Trigonometry 1 & 2

Basic Trig Identities and Angle Values

Here are the basics of Trigonometry 2. You will have to learn this before proceeding into the advanced stuff.


Advanced Trig Identities

Pythagoras states that a²+b²=c²


we know that "a" is the opposite of the triangle and "b" is the adjacent of the triangle and "c" is the hypotenuse. Lets suppose that X is any angle inside this right angle triangle beside the 90 degree angle.

Using trig identities from the previous section we know that opposite = sin(X) * hypotenuse and adjacent = cos(X) * hypotenuse

${(\sin (x) \times hypotenuse)^2} + {(\cos (x) \times hypotenuse)^2} = hypotenus{e^2}$

If we divide Hypotenuse^2 on both side we got this

${\sin ^2}(x) + {\cos ^2}(x) = 1$

If we divide both side with

${\sin ^2}(x)$

We will have


$1 + {\cot ^2}(x) = {\csc ^2}(x)$


If we divide it with


${\cos ^2}(x)$


We will have

${\tan ^2}(x) + 1 = {\sec ^2}(x)$

This are the few variations we can obtain from this

Two Angle Rule

Because

$\sin (a + b) \ne \sin a + \sin b$

so clever mathematicians find an alternative to find it which is the following

$\sin (a \pm b) = \sin a\cos b \pm \sin b\cos a$

They did the same thing with cos and this is what happened

$\cos (a \pm b) = \cos a\cos b \mp \sin a\sin b$

As we know

$\tan (x) = \sin x/\cos x$

so we can deduce that

$\tan (a \pm b) = \frac{{\sin a\cos b \pm \sin b\cos a}}{{\cos a\cos b \mp \sin a\sin b}} = \frac{{\sin a \pm \tan b\cos a}}{{\cos a \mp \sin a\tan b}} = \frac{{\tan a \pm \tan b}}{{1 \mp \tan a\tan b}}$

These are the three "two angle rules"