Working with exact values of sin,cos and tan

[Probability]

I would appreciate any help on understanding how to convert from radian measure to exact values please!

Suppose I have;

sin 1/3 Pi and I was asked to find the exact value, how would I do it?

I know that 1/3 Pi = 60 degrees, and I also know that sin 1/3 Pi = 0.866025404 in decimal notation, but mathematically how would I work out that

sin 1/3 Pi = square root of 3 / 2 ?

I can copy it from books all day, and I know that 1/3 Pi = square root 3/2, but how can it be worked out mathematically so I understand how to do the conversions in future?

 

[DrPhil]

I would appreciate any help on understanding how to convert from radian measure to exact values please!

Suppose I have;

sin 1/3 Pi and I was asked to find the exact value, how would I do it?

I know that 1/3 Pi = 60 degrees, and I also know that sin 1/3 Pi = 0.866025404 in decimal notation, but mathematically how would I work out that

sin 1/3 Pi = square root of 3 / 2 ?

I can copy it from books all day, and I know that 1/3 Pi = square root 3/2, but how can it be worked out mathematically so I understand how to do the conversions in future?

The exact value is as you found,

$\displaystyle \sin(\pi /3) = \sqrt{3}/2$

How do I remember that? much the same way you reasoned it out:
pi/3 is 1/3 the way to 180°, so it is 60°
Imagine that on the unit circle - the 60° angle makes a 60-30-90° triangle, and I know the short side is 1/2
By Pythagorean theorem, the longer side is sqrt[1^2 - (1/2)^2] = sqrt(3/4) = sqrt(3)/2.

OK? Some facts have to be memorized. One easy fact is sin(30°) = 1/2. Another useful fact worth remembering is cos(30°) = sqrt(3)/2. And I manage to remember sin(45°) = cos(45°) = sqrt(2)/2 -- if I forget those I can recall them by using Pythagoras again with the diagonal of a square.

Another essential fact is that a full circle is $\displaystyle 2\pi$ radians - because the circumference of a circle is $\displaystyle \ 2\pi R$. So I know the conversion is $\displaystyle \ \pi\ \text{radians} = 180°$.

 

[Deleted member 4993]

I remember it through equilateral triangle ABC of side 2x.

Suppose the triangle has base BC and the opposite vertex is then A. Drop a perpendicular AD from A onto BC.

Then

AB = BC = CA = 2x

BD = DC = x

AD =√(AC² + DC²) = x√3

mDAC = 30°

mDCA = 60°

and you can now calculate all the trigonometric values. [sin(30°) = sin(DAC) = DC/CA = x√3/(2x) = √3/2, tan(30°) = tan(DAC) = DC/AD = x/(x√3) = 1/√3, etc.]

I do a similar calculation for 45°.

 

[Probability]

Thank you for all replies I really have learned a lot about something I knew very little about this morning.

So to conclude, this is what I have learned. Using circle geometry and triangles in conjunction with Pythagoras Theorem.

Taking an equilateral triangle and including a bisector I have learned where square root of 3 comes from.

Using Pythagoras Theorem and trigonometric ratios this is how I understand the conversion now to exact form.

cos 30* = opposite side AD / hypotenuse AB = square root 3 / 2, therefore cos 30* = square root 3 / 2

I can do sin, cos and tan of other angles by the same method, except for undefined angles.

Thanks again.

 

[Deleted member 4993]

Thank you for all replies I really have learned a lot about something I knew very little about this morning.

So to conclude, this is what I have learned. Using circle geometry and triangles in conjunction with Pythagoras Theorem.

Taking an equilateral triangle and including a bisector I have learned where square root of 3 comes from.

Using Pythagoras Theorem and trigonometric ratios this is how I understand the conversion now to exact form.

cos 30* = opposite side AD / hypotenuse AB = square root 3 / 2, therefore cos 30* = square root 3 / 2

I can do sin, cos and tan of other angles by the same method, except for undefined angles.

Thanks again.

You can get exact values of trigonometric functions of angles 15° , 22.5°, 75° (may be some more that escapes me know) and their multiples (or reverse) using various trigonometric formulas (like cos(A/2) = √[{1+cos(A)}/2])..............edit