Question

What is sine, cosine and tangent?

Answer 1

Underlying principles explained over 3 solution sheets

This is sheet 1

Explanation:

There are two extremely important things to remember about the Trigonometric facts you have been shown.

`color(blue)("Point 1 ")color(brown)("They are ratios of the lengths of sides of a triangle.")`

`color(blue)("Point 2 ")color(brown)("The triangle used for them is a right angled triangle")`
`" "`It has a corner that is `90^o`. A triangle has 3 corners
`" "`and the proper name for a corner in this context is Vertex.`" "`So a triangle has 3 Vertices.

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
There are various saying around that try to help you remember the order and relationships. I use SohCahToa. I will explain this very soon.

Let us look at some numbers. Consider `1/2`

If you multiply by 1 you do not change its value. So `1/2xx1` is still `1/2`

But suppose I write 1 as say, `3/3`. This is still 1

Now I multiply `1/2` by 1 but in the form of:`" "1/2xx3/3`.It is still worth `1/2`

So:`" 1/2xx3/3=(1xx3)/(2xx3)=3/6` Looks different but it is still worth `1/2`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Suppose we had `("Opposite")/("Hypotenuse")` using the abbreviations `o/h`

Suppose we made Opposite worth 3 and Hypotenuse worth 4. Then we would have:

`o/h=3/4` Lets change the numbers but keeping the same ratio

`o/h=3/4 = (3xx2)/(4xx2) =6/8" "`They are all of the same ratio. Lets
`" "`go the other way!

`o/h=3/4=(3-:4)/(4-:4) = (0.75)/1" "`They are all of the same ratio.

What is `(0.75)/1` actually saying?

It is saying that; if you had a triangle where the length of the opposite is 1 then the length of the slope will be 0.75

If triangle 1 had `o/h = 3/4` and triangle 2 had `o/h = 0.75/1`

Then they are the same shape as the ratio is the same but triangle 2 is smaller. The opposite for triangle 1 is 3 long whilst the opposite for triangle 2 is only 0.74 long
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The thing is: Instead of writing `0.75/1` they tend to just wright`" " 0.75`.

Answer 2

Underlying principles explained over 3 solution sheets

This is sheet 2

Explanation:

These ratios are given special names

`"Complete name " " Abbreviated name "" Parts of the triangle"`
`" Sine" " " sin" "("Opposite")/("Hypotenuse") = o/h`

`" Cosine" " cos"" "("Adjacent")/("Hypotenuse") =a/h`

`" Tangent" " tan"" "("Opposite")/("Adjacent")" "= o/a`

By example

Let the angel `/_ABC" be "theta" "` ( value not known)

Then:
`sin(theta)=("opposite")/("hypotenuse") = 4/5 =(4-:5)/(5-:5)= 0.8/1=0.8`

`cos(theta)("adjacent")/("hypotenuse")=3/5=(3-:5)/(5-:5)=0.6/1=0.6`

`tan(theta)("opposite")/("adjacent")=4/3=(4-:3)/(3-:3) =(1.33333..)/1=1.33333..`

Consider: SohCahToa

`color(magenta)("Soh" )->color(magenta)("s")"in" = (color(magenta)("o")"pposite")/(color(magenta)("h")"ypotenuse")`

`color(magenta)("Cah ")->color(magenta)("c")"os"=(color(magenta)("a")"djacent")/(color(magenta)("h")"ypotenuse")`

`color(magenta)("Toa ")->color(magenta)("t")"an"=(color(magenta)("o")"pposite")/(color(magenta)("a")"djacent")`

Answer 3

Underlying principles explained over 3 solution sheets

This is sheet 3 of 3

`color(blue)("Using the trig ratios")`

Explanation:

Soh`-> sin = ("opposite")/("hypotenuse")`

Cah`-> cos=("adjacent")/("hypotenuse")`

Toa`->tan=("opposite")/("adjacent")`

`color(blue)("What is the height of the given triangle if "theta = 30^o" ?")`

Observe that we need the height (0pposite) and that we have the size of the base (adjacent). So we need a ratio that includes these. Notice that this is the tangent (tan)

So this is Toa (form SohCahToa)

So the tangent `-> tan(theta) = ("opposite")/("adjacent")`

We observe that opposite is not given so we use a letter, say `x`
We observe that adjacent = 2.5
We observe that the angle `theta = 30^o`

So we end up with

`tan(theta)= x/2.5`

We look up the value of tan(30) and find that it is 0.5774 to 4 decimal places. We substitute this for `tan(theta)` in the equation giving

`color(brown)(tan(theta)=x/2.5) color(blue)(->0.5774=x/2.5)`

so `color(green)(0.5774/1=x/2.5)` There is our ratio again!

To get `x` on its own multiply both sides by `color(blue)(2.5)`

`color(green)(0.5774/1color(blue)(xx2.5)=x/2.5color(blue)(xx2.5))`

`1.4435=color(green)(x xx color(blue)(2.5)/2.5)`

But `2.5/2.5` is the same as 1

`1.4435=x xx 1`

`x=1.4435` to 4 decimal places

`color(blue)("The height of the triangle is 1.4435 to 4 decimal places")`

Answer 4

In basic trig we work with right-angled triangles.

Trig is all about COMPARING the lengths of the sides of triangles.
One way of comparing is by dividing. This will tell us if one side is 'half' as long as another, or one and a half times as long, and so on..

A well-known triangle has sides of 3cm, 4cm and 5 cm.

We can write down 6 different comparisons/divisions/fractions using these 3 values:

`3/5, 3/4, 4/3, 4/5, 5/3, 5/4`

Each of these comparisons has a different name.
Three of the names are sine, cosine and tangent, depending which sides you divided.

Sine is answer that you get when you divide the length of the side opposite an angle by the hypotenuse. It always gives the same answer for a particular angle, no matter what the size of the triangle is.

For example, Sine 30° = `1/2 =0.5 = 50%`

This tells us that for an angle of 30° in a triangle, the side opposite the 30° will always be half as long (50%) as the hypotenuse.

That is really useful to know.

These 'comparisons' are also known as the trig ratio's.