Trigonometric Identities and Equations

Inverse Trig

100

What is the domain and range of tan^-1x?

domain= -infinity to infinity

range: -𝛑/2 to 𝛑/2

200

What is the domain and range of inverse cosine

Domain [-1, 1]

Range [0, 𝛑]

300

What is the domain and range of sin^-1x?

domain: [-1, 1]

Range: [-𝛑/2, 𝛑/2]

400

What is the cos (sin^-1(7/25))

24/25

500

What is the cos^-1(cos(5𝛑/4))?

3𝛑/4

Solving Trig Equations

100

When you have infinite solutions, what is the general form you would write them in?

solution +2k𝛑  for k=0, 1, 2, -1, -2 etc.

(sometimes is k𝛑 instead of 2k𝛑)

200

Between 0<x<2𝛑 find solution to 

cscx-sqrt2=0

𝛑/4, 3𝛑/4

300

Solve each equation for exact solution 0<x<=2𝛑.

cos 2x = 1 - 3sinx

0, 𝛑, 2𝛑

400

What is the reference angle for the angle 

theta=310^o

50^o

500

What are all possible solutions for 

sin2-sinx=0

0+2kpi,pi/3+2kpi, pi+2kpi,(5pi)/3+2kpi

Sum and Difference Identities

100

What is 

sin (alpha +- beta)

sin alpha cos beta +-cos alpha sin beta

200

what is 

cos alpha +beta

cos alpha cos beta-sin alpha sin beta

300

What is the

tan alpha-beta

(tan alpha-tanbeta)/(1+tan alpha tan beta

400

Write in terms of a single trigonometric function

cosxcos2x+sinxsin2x

cos(x-2x) or cos(-x)

500

What is 

cos(beta-alpha)

cos beta cos alpha+ sin beta sin alpha

Double Angle Identities

100

What is the double angle identity for 

sin (2alpha)

2 sin alpha cos alpha

200

What are the double angle identities for 

cos (2alpha)

cos^2alpha-sin^2 alpha

2cos^2alpha-1

1-2sin^2alpha

300

Solve for x: 

0<x<=2pi

4 sinxcosx = -1

(7pi)/12, (11pi)/12, (19pi)/12, (23pi)/12

400

Prove:

(1-cos2theta)/2 = sin^2theta

Yep! Explain how you got the answer.

500

A value of θ is defined. 

Evaluate the expressions sin 2θ, cos 2θ, and tan 2θ.

tan θ = 0.75 and pi < x < (3pi)/2

24/25, 7/25, 24/7

Verifying Trig Identities

100

What are the three fundamental identities

sin²x+cos²x=1

1+tan²x=sec²x

1+cot²x=csc²x

200

What are the six ratio identities?

tan=sin/cos

cot=cos/sin

sin=1/csc

cos=1/sec

sec=1/cos

csc=1/sin

300

What are some tips for solving trig identities?

Answers include

-work on one side only

-start with more complicated side

-treat each side seperately

-Factoring, adding or subtracting fractions, simplifying complex fractions

-Use basic fundamental identities

-convert to sines and cosines

400

How is the verification of identities different than solving trig equations?

Verification you are working with one side to manipulate it to look like the other side.

Solving equations you are still kind of manipulating but you can work with both sides, and there are specific answers you are looking for. 

500

Is this an identity? Prove it.

cos^3x+cosxsin^2x=cosx

Yep! (Explain how you got that answer?)