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

The **inverse sine** function sin-1 takes the ratio *opposite***hypotenuse ** và gives angleθ

And cosine và tangent follow a similar idea.

### Example (lengths are only to one decimal place):

### And now for the details:

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

They are very similar functions ... So we will look at the **Sine Function** and then **Inverse Sine** lớn learn what it is all about.

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## Sine Function

TheSineofangle**θ**is:

**length of the side Opposite**angle

**θ**divided by the

**length of the Hypotenuse**

Or more simply:

sin(θ) = Opposite / Hypotenuse

### Example: What is the sine of 35°?

Using this triangle (lengths are only to lớn one decimal place): sin(35°) = Opposite / Hypotenuse |

### Example: Use the **sine function** lớn find **"d"**

We know

The angle the cable makes with the seabed is 39° The cable"s length is 30 m.and we want to lớn know "d" (the distance down).

## Inverse Sine Function

But sometimes it is the **angle** we need to lớn find.

This is where "Inverse Sine" comes in.

It answers the question "what **angle** has sine equal khổng lồ opposite/hypotenuse?"

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

### Example: Find the angle **"a"**

We know

The distance down is 18.88 m.The cable"s length is 30 m.& we want to know the angle "a"

sin takes an

**angle**and gives us the

**ratio**"opposite/hypotenuse"sin-1 takes the

**ratio**"opposite/hypotenuse" and gives us the

**angle.**

## Calculator

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

On your calculator, try using sin & then sin-1 khổng lồ see what happens

## More Than One Angle!

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

### Example: Here are two angles where opposite/hypotenuse = 0.5

**In fact there are infinitely many angles**, because you can keep adding (or subtracting) 360°:

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

## Summary

The Sine of angle **θ** is:

sin(θ) = Opposite / Hypotenuse

And Inverse Sine is :

sin-1 (Opposite / Hypotenuse) = θ

## What About "cos" & "tan" ... ?

Exactly the same idea, but different side ratios.

CosineThe Cosine of angle **θ** is:

cos(θ) = Adjacent / Hypotenuse

And Inverse Cosine is :

cos-1 (Adjacent / Hypotenuse) = θ

### Example: Find 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) = Opposite / Adjacent

So Inverse Tangent is :

tan-1 (Opposite / Adjacent) = θ

### Example: Find the kích cỡ of angle x°

tan x° = Opposite / Adjacent

tan x° = 300/400 = 0.75

x° = **tan-1** (0.75) = **36.9°** (correct lớn 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 called atan** or **arctan**

### Examples:

**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 of Sine, Inverse Sine, Cosine & Inverse Cosine:

Sine

Inverse Sine

Cosine

Inverse Cosine

Did you notice anything about the graphs?

They look similar somehow, right?But the Inverse Sine and Inverse Cosine don"t "go on forever" like Sine và Cosine vị ...Let us look at the example of Cosine.

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

**Cosine & Inverse Cosine**

They are mirror images (about the diagonal)

But why does Inverse Cosine get chopped off at top & bottom (the dots are not really part of the function) ... ?

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

**One Answer or Infinitely Many Answers**

**But we saw earlier that there are infinitely many answers**, & the dotted line on the graph shows this.

So yes there **are** infinitely many answers ...

... But imagine you type 0.5 into your calculator, press cos-1 và it gives you a never ending danh mục of possible answers ...

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

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So, by chopping it off lượt thích that we get just one answer, but **we should remember that there could be other answers**.

## Tangent và Inverse Tangent

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

Tangent

Inverse Tangent

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