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

The **inverse sine** duty sin-1 bring away the proportion *opposite***hypotenuse ** 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

TheSineofangle**θ**is:

**length of the side Opposite**angle

**θ**divided 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 |

### 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.**

## Calculator

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!

## Summary

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.

CosineThe 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)

Tangent

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 **arcsin****Likewise cos-1 is called acos** or **arccos****And tan-1 is referred to as atan** or **arctan**

### Examples:

**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:

Sine

Inverse Sine

Cosine

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) ...?

Tangent

Inverse Tangent

arbitrarily Trigonometry The legislation of Sines The law of Cosines resolving Triangles Trigonometry index Algebra table of contents