Trig Equations using Identities

Trig. Identities

100

If cos x = (2/3) find sec x

What is sec x = (1/cos x)

=(1/(2/3))

=(3/2)

200

If tan a = (1/5) find cot a

What is cot a = (1/tan a )

=(1/(1/5))

=(5/1)

=5

300

if cos x = (1/6) and sin x = ((√35)/6) find cot x

What is cot x = ((cos x)/(sin x))

=((1/6)/(√35)/6)

=(1/(√35)) * ((√35)/(√35))

=((√35)/35)

400

Simplify:

csc x - cos x cot x

What is csc x-cosx cot x = (1/sinx)-cosx(cosx/sinx)

=(1/sin x)-((cos²x)/(sinx))

=((1-cos²x)/(sinx))

=((sin²x)/(sinx))

=sinx

500

Simplify:

csc x sec x - tan x

What is csc x sec x - tan x = (1/sinx)*(1/sinx)-(sinx/cosx)

=(1/(sinxcosx))-(sinx/cosx)

=(1/(sinxcosx))-(sinx/sinx)(sinx/cosx)

=((1-sin²x)/(sinxcosx)

=((cos²x)/(sinxcosx)

=(cosx/sinx)

=cotx

Verifying/Simplifying Trig. Identities

100

(sec²x-1)cos²x = sin²x

What is (sec²x-1)cos²x

=(tan²x)cos²x    Pythagorean Identity

=((sin²x)/(cos²x))cos²x  Quotient Identity

=sin²x   Multiply and divide out common factor

200

sec²x(1-cos²x) = tan²x

What is sec²x(1-cos²x)

=sec²x-sec²xcos²x   Distributive Property

=sec²x-(1/(cos²x))*cos²x Reciprocal Identity

=sec²x-1 Multiply and divide out common factor

=tan²x Pythagorean Identity

300

sinx-sinx cos²x=sin³x

What is sinx-sinx cos²x

=sinx(1-cos²x)  Factor

=sinxsin²x  Pythagorean Identity

=sin³x Multiply

400

cot²x csc²x-cot²x=cot⁴x

What is cot²xcsc²x-cot²x

=cot²(csc²x-1)  Factor

=cot²xcot²x  Pythagorean Identity

=cot⁴x Multiply and add exponents 

500

tanx csc²x-tanx = cot x

What is tanx csc²x-tanx

=tanx(csc²x-1) Factor

=tanx cot²x Pythagorean Identity

=((sinx)/(cosx))*((cos²x)/(sin²x)) Quotient Identities

=(cosx)/(sinx)  Multiply and divide common

=cot x quotient identity

Trig Equations

100

Find the solutions on the unit circle in degrees:

4=-8cos(x)

(2pi)/3, (4pi)/3 + 2pin

200

Find the solutions on the unit circle in radian:

5=1+2csc(x)

pi/6, (5pi)/6 + 2pin

300

Find the solutions on the unit circle in radians:

-4=-2+2tan(x)

(3pi)/4 + pin

400

Find the solutions on the unit circle in radians:

-csc(x)+cot(x)=-1

pi/2 +2pin

500

Find the solutions on the unit circle in degrees:

0=-2cot(x)-csc²(x)

(3pi)/4 + pin

Solving Trig. Equations

100

5sinx+2=sinx [0,2pi]

What is 5 sinx+2=sinx

=4sinx+2=0

=2(2sinx+1)=0

=2sinx=-1

= sinx=(1/2)  

((7pi)/6) and ((11pi)/6)

. 

200

2-10secx=4-9secx [0, 2pi]

What is 2-10secx=4-9secx

=-10secx=2-9secx

=-secx=2

=secx=-2  = cosx=-(1/2)

((2pi)/3) and ((4pi)/3)

300

5=sec²x+3 [0,2pi]

What is 5=sec²x+3

= 2=sec²x

=(+/-)(√2)=(√sec²x)

= secx=(+/-)(√2)

= cosx=(+/-)(1/(√2))*((√2)/(√2))

= cosx=(+/-)((√2)/2)

=

(pi/4),(3pi/4),(5pi/4),(7pi/4)

400

3cscx=2cscx+(√2) [0,2pi]

What is 3cscx=2cscx+(√2)

= cscx=(√2)

= sinx=(1/(√2))*((√2)/(√2))

= sinx=(√2)/2)

=

(pi/4), ((3pi)/4)

500

11=3csc²x+7

What is 11=3csc²x+7

= 4=3csc²x

=(4/3) = csc²x

= (+/-)((√(4/3))=(√csc²x)

=cscx = (+/-)(2/(√3))or(+/-)(2(√3)/3)

= sinx=(+/-)(√3/2)

= 

(pi/3), ((2pi)/3), ((4pi)/3), ((5pi)/3)

Random -GenZ Trivia

100

What is Eleven's real name in Stranger Things?

Millie Bobby Brown. 

200

What is the modern term for ending all communication without explanation?

Ghosting.  

300

Who is the Gen Z singer known for hits like “bad guy” and “Happier Than Ever”?

Billie Eilish

400

What year did the first iPhone release, marking the beginning of the smartphone era for Gen Z?

2007

500

GenZ's are scientifically known as the biggest_______

Crybabies