The sine role sin takes edge θ and gives the proportion opposite hypotenuse

The inverse sine function sin-1 take away the proportion oppositehypotenuse and gives angleθ

And cosine and tangent monitor a similar idea.

Example (lengths are just to one decimal place):

And now for the details:

Sine, Cosine and also Tangent room all based upon a Right-Angled Triangle

They space very similar functions ... For this reason we will look at the Sine Function and also then Inverse Sine to learn what the is every about.

You are watching: Tan(cos^-1(x))

Sine Function


the length of the next Opposite edge θdivided by the length of the Hypotenuse

Or more simply:

sin(θ) = the contrary / Hypotenuse

Example: What is the sine that 35°?

Using this triangle (lengths are only to one decimal place):

sin(35°) = the opposite / Hypotenuse = 2.8/4.9 = 0.57...


Example: use the sine function to uncover "d"

We know

The edge the cable provides with the seabed is 39° The cable"s size is 30 m.

and we want to know "d" (the street down).

Inverse Sine Function

But periodically it is the angle we must find.

This is whereby "Inverse Sine" come in.

It answer the question "what angle has sine equal to opposite/hypotenuse?"

The symbol for inverse sine is sin-1, or periodically arcsin.


Example: discover the angle "a"

We know

The distance down is 18.88 m.The cable"s size is 30 m.

and also we desire to know the angle "a"

sin takes an angle and gives united state the ratio "opposite/hypotenuse"sin-1 takes the ratio "opposite/hypotenuse" and gives united state the angle.


On the calculator friend press among the adhering to (depending on your brand of calculator):either "2ndF sin" or "shift sin".

On her calculator, try using sin and also then sin-1 to check out what happens

More 보다 One Angle!

Inverse Sine only reflects you one angle ... But there are more angles that could work.

Example: here are two angles wherein opposite/hypotenuse = 0.5


In fact there room infinitely numerous angles, since you deserve to keep adding (or subtracting) 360°:


Remember this, due to the fact that there room times when you actually require one the the other angles!


The Sine of angle θ is:

sin(θ) = opposite / Hypotenuse

And station Sine is :

sin-1 (Opposite / Hypotenuse) = θ

What about "cos" and also "tan" ... ?

Exactly the exact same idea, yet different next ratios.


The Cosine of angle θ is:

cos(θ) = surrounding / Hypotenuse

And train station Cosine is :

cos-1 (Adjacent / Hypotenuse) = θ


Example: uncover the size of edge a°

cos a° = adjacent / Hypotenuse

cos a° = 6,750/8,100 = 0.8333...

a° = cos-1 (0.8333...) = 33.6° (to 1 decimal place)


The Tangent of angle θ is:

tan(θ) = the contrary / Adjacent

So inverse Tangent is :

tan-1 (Opposite / Adjacent) = θ


Example: find the size of angle x°

tan x° = the contrary / adjacent

tan x° = 300/400 = 0.75

x° = tan-1 (0.75) = 36.9° (correct to 1 decimal place)

Other Names

Sometimes sin-1 is referred to as asin or arcsinLikewise cos-1 is referred to as acos or arccosAnd tan-1 is called atan or arctan


arcsin(y) is the same as sin-1(y) atan(θ) is the same as tan-1(θ)etc.

The Graphs

And lastly, here are the graphs the Sine, station Sine, Cosine and also Inverse Cosine:

Inverse Sine
Inverse Cosine

Did you notification anything about the graphs?

They look similar somehow, right?But the train station Sine and also Inverse Cosine don"t "go ~ above forever" choose Sine and also Cosine perform ...

Let us look in ~ the example of Cosine.

Here is Cosine and Inverse Cosine plotted on the exact same graph:

Cosine and Inverse Cosine

They space mirror images (about the diagonal)

But why walk Inverse Cosine acquire chopped turn off at top and bottom (the dots are not really component of the function) ... ?

Because to be a function it can only give one answer when we ask "what is cos-1(x) ?"

One price or Infinitely many Answers

But us saw earlier that there room infinitely plenty of answers, and the dotted heat on the graph shows this.

So yes there are infinitely many answers ...

... Yet imagine you type 0.5 right into your calculator, press cos-1 and also it offers you a never finishing list of possible answers ...

So we have this rule that a function can only provide one answer.

See more: Fantastic Beasts And Where To Find Them Fandango, Fantastic Beasts And Where To Find Them

So, through chopping the off favor that we get just one answer, but we must remember the there can be various other answers.

Tangent and also Inverse Tangent

And here is the tangent function and inverse tangent. Have the right to you see just how they are mirror images (about the diagonal) ...?

Inverse Tangent
random Trigonometry The law of Sines The law of Cosines fixing Triangles Trigonometry table of contents Algebra index