قوانيين النسب المثلثية

قوانيين النسب المثلثية                                قوانيين النسب المثلثية 

Sine Function:	sin(θ) = Opposite / Hypotenuse
Cosine Function:	cos(θ) = Adjacent / Hypotenuse
Tangent Function:	tan(θ) = Opposite / Adjacent

For a given angle θ each ratio stays the same
no matter how big or small the triangle is

When we divide Sine by Cosine we get:

So we can say:

tan(θ) = sin(θ)/cos(θ)

That is our first Trigonometric Identity.

Cosecant, Secant and Cotangent

We can also divide "the other way around" (such as Adjacent/Opposite instead of Opposite/Adjacent):

Cosecant Function:	csc(θ) = Hypotenuse / Opposite
Secant Function:	sec(θ) = Hypotenuse / Adjacent
Cotangent Function:	cot(θ) = Adjacent / Opposite

Example: when Opposite = 2 and Hypotenuse = 4 then

sin(θ) = 2/4, and csc(θ) = 4/2

Because of all that we can say:

sin(θ) = 1/csc(θ)

cos(θ) = 1/sec(θ)

tan(θ) = 1/cot(θ)

And the other way around:

csc(θ) = 1/sin(θ)

sec(θ) = 1/cos(θ)

cot(θ) = 1/tan(θ)

And we also have:

cot(θ) = cos(θ)/sin(θ)

Pythagoras Theorem

For the next trigonometric identities we start with Pythagoras' Theorem:

The Pythagorean Theorem says that, in a right triangle, the square of a plus the square of b is equal to the square of c: a2 + b2 = c2

Dividing through by c² gives

a²c² = b²c² = c²c²

This can be simplified to:

(ac)² + (bc)² = 1

Now, a/c is Opposite / Hypotenuse, which is sin(θ)

And b/c is Adjacent / Hypotenuse, which is cos(θ)

So (a/c)² + (b/c)² = 1 can also be written:

sin² θ + cos² θ = 1

Note:

• sin² θ means to find the sine of θ, then square the result, and
• sin θ² means to square θ, then do the sine function

Example: 32°

Using 4 decimal places only:

• sin(32°) = 0.5299...
• cos(32°) = 0.8480...

Now let's calculate sin² θ + cos² θ:

0.5299² + 0.8480²
= 0.2808... + 0.7191...
= 0.9999...

We get very close to 1 using only 4 decimal places. Try it on your calculator, you might get better results!

Related identities include:

sin² θ = 1 − cos² θ
cos² θ = 1 − sin² θ
tan² θ + 1 = sec² θ
tan² θ = sec² θ − 1
cot² θ + 1 = csc² θ
cot² θ = csc² θ − 1

How Do You Remember Them? The identities mentioned so far can be remembered using one clever diagram called the Magic Hexagon:

But Wait ... There is More!

There are many more identities ... here are some of the more useful ones:

Opposite Angle Identities

sin(−θ) = −sin(θ)

cos(−θ) = cos(θ)

tan(−θ) = −tan(θ)

Double Angle Identities

                       

sin(−θ) = −sin(θ)

cos(−θ) = cos(θ)

tan(−θ) = −tan(θ)