Inverse Sine, Cosine, Tangent
Sine, Cosine and Tangent are all based on a Right-Angled Triangle
And 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:
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 = 2.8/4.9 = 0.57... |
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).
Start with: | sin 39° = opposite/hypotenuse = d/30 | |
Swap Sides: | d/30 = sin 39° | |
Use a calculator to find sin 39°: | d/30 = 0.6293… | |
Multiply both sides by 30: | d = 0.6293… x 30 = 18.88 to 2 decimal places. |
The depth "d" is 18.88 m
Inverse Sine
But what if it is the angle we don't know?
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}
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"
Start with: | sin a° = opposite/hypotenuse = 18.88/30 | |
Calculate 18.88/30: | sin a° = 0.6293... | |
What angle has sine equal to 0.6293...? The Inverse Sine will tell us. |
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Inverse Sine: | a° = sin^{-1}(0.6293...) | |
Use a calculator to find sin^{-1}(0.6293...): | a° = 39.0° (to 1 decimal place) |
The angle "a" is 39.0°
They Are Like Forward and Backwards!
- The Sine function sin takes an angle and gives us the ratio “opposite/hypotenuse”
- Inverse Sine sin^{-1} takes the ratio “opposite/hypotenuse” and gives us the angle.
Example:
Sine Function: | sin(30°) = 0.5 | |
Inverse Sine: | sin^{-1}(0.5) = 30° |
Calculator
On the calculator you would 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.
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)
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.
So can be written ,
and is also
and is also
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? (I have showed it as dots, but it's not really part of it).
Because to be a function it must 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, and you are supposed to know 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 |