How do I find the value of sin ( (tax ^ -1) (1) ) ?

Question

mathstudent55 · Answer

You mean $\sin(\tan ^{-1} 1)$ right?

mathstudent55 · Answer

$\tan^{-1} 1$ is the inverse tangent of 1.
That means what angle has a tangent of 1?

anonymous · Answer

90 degrees?

mathstudent55 · Answer

No.
Let's look at our friendly 45-45-90 triangle.

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mathstudent55 · Answer

As you can see, it's the 45 deg angle that has a tangent of 1.

mathstudent55 · Answer

Now we know that the inverse tangent of 1 is 45.
Now we can continue:

$\sin(\tan ^{-1} 1)$

$= \sin 45^o $

Now we can refer to the figure again to find the sine of 45 deg.
Remember that sine = opp/hyp
In the case of a 45 deg angle, opp = 1 and hyp = $\sqrt 2$

$= \dfrac{1}{\sqrt 2} $

Now all you need to do is rationalize that fraction.
Multiply the numerator and denominator by sqrt(2).

anonymous · Answer

Okay. But does that help me? Because my final answer has to be in radians, right? Because then it would be sin(45)(1), and I don't know what I would do from there.

anonymous · Answer

Oh whoops, sorry. Your reply just came through.

anonymous · Answer

So then the answer is 1/2?

mathstudent55 · Answer

Your answer can't be in radians because your answer is the sine of an angle. The sine of an angle is a unitless number since it is a ratio of two lengths. A radian measure is unitless, but it is used to measure angles. Your answer is not an angle measure.

mathstudent55 · Answer

$\dfrac{\sqrt 2}{2} $

anonymous · Answer

Oh, I understand. Since I'm using trigonomic values, it doesn't make sense to put my answer in radians.
So root 2 / 2 is my final answer?

mathstudent55 · Answer

The answer to your second question is yes. The answer is sqrt(2)/2

anonymous · Answer

Thank you so much for your help! I think I understand the concepts a little bit better now.

mathstudent55 · Answer

Let me explain again about radians.
An angle can be measured using several types of units.
The most common unit that is familiar to most people is a degree.
Another very important unit is the radian.
Once again, degrees and radians are used as units for angle measure.

Now let's look at this problem again.
The problem is asking: "What is the sine of the angle whose tangent is 1?"
In order to find the value of the sine, we need to know the angle whose sine we want.
That angle is the angle which has a tangent of 1.
It turns out that in degrees, the angle is 45 degrees.
In radians, a 45-deg angle is a pi/4 radian angle.
Now we take the sine of 45 degrees or the sine of pi/4 radians.
The answer is sqrt(2)/2.
This final answer of sqrt(2)/2 is the sine of an angle. It is not an angle measure, so it cannot be in degrees or in radians.