# Maths -Exact trig values

#1
How do you work out exact trig values using the ones you already know ,without a calculator ?
e.g- the exact trig value of cos 360°
Any videos regarding this would be helpful
Last edited by Sb2005; 3 months ago
0
3 months ago
#2
(Original post by Sb2005)
How do you work out exact trig values using the ones you already know ,without a calculator ?
e.g- the exact trig value of cos 360°
Any videos regarding this would be helpful
Bunch of things to keep in mind.

Firstly, know your usual sines/cosines/tangents of typical angles 0,30,45,60,90,180,360.

Then, keep in mind periodicity of these functions. Sine & cosine repeat themselves every 360 degrees, so you can freely add/subtract multiples of 360 onto/from the argument ... so cos(420) = cos(420-360) = cos(60). Tangent repeats itself every 180 degrees so the same idea holds just with multiples of 180 instead.

Secondly, keep in mind some identities such as:

* sin(-x) = -sin(x) [ e.g. we can say that sin(-30) = -sin(30) ]
* cos(-x) = cos(x) [ e.g. we can say that cos(-45) = cos(45) ]

* sin(x) = sin(180-x) [ e.g. we can say that sin(120) = sin(180-120) = sin(60) ]
* cos(x) = cos(360-x) [ e.g. we can say that cos(315) = cos(360-315) = cos(45) ]

* tan(x) is ....(by definition) ... sin(x)/cos(x) ... so knowing the above sine and cosine tricks is sufficient to also adress tangent angles [ e.g. tan(-330) = sin(-330)/cos(-330) = sin(-330 + 360)/cos(-330 + 360) = sin(30)/cos(30) = tan(30)]

* sin(x) = cos(90-x) ... [ e.g. sin(120) = cos(90-120) = cos(-30) = cos(30) ]
* cos(x) = sin(90-x) ... these two are useful if you want to convert from sine to cosine and vice versa.
Last edited by RDKGames; 3 months ago
1
3 months ago
#3
I'd like to suggest a different approach to remembering the graphs of and . Namely, the idea of the unit circle.

That is, consider a circle centred on the origin with a radius of 1:

What's so important about the unit circle?

There is one very simple thing to remember about the unit circle which makes deriving trigonometric values and identities very easy.

That is: If you draw a line from the origin to a point on the circle such that the angle between the line you drew and the x-axis is (anti-clockwise) then the point on the circle will have coordinates .

As the say, a picture's worth a thousand words:

Let's see how this diagram can help us determine some common trigonometric values.

Let's say we want to work out and .

Let's draw a line at 90° to the x-axis:

Clearly, the point (on the circle) we touch is so that means we can conclude that and .

Now, let's say we want to work out and .

Let's draw a line at 0° to the x-axis (i.e. along the x-axis):

In this case, the point (on the circle) we touch is so that means we can conclude that and .

Have a go at determining , , , using the unit circle.

Eventually, you'll likely just be able to recall these values due to familiarity but I found the unit circle an invaluable tool to derive values to build that familiarity.

It's also useful for deriving trigonometric identities:
Spoiler:
Show
For example, suppose we have an angle and want to determine which other angle "has the same ".

Well, we know that is given by the y-coordinate of a point on the circle. Suppose is acute:
We can see that, by symmetry, there are at most two points on the circle which have the same y-coordinate:

The point at the end of the red line and the point at the end of the blue line. The anti-clockwise angles are and .

This gives us the identity (we assumed that was acute for the diagram's sake but this identity holds more generally).

Have a go at:
(i) Deriving a similar identity for using symmetry
(ii) Determining a relationship between and
Last edited by ElMoro; 3 months ago
1
3 months ago
#4
(Original post by Sb2005)
How do you work out exact trig values using the ones you already know ,without a calculator ?
e.g- the exact trig value of cos 360°
Any videos regarding this would be helpful
This is a GCSE topic - post #2 is far too complex.

0
#5
Thank you very much to all of you!
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