# Maths -Exact trig values

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Sb2005

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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

e.g- the exact trig value of cos 360°

Any videos regarding this would be helpful

Last edited by Sb2005; 3 months ago

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RDKGames

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(Original post by

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

**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

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

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ElMoro

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I'd like to suggest a different approach to remembering the graphs of and . Namely, the idea of the

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:

**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

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

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Muttley79

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**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

What have you been taught about this?

https://mathsmadeeasy.co.uk/gcse-mat...nd-worksheets/

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Sb2005

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