Trigonometry Test #03 Review Sheet
Section 5.1: Fundamental Identities
Reciprocal Identities:
1
cot   
tan  
Quotient Identities:
tan   
Pythagorean Identities:
sin 2    cos2    1
Negative-Angle Identities:
sin      sin  
csc      csc  
sec   
sin  
cos  
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1
cos  
cot   
csc   
1
sin  
cos  
sin  
tan 2    1  sec2  
1  cot 2    csc2  
cos     cos  
tan      tan  
sec     sec  
cot      cot  
Trigonometry Test #03 Review Sheet
Page 2 of 12
Section 5.2: Verifying Trigonometric Identities
Hints for Verifying Identities
 Learn (i.e., memorize) the fundamental identities from section 5.1, and be aware of their
equivalent forms (like a re-arranged Pythagorean identity.
Reciprocal Identities:
1
1
1
cot   
sec   
csc   
tan  
cos  
sin  
Quotient Identities:
sin  
cos  
tan   
cot   
cos  
sin  
Pythagorean Identities:
sin 2    cos2    1
tan 2    1  sec2  
1  cot 2    csc2  
Negative-Angle Identities:
sin      sin  
cos     cos  
tan      tan  
csc      csc  
sec     sec  
cot      cot  



Try to simplify the more complicated side until it looks like the simpler side.
Sometimes it is helpful to express all trig functions in terms of sine and cosine, and then
simplify.
Usually it helps to factor when possible, and perform any indicated algebraic operations.
Example: replace sin 2  x   2sin  x   1 with its factored form of  sin  x   1
Example: replace


2
cos  x   sin  x 
1
1
with the fraction of

sin  x  cos  x 
sin  x  cos  x 
As you make substitutions to convert one side into the other, always work toward the
goal of the other side.
A common trick is to multiply expressions like 1  sin  x  by the following fraction,
because then a Pythagorean identity can be used to simplify:
1  sin  x  1  sin  x   1  sin  x  
1  sin  x  

1  sin  x 
1  sin  x 


1  sin 2  x 
1  sin  x 
cos 2  x 
1  sin  x 
Trigonometry Test #03 Review Sheet
Section 5.3: Sum and Difference Identities for Cosine
Cosine of a Sum or Difference
cos  A  B   cos  A cos  B   sin  A sin  B 
cos  A  B   cos  A cos  B   sin  A sin  B 
Cofunction Identities
cos  90  A  sin A
sec  90  A  csc A
sin  90  A  cos A
csc  90  A  sec A
tan  90  A  cot A
cot  90  A  tan A
Section 5.4: Sum and Difference Identities for Sine and Tangent
Sine of a Sum or Difference
sin  A  B   sin  A cos  B   cos  A sin  B 
sin  A  B   sin  A cos  B   cos  A sin  B 
Tangent of a Sum or Difference
tan  A   tan  B 
tan  A  B  
1  tan  A  tan  B 
tan  A  B  
tan  A   tan  B 
1  tan  A  tan  B 
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Trigonometry Test #03 Review Sheet
Section 5.5: Double-Angle Identities
Double Angle Identities
cos  2 A  cos2  A  sin 2  A
cos  2 A  2cos2  A 1
cos  2 A  1  2sin 2  A
sin  2 A  2sin  A cos  A
tan  2 A  
2 tan  A 
1  tan 2  A 
Product-To-Sum Identities
1
cos  A  cos  B   cos  A  B   cos  A  B  
2
1
sin  A  sin  B   cos  A  B   cos  A  B  
2
1
sin  A cos  B   sin  A  B   sin  A  B  
2
1
cos  A sin  B   sin  A  B   sin  A  B  
2
Sum-To-Product Identities
 A B 
 A B 
sin  A   sin  B   2sin 
 cos 

 2 
 2 
 A B   A B 
sin  A   sin  B   2cos 
 sin 

 2   2 
 A B
 A B
cos  A   cos  B   2 cos 
 cos 

 2 
 2 
 A B  A B
cos  A   cos  B   2sin 
 sin 

 2   2 
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Trigonometry Test #03 Review Sheet
Page 5 of 12
Section 5.6: Half-Angle Identities
Half-Angle Identities
1  cos  A
 A
cos    
2
2
1  cos  A
 A
sin    
2
2
1  cos  A
 A
tan    
1  cos  A
2
sin  A 
 A
tan   
 2  1  cos  A 
 A  1  cos  A 
tan   
sin  A 
2
Section 6.1: Inverse Circular Functions
Vertical Line test
Any vertical line will intersect the graph of a function in at most one point.
Horizontal Line Test
Any horizontal line will intersect the graph of a one-to-one function in at most one point.
Inverse Function
The inverse function of a one-to-one function f is defined as f 1   y, x  |  x, y  belongs to f  .
In other words, switch the x and y values of points on the graph of a function to obtain the graph
of an inverse function.
Summary of Inverse Functions
 For a one-to-one function, each x-value corresponds to only one y-value, and each yvalue corresponds to only one x-value (i.e., the function passes the vertical line test and
the horizontal line test).
 If a function f is one-to-one, then f has an inverse function, which we write as f-1.
 The domain of f is the range of f-1, and the range of f is the domain of f-1.
 The graphs of f and f-1 are reflections of each other across the line y = x.
 To find f-1(x) from an algebraic function f(x), follow these steps:
o Interchange x and y in the equation y = f(x).
o Solve for y.
o The resulting expression that y is equal to is f-1(x).
Trigonometry Test #03 Review Sheet
Page 6 of 12
Inverse Sine Function
y  sin 1  x  or y  arcsin  x  means that x  sin  y  , for  2  y  2 .

Note that the domain of the sine function has to be restricted to make it a one-to-one
function in order to define an inverse.
Graph of the Inverse Sine Function
y  sin 1  x  OR y  arcsin  x 
Domain:
Range:
Table of Values:
x
y  sin 1  x 
Notes on the graph of the inverse sine function:
 The inverse sine function is increasing and continuous on its domain.
 Both the x- and y- intercepts are 0.
 The inverse sine function is an odd function.
Trigonometry Test #03 Review Sheet
Page 7 of 12
Inverse Cosine Function
y  cos1  x  or y  arccos  x  means that x  cos  y  , for 0  y   .

Note that the domain of the cosine function has to be restricted to make it a one-to-one
function in order to define an inverse.
Graph of the Inverse Cosine Function
y  cos1  x  OR y  arccos  x 
Domain:
Range:
Table of Values:
x
y  cos1  x 
Notes on the graph of the inverse cosine function:
 The inverse cosine function is decreasing and continuous on its domain.
 Its x-intercept is 1, and its y-intercept is /2.
 The inverse cosine function is neither odd nor even.
Trigonometry Test #03 Review Sheet
Page 8 of 12
Inverse Tangent Function
y  tan 1  x  or y  arctan  x  means that x  tan  y  , for  2  y  2 .

Note that the domain of the tangent function has to be restricted to make it a one-to-one
function in order to define an inverse.
Graph of the Inverse Tangent Function
y  tan 1  x  OR y  arctan  x 
Domain:
Range:
Table of Values:
x
y  tan 1  x 
Notes on the graph of the inverse tangent function:
 The inverse tangent function is increasing and continuous on its domain.
 Both the x- and y- intercepts are 0.
 The inverse tangent function is odd.
 The lines y   2 are horizontal asymptotes.
Trigonometry Test #03 Review Sheet
Inverse Cotangent, Secant, and Cosecant Functions
Page 9 of 12
Trigonometry Test #03 Review Sheet
Page 10 of 12
Finding Inverse Trigonometric Functions with a Calculator
1
1
1
cot 1  u   tan 1   sec 1  u   cos 1   csc1  u   sin 1  
u
u
u
Finding Trigonometric Functions of Inverse Trigonometric Functions
(Note: there are restrictions to the domains and ranges of the formulas below that are being
glossed over…)
u
cos sin 1  u   1  u 2
sin  sin 1  u    u
tan sin 1  u  
1 u2


csc sin 1  u  
1
u




sec sin 1  u  
sin  cos 1  u    1  u 2




1 u






1 u







2
csc sec1  u  




2



cos sec1  u  


cot cos 1  u  
1 u
u2 1




1
u


1
u


cot tan 1  u  
u
tan cot 1  u  
1 u2
1 u2
u
1
u
cot cot 1  u   u
tan  sec1  u    u 2  1





cot sec1  u  
u2 1
u
u
tan csc1  u  




cos csc1  u  
csc csc1  u   u
sec csc1  u  
u 1
2
1 u2
tan tan 1  u   u
2
sec sec 1  u   u
1
u
sin csc 1  u  

1
sec cot 1  u  
u2 1
u
u
sin sec1  u  

1
u
cos cot 1  u  
1
1 u


1 u2
u
u
tan cos 1  u  
sec  tan 1  u    1  u 2
1 u2
csc cot 1  u  

cos tan 1  u  
1
sin cot 1  u  

2
1 u2
u
csc tan 1  u  

sec cos 1  u  
u
sin tan 1  u  


1 u2
u
cot sin 1  u  
1 u2



1
cos cos 1  u   u
1
csc cos 1  u  

1
u2 1
1
u2 1
cot  csc1  u    u 2  1
Trigonometry Test #03 Review Sheet
Page 11 of 12
Section 6.2: Trigonometric Equations I
To solve a trigonometric equation:
 If possible, graph the equation first so you can see what kind of answers to expect.
 Use identities so that only one of the trig functions occurs in the equation, and every
occurrence of that trig function has the same argument.
o Sometimes factoring first can lead to “mini equations,” each of which has only
one trig function, which means you don’t need to use trig identities.
 Use algebra to isolate the trig function
 Use inverse trig functions and the unit circle to find the angle or angles that solve the
equation.
 Check your answers; sometimes an extraneous solution is introduced when you square
both sides of the equation.
Section 6.3: Trigonometric Equations II
To solve a trigonometric equation:
 If possible, graph the equation first so you can see what kind of answers to expect.
 Use identities so that only one of the trig functions occurs in the equation, and every
occurrence of that trig function has the same argument.
o Sometimes factoring first can lead to “mini equations,” each of which has only
one trig function, which means you don’t need to use trig identities.
 Use algebra to isolate the trig function
 Use inverse trig functions and the unit circle to find the angle or angles that solve the
equation.
 Check your answers; sometimes an extraneous solution is introduced when you square
both sides of the equation.
Trigonometry Test #03 Review Sheet
Page 12 of 12
Section 6.4: Equations Involving Inverse Trigonometric Functions
To solve an inverse trigonometric equation:
 If possible, graph the equation first so you can see what kind of answers to expect.
 Use algebra to isolate one of the inverse trig functions.
 Take the corresponding trig function of both sides of the equation.
o If necessary, use angle sum or difference formulas.
o If necessary, use the trick from section 6.1 to compute an exact algebraic
expression for the composition of an inverse trig function and a trig function.
 Check your answers; sometimes an extraneous solution is introduced when you square
both sides of the equation.