# How do you express trigonometric expressions in simplest form?

Apr 12, 2015

Examples of trig expressions:  f(x) = sin 2x + cos x;
$f \left(x\right) = \sin x + \sin 2 x + \sin 3 x$
Examples of trig equations: $f \left(x\right) = \sin 2 x + \cos x = 0$
$f \left(x\right) = \sin x + \sin 2 x + \sin 3 x = 0$
Examples of trig inequalities $f \left(x\right) = \sin 2 x + \cos x > 0$
$f \left(x\right) = \sin x + \sin 2 x + \sin 3 x < 0$

Use trig Transformation Identities to transform these above trig expressions into trig basic expressions, or expressions in simplest form.
Example: Transform $f \left(x\right) = \sin 2 x + \cos x$. Use Identity $\left(\sin 2 a = 2 \sin a \cdot \cos a\right)$ to transform $f \left(x\right) .$
$f \left(x\right) = 2 \cdot \sin x \cdot \cos x + \cos x = \cos x \cdot \left(2 \sin x + 1\right)$
This is f(x) expressed in simplest form.
Trig equation in simplest form: $f \left(x\right) = \cos x \cdot \left(\sin 2 x + 1\right) = 0$
Trig inequality in simplest form: $f \left(x\right) = \cos x \cdot \left(2 \sin x + 1\right) > 0$