What is inverse trigonometrics exactly? That is a little hard to explain, to understand that you need to understand what “trigonometrics” is. Short hand being “trig” it is simply using the functions “Sin, Cos, and Tan” to find lengths when you are given an angle of a right triangle. Sin being opposite/ hypotenuse, Cos being adjacent/ hypotenuse and finally Tan being opposite/ adjacent. If you understand this then the rest should just be like an update to this prior knowledge. Now inverse trig is just doing this backwards and by that I mean using Sin, Cos, and Tan to use a length to find an angle. And in order to do this you need one of these two options one being using a calculator, fairly simple but you unless you already are certified mathematician you won’t be able to fully understand why the answer you get is correct. The second option being the unit circle, this is a bit more complicated. In the unit circle you are given a circle and on the inside are the most common angles and lengths used in these kinds of problems. Depending on which of the three functions you are using will decide whether your answer is 30 degrees or 210 degrees. Sin correspondents with the Y axis while Cos corresponds with the X axis and Tan corresponds with Y/X. Now I know this is a bit confusing but bare with me it gets a little bit more confusing, one being how the questions are written. Now Sin^-1(x)=(Θ)is just saying inverse Sin of blank equals blank, but arcSin(x)=(Θ) is the exact same thing. The “Arc” is just another way of saying inverse in these kinds of problems and it is mainly used because many people will think that Sin^-1 is Sin raised to the power of -1 rather than inverse of Sin. This stands true for all the other functions as well. Now back to the unit circle, this is very important without it you won’t be able to solve the problem unless you are a highly skilled mathematician or have a calculator. Now this circle is complex and is out of 360 degrees and you need to memorize it as well. With this update to what trig and inverse trig is your mind must be spinning a little but don’t worry because all will be explained in detail as you read on.
This is “the unit circle” it is extremely helpful for solving inverse trig without a calculator, in fact it would prove to be extremly difficult to solve these functions without the assistance of either “the unit circle” or a calculator. With this in mind it would be more than helpful to memorize this diagram. In the unit circle “Sin” corresponds with quadrents 1 and 3, only this does not seem to make sense at first glance but if you look closely you will see how in quadrents both 1 and 2 there is a 1/2 in the Y axis part of the lengths, not only this but both 1/2’correspond with different angles. So each function corresponding with its own set of quadrents is helpful one being instead of searching the whole circle for your answer and finding more than one answer you reduce it to 2 quadrents instead of 4 and reduced to one answer. Now Cos corresponds with quadrent 2 and 4 and finally leaving Tan corresponding with 1 and 4. To use this you need to know what function you are using which is fairly simple but depending on the function will determine wether you look at the X, Y axis or Y/X for example say you were given “arcSin(1/2)=x”, sin corresponds with the Y axis so you look in the either the 1 or 3 quantrent and look for 1/2 in the Y axis of one of the pairs and you have you answer. The 1/2 for this question is in the 1 quadrent angle it goes with is 30 degrees. And boom just like that you have your answer. It’s fairly simple to do, it just looks and sounds complicated to use. Now remeber sin corresponds with Y cos corresponds with X and tan corresponds with Y/X.
ArcSin in the inverse of sin
It can be written in about 2 different ways
What does arcSin look like graphed
Sin corresponds with the Y value in a coordinate pair on the unit circle. Meaning if the measurement givein is in the X value you have the wrong answer
Lastly the domain and range for arcSin is (X=[-1,1]), (Y=[-1/2π, 1/2π])
Arccosine is the inverse function of cosine.
Arccosine can be written in a few different forms:
This is what Arccosine looks like when it’s graphed.
When looking for the arccosine of an equation, your answers are restricted to the first and second quadrants.
Arccosine essentially represents the “x” in a coordinate pair, if you are looking at the coordinates on the unit circle.
Lastly the domain and range for Arccos is (X=[-1,1]), (Y=[0,π])
This is inverse tan or arctan graphed (arctan or tan^-1)
When you are looking for coordinates for arctangent on a unit circle the only quadrants that can be viable for arctan is on the 1st and 4th quartile
Also the domain and range for ArchTan is (X= [- ∞, ∞]),
(Y=[-1/2π, 1/2π]) and lastly Arctan has two asymtopes and they are (X=π/2, X=-π/2)
- cos‾¹ (1/2)
Use the unit circle to find a coordinate with 1/2 as the “x” (remember that arccos is restricted to the first and second quadrant).
Now that we have located where 1/2 is “x”, we can see that the angle corresponding with that set of coordinates is 60°.
This is asking what the sin of cos‾¹ is. This is the exact same process as finding the angle that corresponds with cos‾¹(√2/2) except the answer will be the sin or the “y” coordinate that goes with (√2/2).
Again, use the unit circle to find where √2/2 is “x”.
Now we know that sin(cos‾¹(√2/2)) is equal to √2/2