³x)/(1-tan²x-2tan²x)

>=(3tanx-tan³x)/(1-3tan²x)

 tanx(3-tan²x)         tanx(3-tan²x)

____________  =  ____________ 

   (1-3tan²x)             (1-3tan²x)

VERIFIED!!!

500

VERIFY: cosx-sinx=(cos²x-sin²x)/cosx+sinx

{cosx-sinx=(cos²x-sin²x)/cosx+sinx}

= cosx-sinx=(cos(2x))/cosx+sinx

>= (cosx-sinx)*(cosx+sinx)={(cos(2x))/cosx+sinx}*(cosx+sinx)

>= cos²x-sin²x=cos(2x)

>= cos²x-sin²x=cos²x-sin²x

VERIFIED!!!

General Solution

100

tan(x)=1

x=pi/4+npi

x=5pi/4+npi

200

cos(2x)=1/2

x=pi/12+npi

x=5pi/6+npi

300

sin(x/2)=sqrt2/2

x=pi/2+4npi
x=3pi/2+4npi

400

cot²(3x)= 3

x=pi/18+2/3npi

x=5pi/18+2/3npi

x=7pi/18+2/3npi

x=11pi/18+2/3npi

500

sin(1/3x-pi/6)=1

x=9pi/2+6npi

Specific Solution

100

sin2θ=1 on the interval of [0, 2pi]

pi/4

3pi/4

5pi/4

7pi/4

200

cos4θ=-1 on the interval of [0, pi]

3pi/16

7pi/16

11pi/16

15pi/13

300

sin(4θ-pi/2)=-1 on the interval of [0, 2pi]

0

pi/2

pi

3pi/2

2pi

400

sin2θsinθ=cosθ on the interval of [0, 2pi]

pi/4

2pi/4

3pi/4

5pi/4

6pi//4

7pi/4

500

csc²2θ-4=0 on the interval of [0,2pi]

pi/12

5pi/12

7pi/12

11pi/12

13pi/12

17pi/12

19pi/12

23pi/12

eval. values of trig func.

100

If sinθ=7/25 and π/2<θ<π, what is cosθ ?

-24

___

 25

200

If sinθ =-7/25 and π/2<θ<π, what is tan2θ ?

336

___

527

300

cosβ=80/89  , 3π/2<β<2π
sinθ=7/25     , π/2<θ<π

What is cos^2(β+θ)  ?

141376/4950625

400

tanα=-8/15 , π/2<α<π

cos(4α)  ?

-31679/83521

500

tanα=-65/72 , π/2<α<π
cosβ=80/89  , 3π/2<β<2π
sinθ=7/25     , π/2<θ<π

What is {tan(α-β+θ)}{sin(α+θ)}?

-1972282781850878813122296
__________________________
2235760260012388477485000

evaluate sum and diff

100

cos(θ)cos(a)+sin(θ)sin(a)

cos(θ-a)

200

sin(a-θ)

sin(a)cos(θ)-cos(a)sin(θ)

300

tan(θ+a)

tan(θ)+tan(a)

____________

1-tan(θ)tan(a)

400

tan{ (pi/6) + (3pi/2) }

1-{^sqrt(-3)}

_________

1+{^sqrt(-3)}

500

cos(465^o)

{^sqrt(2)} - {^sqrt(6)}

_______________

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