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

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

And cosine and tangent follow a similar idea.

And now for the details:

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

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

Bạn đang xem: Find value of sin 1 degrees

Sine Function

TheSineofangleθis:

the length of the side Opposite angle θdivided 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 = 2.8/4.9 = 0.57...

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 khổng lồ 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 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.

và we want khổng lồ 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 lớn 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.

Cosine

The Cosine of angle θ is:

And Inverse Cosine is :

cos-1 (Adjacent / Hypotenuse) = θ

Example: Find the kích cỡ 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:

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 khổng lồ 1 decimal place)

Other Names

Sometimes sin-1 is called asin or arcsinLikewise cos-1 is called acos or arccosAnd 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 & Inverse Cosine don"t "go on forever" like Sine and Cosine vị ...

Let us look at the example of Cosine.

Here is Cosine & Inverse Cosine plotted on the same graph:

Cosine và Inverse Cosine

They are mirror images (about the diagonal)

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

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

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 sách of possible answers ...

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

Xem thêm: Các Câu Chúc 20 11 Hay Nhất, Lời Chúc Ngày Nhà Giáo Việt Nam 20/11 Hay Nhất

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 và inverse tangent. Can you see how they are mirror images (about the diagonal) ...?

Tangent
Inverse Tangent
Random Trigonometry The Law of Sines The Law of Cosines Solving Triangles Trigonometry Index Algebra Index