Pre-Calculus 40S Winter 2013 Section DB: May 2013

Trigonometric Identities

Reciprocal identities

Quotient Identities

Co-Function Identities

Pythagorean Identities

Sum and Difference Formulas

Double Angle Formulas

Function	Abbreviation	Description	Trigonometric Identities using Radians
Sine	sin	$\frac {\textrm{opposite}} {\textrm{hypotenuse}}$	$\sin \theta = \cos \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\csc \theta}$
Cosine	cos	$\frac {\textrm{adjacent}} {\textrm{hypotenuse}}$	$\cos \theta = \sin \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\sec \theta}\,$
Tangent	tan	$\frac {\textrm{opposite}} {\textrm{adjacent}}$	$\tan \theta = \frac{\sin \theta}{\cos \theta} = \cot \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cot \theta}$
Cotangent	cot	$\frac {\textrm{adjacent}} {\textrm{opposite}}$	$\cot \theta = \frac{\cos \theta}{\sin \theta} = \tan \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\tan \theta}$
Secant	sec	$\frac {\textrm{hypotenuse}} {\textrm{adjacent}}$	$\sec \theta = \csc \left(\frac{\pi}{2} - \theta \right) =\frac{1}{\cos \theta}$
Cosecant	csc	$\frac {\textrm{hypotenuse}} {\textrm{opposite}}$	$\csc \theta = \sec \left(\frac{\pi}{2} - \theta \right) =\frac{1}{\sin \theta}$

Non-permissible values

Sin x/Cos x Cos x cannot be equal to zero

Therefore the non-permissible values of Sin x/Cos x is where Cos x is equal to zero on the unit circle.

Cos x is equal to zero at [ (0,1) 90°   π/2 ] And [ (0,-1)   270° 3π/2   ]

general formula for non-permissible values: X ≠   π/2 +   πn , n  ∈ I

Tan x/Sec x Tan x/Csc x=   Sin x/Cos x / 1/Sin x

Cos x cannot be equal to zero

Sin x cannot be equal to zero

Therefore the non-permissible values of Tan x/Sec x is where cos x and sin x are equal to zero.

Cos x is equal to zero at [ (0,1) 90° π/2 ] And [ (0,-1) 270° 3π/2 ]

Sin x is equal to zero at [ (1,0) 0°, 360° 0,2π ] And [ (-1,0) 180° π ]