# Inverse Sine, Cosine, Tangent

### Quick Answer:

For a right-angled triangle:

The **sine** function sin takes angle θ and gives the ratio \frac{opposite}{hypotenuse }

The **inverse sine** function sin^{-1} takes the ratio \frac{opposite}{hypotenuse } and gives angle θ

And cosine and tangent follow a similar idea.

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

**sin(35°)**= Opposite / Hypotenuse

**sin**= sin

^{-1}(Opposite / Hypotenuse)^{-1}(0.57...)

### 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** to learn what it is all about.

## Sine Function

The Sine of angle ** θ** is:

- the
**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 one decimal place): sin(35°) = Opposite / Hypotenuse |

The Sine Function can help us solve things like this:

### Example: Use the **sine function** to 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 know "d" (the distance down).

**18.88**to 2 decimal places

The depth "d" is **18.88 m**

## Inverse Sine Function

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

This is where "Inverse Sine" comes in.

It answers the question "what **angle** has sine equal to 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.

And we want to know the angle "a"

What **angle** has sine equal to 0.6293...?

The **Inverse Sine** will tell us.

**sin**(0.6293...)

^{−1}**sin**(0.6293...):a° =

^{−1}**39.0°**(to 1 decimal place)

The angle "a" is **39.0°**

## They Are Like Forward and Backwards!

- sin takes an
**angle**and gives us the**ratio**"opposite/hypotenuse" - sin
^{-1}takes the**ratio**"opposite/hypotenuse" and gives us the**angle.**

### Example:

**30°**) =

**0.5**

^{−1}(

**0.5**) =

**30°**

## 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 and then sin^{-1} to 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" and "tan" ... ?

Exactly the same idea, but different side ratios.

#### Cosine

The 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 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 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 and 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 and Cosine do ...

Let us look at the example of Cosine.

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

Cosine and Inverse Cosine

They are mirror images (about the diagonal)

But why does Inverse Cosine get chopped off at top and 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**, and 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} and it gives you a never ending list of possible answers ...

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

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

## Tangent and 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