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Here are a few recommended readings before getting started with this lesson.
On Sunday, Magdalena and her younger sister Paulina went to the amusement park Adventurally with their father. They all had a lot of fun going on numerous rides, including a Ferris Wheel. When they first saw it close up, the girls were so amazed by its size that they asked one of the workers for more details about it.
The worker replied that it has a diameter of $46$ meters and it turns at a rate of $1.5$ revolutions per minute. When Magdalena's and Paulina's father heard this, he said that the height of their seat $h$ above the ground in meters after $t$ minutes can be modeled by the following function.There are functions that undo the trigonometric functions, so to speak. These functions are called inverse trigonometric functions.
The inverse trigonometric functions are the inverse functions of the trigonometric functions. For example, the inverse sine is the inverse function of the sine function. The main inverse trigonometric functions are shown in the table below.
Trigonometric Function  Inverse Trigonometric Function 

$f(x)=sinx$  $f_{1}(x)=sin_{1}x$ 
$f(x)=cosx$  $f_{1}(x)=cos_{1}x$ 
$f(x)=tanx$  $f_{1}(x)=tan_{1}x$ 
Inverse Trigonometric Function  Domain  Range 

$y=sin_{1}x$  $[1,1]$  $2π ≤x≤2π $ 
$y=cos_{1}x$  $[1,1]$  $0≤x≤π$ 
$y=tan_{1}x$  All real numbers  $2π ≤x≤2π $ 
Contrary to trigonometric identities — which are true for all values of the variable for which both sides are defined — some equations involving trigonometric functions are true only for certain values of the variable. Now such equations will be presented.
To sum up, the following facts can be used to solve trigonometric equations. Note that $n$ and $m$ written below are integers.
Equation  Solutions 

$sinθ=sinα$  $θ=nπifnis odd,ifnis even, +(1)_{n}α⇓θ=(2m+1)π−αθ=2mπ+α $

$cosθ=cosα$  $θ=2nπ±α$ 
$tanθ=tanα$  $θ=nπ+α$ 
$cos(2θ)=1−2sin_{2}(θ)$
$LHS+2sin_{2}θ=RHS+2sin_{2}θ$
$LHS−1=RHS−1$
Rearrange equation
Commutative Property of Addition
Rewrite $15sinθ$ as $14sinθ+sinθ$
Distribute $1$
$LHS+1=RHS+1$
$LHS/2=RHS/2$
$sin_{1}(LHS)=sin_{1}(RHS)$
$f_{1}(f(x))=x$
$cos(2θ)=2cos_{2}(θ)−1$
$LHS+3=RHS+3$
Commutative Property of Addition
$LHS/2=RHS/2$
$a_{2}+2ab+b_{2}=(a+b)_{2}$
$LHS =RHS $
$LHS−1=RHS−1$
$sec_{2}θ=1+tan_{2}θ$
$LHS−tan_{2}θ=RHS−tan_{2}θ$
$LHS+1=RHS+1$
$LHS/2=RHS/2$
$LHS =RHS $
The teacher gave the class a couple of exercises to solve for homework. She also warned that one of the equations has extraneous solutions and that the students should identify them. To make things interesting, Magdalena and Davontay decided to make a bet about which equation has extraneous solutions.
After each chose an equation, they started solving them to see who guessed correctly. The winner will get the last piece of cake left in the fridge. To solve the equations, they must write all the solutions in radians such that $0≤θ≤2π.$
$cos_{2}θ=1−sin_{2}θ$
Distribute $2$
Commutative Property of Addition
$LHS−3=RHS−3$
$LHS⋅(1)=RHS⋅(1)$
Rewrite $3sinθ$ as $2sinθ+sinθ$
Distribute $1$
Factor out $2sinθ$
Factor out $1$
Factor out $sinθ−1$
$(I), (II):$ $LHS+1=RHS+1$
$(I):$ $LHS/2=RHS/2$
$θ=6π $
$(ba )_{m}=b_{m}a_{m} $
$a⋅cb =ca⋅b $
$ba =b/2a/2 $
Add fractions
Solution  Substitute  Evaluate  True or False 

$θ=6π $  $2cos_{2}(6π )+3sin(6π )=?3$  $2(23 )_{2}+3(21 )=?3$  $3=3✓$ 
$θ=65π $  $2cos_{2}(65π )+3sin(65π )=?3$  $2(23 )_{2}+3(21 )=?3$  $3=3✓$ 
$θ=2π $  $2cos_{2}(2π )+3sin(2π )=?3$  $2(0)_{2}+3(1)=?3$  $3=3✓$ 
Therefore, there are three solutions to the equation and none of them are extraneous.
$cos_{2}θ=1−sin_{2}θ$
$LHS+sin_{2}θ=RHS+sin_{2}θ$
$LHS/2=RHS/2$
$LHS =RHS $
$ba =b a $
$ba =b⋅2 a⋅2 $
Solution  Substitute  Evaluate  True or False 

$θ_{1}=4π $  $sin(4π )−cos(4π )=?0$  $22 −22 =?0$  $0=0✓$ 
$θ_{2}=43π $  $sin(43π )−cos(43π )=?0$  $22 −(22 )=?0$  $2 =0×$ 
$θ_{3}=45π $  $sin(45π )−cos(45π )=?0$  $22 −(22 )=$ 