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

The inverse sine function sin-1 takes the ratio oppositehypotenuse and gives angleθ

And cosine and also tangent monitor a similar idea.

Example (lengths are just to one decimal place):

And now for the details:

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

They are very comparable functions ... Therefore we will certainly look at the Sine Function and then Inverse Sine to learn what the is every about.

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Sine Function


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

Or much more simply:

sin(θ) = opposite / Hypotenuse

Example: What is the sine the 35°?

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

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


Example: usage the sine function to uncover "d"

We know

The edge the cable renders with the seabed is 39° The cable"s length is 30 m.

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

Inverse Sine Function

But periodically it is the angle we have to find.

This is where "Inverse Sine" come in.

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

The symbol because that inverse sine is sin-1, or sometimes arcsin.


Example: discover the edge "a"

We know

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

and we desire to understand the edge "a"

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


On the calculator you press among the complying with (depending on your brand the calculator):either "2ndF sin" or "shift sin".

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

More 보다 One Angle!

Inverse Sine only mirrors you one angle ... However there are more angles that can work.

Example: below are 2 angles wherein opposite/hypotenuse = 0.5


In fact there are infinitely many angles, due to the fact that you deserve to keep including (or subtracting) 360°:


Remember this, due to the fact that there are times as soon as you actually require one that the various other angles!


The Sine of edge θ is:

sin(θ) = opposite / Hypotenuse

And station Sine is :

sin-1 (Opposite / Hypotenuse) = θ

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

Exactly the exact same idea, however different side ratios.


The Cosine of edge θ is:

cos(θ) = adjacent / Hypotenuse

And inverse Cosine is :

cos-1 (Adjacent / Hypotenuse) = θ


Example: uncover 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 edge θ is:

tan(θ) = the contrary / Adjacent

So inverse Tangent is :

tan-1 (Opposite / Adjacent) = θ


Example: discover the size of edge x°

tan x° = the opposite / nearby

tan x° = 300/400 = 0.75

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

Other Names

Sometimes sin-1 is dubbed asin or arcsinLikewise cos-1 is called acos or arccosAnd tan-1 is called 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, right here are the graphs the Sine, train station Sine, Cosine and Inverse Cosine:

Inverse Sine
Inverse Cosine

Did you notification anything about the graphs?

They look similar somehow, right?But the inverse Sine and also Inverse Cosine don"t "go top top forever" favor Sine and also Cosine execute ...

Let us look in ~ the instance of Cosine.

Here is Cosine and Inverse Cosine plotted ~ above the same graph:

Cosine and also Inverse Cosine

They room mirror pictures (about the diagonal)

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

Because to be a function it have the right to only offer one answer when us ask "what is cos-1(x) ?"

One prize or Infinitely many Answers

But we saw previously that there are infinitely many answers, and the dotted line on the graph shows this.

So yes over there are infinitely numerous answers ...

... But imagine you form 0.5 into your calculator, push cos-1 and also it gives you a never finishing list of possible answers ...

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

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So, by chopping it off prefer that we acquire just one answer, yet we must remember the there could be other answers.

Tangent and Inverse Tangent

And below 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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