trigonometry

All about the unit circle and the trigonometric functions.

In a right angled triangle with an angle, all three sides has a relation to the angle: hypotenuse, opposite and adjacent.

A point on the unit circle that corresponds to an angle $\theta$ must have the coordinates $(\cos (\theta ),\sin (\theta ))$.

An angle can be measured in degrees or radians… Since an angle without the degree symbol shown is radians by default, it is always important to add the degrees symbol when you use degrees.

A full revolution of a circle is 360 degrees ($360°$) , and is equal to $2\pi$ radians ($2{\pi }^c$). use this to convert between radians and degrees!

In a unit circle, a radian is the angle you get if you follow the circumference for 1 unit.

how to evaluate trig functions
how to graph trig functions
how to evaluate reciprocal trig functions
how to graph reciprocal trig functions
how to evaluate inverse trig functions
how to graph inverse trig functions

trig identities

trig identities are just equations that can help you evaluate trig expressions

essentially, trig identities provide us the ability to convert between sine and cosine and to express a trig expression in different ways

trig identity

by forming a right angle triangle on a unit circle and using Pythagorean theorem, we get the Pythagorean identity: ${\sin }^2(a)+{\cos }^2(a)=1$

the Pythagorean identity can be rearranged to be very useful
${\sin }^2(a)=1-{\cos }^2(a)=(1-\cos (a))(1+\cos (a))$
${\cos }^2(a)=1-{\sin }^2(a)=(1-\sin (a))(1+\sin (a))$

example problem: in the first quadrant, if $\sin (x)=\frac{4}{5}$, find $\cos (x)$ and $\tan (x)$
note that $x$ in this problem is an angle and we don’t need to know what it is at all
apply the Pythagorean identity
$\frac{16}{25}+{\cos }^2(x)=1$
${\cos }^2(x)=1-\frac{16}{25}$
${\cos }^2(x)=\frac{9}{25}$
$\cos (x)=\frac{3}{5}$

$\tan (x)=\frac{4}{3}$

example problem: in the fourth quadrant, if $\cos (x)=\frac{5}{13}$, find $\sin (x)$ and $\tan (x)$

we apply the Pythagorean identity, but because this is not the first quadrant we need to consider sign
remember the sign convention table? in the fourth quadrant, the cosine is positive and the sine and tangent are negative

$\frac{25}{169}+{\sin }^2(x)=1$
${\sin }^2(x)=\frac{144}{169}$
$\sin (x)=\frac{12}{13}$

in the fourth quadrant, sine is negative

$\sin (x)=-\frac{12}{13}$

$\tan (x)=-\frac{12}{5}$