The sine duty sin takes angle θ and also gives the ratio opposite hypotenuse

The inverse sine duty sin-1 bring away the proportion oppositehypotenuse and also gives angleθ

And cosine and tangent monitor a comparable idea.

Example (lengths are only to one decimal place):

And now for the details:

Sine, Cosine and also Tangent are all based on a Right-Angled Triangle

They are very comparable functions ... Therefore we will look in ~ the Sine Function and then Inverse Sine to find out what that is every about.

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Sine Function


the length of the side Opposite angle θdivided through the length of the Hypotenuse

Or much more simply:

sin(θ) = opposite / Hypotenuse

Example: What is the sine that 35°?

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

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


Example: usage the sine function to find "d"

We know

The angle the cable renders with the seabed is 39° The cable"s size is 30 m.

and we desire to know "d" (the distance down).

Inverse Sine Function

But periodically it is the angle we should find.

This is whereby "Inverse Sine" comes in.

It answers the question "what angle has actually sine same to opposite/hypotenuse?"

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


Example: find the angle "a"

We know

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

and we want to understand the angle "a"

sin take away an angle and also gives us the ratio "opposite/hypotenuse"sin-1 takes the ratio "opposite/hypotenuse" and also gives us the angle.


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

On your calculator, shot using sin and also then sin-1 to watch what happens

More than One Angle!

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

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


In reality there are infinitely plenty of angles, because you deserve to keep adding (or subtracting) 360°:


Remember this, because there space times when you actually need one of the other angles!


The Sine of edge θ is:

sin(θ) = the contrary / Hypotenuse

And inverse Sine is :

sin-1 (Opposite / Hypotenuse) = θ

What around "cos" and "tan" ... ?

Exactly the same idea, but different side ratios.


The Cosine of edge θ is:

cos(θ) = nearby / Hypotenuse

And train station Cosine is :

cos-1 (Adjacent / Hypotenuse) = θ


Example: discover the size of angle 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 train station Tangent is :

tan-1 (Opposite / Adjacent) = θ


Example: uncover the size of angle x°

tan x° = opposite / 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 called asin or arcsinLikewise cos-1 is called acos or arccosAnd tan-1 is referred to as atan or arctan


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

The Graphs

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

Inverse Sine
Inverse Cosine

Did you notice anything about the graphs?

They look comparable somehow, right?But the station Sine and also Inverse Cosine don"t "go top top forever" prefer Sine and Cosine perform ...

Let us look at the example of Cosine.

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

Cosine and Inverse Cosine

They space mirror pictures (about the diagonal)

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

Because to it is in a duty it can only offer one answer when we ask "what is cos-1(x) ?"

One prize or Infinitely many Answers

But we saw earlier that there space infinitely numerous answers, and also the dotted heat on the graph reflects this.

So yes over there are infinitely countless answers ...

... But imagine you type 0.5 right into your calculator, push cos-1 and it gives you a never ending list of possible answers ...

So we have this preeminence that a role can only give one answer.

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So, through chopping the off prefer that we gain just one answer, but we have to remember that there might be other answers.

Tangent and Inverse Tangent

And here is the tangent role and train station tangent. Can you see how they space mirror images (about the diagonal) ...?

Inverse Tangent
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