The inverse trigonometric functions are called the arcsine, arccosine, and arctangent. They are the opposite of sine, cosine, and tangent. That means that the range of one is the domain of the other, and the other way round. Note to be very careful with the notation of the inverse functions. There are some ambiguous notations that could be misleading. In this lecture you'll also learn how to graph these inverse functions. If you look closely, you'll see that the transformation of the sine graph doesn't pass the horizontal line test, so we'll need to use some restrictions. The same rule will apply to the other inverse functions.
The arcsine function, also known as the inverse sine function, is a function that, for each x between -1 and 1, produces an angle θ
/2) and (π
/2) whose sine is x.
If arcsinx = θ
, then sinθ
It is sometimes written sin−
1x, but this notation is misleading because it could also be interpreted as [1/(sinx)], which is different from arcsinx.
The arccosine function, also known as the inverse cosine function, is a function that, for each x between -1 and 1, produces an angle θ
between 0 and π
whose cosine is x.
If arccosx = θ
, then cosθ
It is sometimes written cos−
1x, but this notation is misleading because it could also be interpreted as [1/(cosx)], which is different from arccosx.
The arctangent function, also known as the inverse tangent function, is a function that, for each x, produces an angle θ
(π/2) and (π/2) whose tangent is x.
If arctanx = θ
, then tanθ
It is sometimes written tan−
1x, but this notation is misleading because it could also be interpreted as [1/(tanx)], which is different from arctanx.
Identify the domain and range of the arcsine function, and graph the function.
)/6])), and arctan(tan[(3π
Identify the domain and range of the arctangent function, and graph the function. Identify the asymptotes of the graph.
Identify the domain and range of the arccosine function, and graph the function.
)/4])), and arctan(tan[(7π
Inverse Trigonometric Functions
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Finally we have arctan(tan 7pi/6), 7pi/6 is just bigger than pi, there it is down there 7pi/6.0196
We want to find an angle that has the same tan as that.0209
Remember with tan, if we reflect across the origin we will get an angle that has the same sin and cos except they switched from being negative to positive.0213
But they both switched which mean tan(sin/cos) will actually end up the same.0233
This angle is pi/6 will have the same tan as 7pi/6, that is good because arctan is forced to be between –pi/2 and pi/2.0240
We got this angle pi/6 that has the same tan as what we are looking for 7pi/6.0259
The key to solving that problem is to take each one of these angles and find them on the unit circle.0280
Figure out what the range of our function is we are looking for, arctan between –pi/2 and pi/2, arcsin in the same range, arcos little different between 0 and pi.0287
Then we try to find an angle within that range that has the same sin, cos, or tan, as the angle that we started with.0302
That is why I drawn this little dotted lines here on the unit circle, I’m trying to find angles with same cos or same sin, or same tan as what we started with.0313
That what gave us our answers –pi/3, pi/4 and pi/6, those are angles that has the same sin, cos, and tan as the ones we are given.0323
That was our first lecture on inverse trigonometric functions, these are the trigonometry lectures for www.educator.com.0334
Hi, these are the trigonometry lectures for educator.com, and we're here to learn about the inverse trigonometric functions.0000
We've already learned about sine, cosine and tangent, and today we're going to learn about the inverse sine, inverse cosine, and inverse tangent, probably better known as arc sine, arc cosine, and arc tangent.0008
The arc sine function is also known as the inverse sine function, basically, it's kind of the opposite of the sine function.0022
The idea is that you're given a value of x, and you want to find an angle whose sine is that value of x.0030
In order to make this work, sines only occur between -1 and 1, so you have to be given a value of x between -1 and 1.0041
You're going to try to give an angle between -π/2 and π/2.0050
You'll try to give an angle between -π/2 and π/2 that has the given value as a sine.0060
If arcsin(x)=θ, what that really means is that the sin(θ)=x.0078
Now, there's some unfortunate notation in mathematics which is that arcsin is sometimes written as sin to the negative 1 of x.0087
This is very unfortunate because people talk about, for example, sin2x means (sin(x))2.0095
You might think that sin-1(x), would be (sin(x))-1, which would be 1/sin(x).0106
Now, that's not what it means, the sine inverse of x doesn't mean 1/sin(x), it means arcsin(x).0119
This notation really is very ambiguous because this inverse sine notation could mean arcsin(x) or it could mean 1/sin(x), and so this notation is ambiguous because arcsin(x) and 1/sin(x) are not the same.0129
The safest thing to do is to not use this notation sin-1 at all.0151
Let me just say, avoid this notation completely because it is ambiguous, it could be interpreted to mean these two different things that are not equal to each other.0159
Instead, it's probably safer to use the notation arcsin(x), which definitely means inverse sin(x), and cannot be confused.0175
The arccos(x), the arc cosine function is sort of the opposite of the cosine function.0187
You're given a value of x, and you have to find an angle whose cosine is that value of x.0194
Again, the value of x you must be given would have to be between -1 and 1, because those are the only values that come up as answers for cosine.0201
What you try to do is produce an angle between 0 and π, so there's 0, π/2, π.0215
You try to produce an angle between 0 and π that has that value as its cosine.0229
Just like we have with sine, there's the problem of this misleading notation cos-1(x), it could be interpreted as 1/cos(x) or it could be interpreted as arccos(x).0233
The best thing to do is to avoid using this notation completely, cos-1(x) is just misleading, it could be interpreted either way.0247
Try not to use it at all, instead stick to the notation arccos(x).0263
Finally, arc tangent is known as the inverse tangent function.0270
You're given a value of x, and you want to find an angle whose tangent is x.0275
For arc tangent, we're going to try to find the angles between -π/2 and π/2 because that covers all the possible tangents we could get.0282
If arctan(x)=θ, that really means that tan(θ)=x.0297
Just like with sine and cosine, we have this potentially misleading notation, tan-1(x), could be interpreted to mean arctan(x) or 1/tan(x).0302
Those are both reasonable interpretations but they mean two different things.0314
Again, let's try to avoid this notation completely because it could mean two completely different things.0317
We'll just try not to use that at all when we're talking about inverse tangents we'll say arctan instead of tan-1.0337
The domain here, it's all the numbers that you can plug into the arcsin, that's all values of x with -1 less than or equal to x, less than or equal to +1.0532
We can't plug any other values of x into arcsin, and that's really because in the other direction the only values that come out of the sine function are between -1 and 1.0560
The only values that you can plug into the arcsin function are between -1 and 1.0571
The range, the numbers that come out of the arcsin, all values of x between -π/2 and π/2.0580
Let me write that as y because when we think of those are the values coming out of the arcsin function.0596
Those are all the y values you see here and you see that the smallest y-value is -π/2 and the biggest value we see is π/2.0605
Arcsin takes in a number between -1 or 1, gives you a number between -π/2 and π/2 and its graph looks like a sort of chopped of piece of the sine graph.0615
Remember, the reason we had to chop it off is to get a piece of the sine graph that would satisfy the horizontal line test, so that the arcsin graph satisfies the vertical line test and really is a function.0629
In our second example here, we have to find the arcsin of the sin(2π/3) and then some similar values for our cosine and our tangent.0647
The first thing to do here is really to figure out where we are on the unit circle.0663
We look at the sine of 2π/3, and then we want to find another angle that has that sine but it has to be in the specified range for arcsin.0981
We're really trying to find what's an angle between -π/2 and π/2 whose sine is the same as sin(2π/3).0993
That's why we did this reversal to find the angle π/3 that has the same sine as the sin(2π/3).1003
That was kind of the same process of all three of them, finding angles that have the same cosine as -5π/6, but is in the specified range, finding an angle that has the same tangent as 3π/4 but is in the specified range.1017
Our third example here, we're asked to identify the domain and range of the arctan function, and graph the function.1037
Remember when we graphed arcsin, we started out with a graph of sine, so let me start out here with a graph of tangent.1044