2-Trigonometry

= Trigonometry =

Revise
 * Exact Values
 * Using 4 Quadrants
 * Solving Trig Equations
 * Sequence Notation
 * Sine Rule
 * Cosine Rule
 * Rules for Area of Triangles

2A Reciprocal Trig Functions


 * The reciprocal trig functions are defined as:

math \\ . \qquad \qquad \text{Cosecant:} \quad \text{cosec } \theta = \dfrac{1}{sin \, \theta} = \dfrac {HYP}{OPP} \\. \\ . \qquad \qquad \text{Secant:} \qquad \quad \text{sec } \theta = \dfrac{1}{cos \, \theta} = \dfrac {HYP}{ADJ} \\. \\ . \qquad \qquad \text{Cotangent}: \quad \text{cot } \theta = \dfrac{1}{tan \, \theta} = \dfrac {ADJ}{OPP} math

2B Graphs of Reciprocal Trig Functions


 * The graphs of the reciprocal trig functions can be obtained by
 * drawing the graphs of the original trig functions
 * and taking the reciprocal of the ordinates.

2C Trig Identities


 * The pythagorean trig identities are:

math . \qquad \qquad \sin^2 \theta + \cos^2 \theta = 1 math

math . \qquad \qquad \tan^2 \theta + 1 = \sec^2 \theta \qquad \; \theta \in R \backslash \Big\{ \dfrac {\pi}{2} + n\pi, \; n \in Z \Big\} math

math . \qquad \qquad 1 + \cot^2 \theta = \text{cosec}^2 \theta \qquad \theta \in R \backslash \{ n\pi, \; n \in Z \} math

2D Compound and Double Angle Formulas


 * The compound angle formulas are:

... ... ... ... sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

... ... ... ... sin(A – B) = sin(A)cos(B) – cos(A)sin(B)

and

... ... ... ... cos(A + B) = cos(A)cos(B) – sin(A)sin(B)

... ... ... ... cos(A – B) = cos(A)cos(B) + sin(A)sin(B)

and

math \\ . \qquad \qquad \tan(A+B) = \dfrac {\tan(A) + \tan (B)}{1 - \tan(A)\tan(B)} \\. \\ . \qquad \qquad \tan(A-B) = \dfrac {\tan(A) - \tan (B)}{1 + \tan(A)\tan(B)} math


 * The double angle formulas are:

... ... ... ... sin(2A) = 2sin(A)cos(A)

and

... ... ... ... cos(2A) = cos 2 (A) – sin 2 (A)

... ... ... ... cos(2A) = 2cos 2 (A) – 1

... ... ... ... cos(2A) = 1 – 2sin 2 (A)

and

math . \qquad \qquad \tan(2A) = \dfrac {2\tan(A)}{1-\tan^2(A)} math

2E Inverse Trig Functions


 * Be careful not to confuse Inverse Trig Functions with Reciprocal Trig Functions :


 * ** The trig functions, within the __Principal Domains__, are defined as: **
 * ** Note the capital letters **

math . \qquad \qquad y=Sin(x), \; x \in \left[ -\dfrac{\pi}{2}, \; \dfrac{\pi}{2} \right] \quad y \in [-1, \, 1] math

and

math . \qquad \qquad y=Cos(x), \; x\in [0, \, \pi] \quad y\in [-1, \, 1] math

and

math . \qquad \qquad y=Tan(x), \; x \in \left( -\dfrac {\pi}{2}, \; \dfrac {\pi}{2} \right) \quad y \in R math


 * ** The Inverse Trig Functions are therefore: **

math . \qquad \qquad y=Sin^{-1}(x), \; x \in [-1, \, 1] \quad y \in \left[ -\dfrac{\pi}{2}, \; \dfrac{\pi}{2} \right] math

and

math . \qquad \qquad y=Cos^{-1}, \; x \in [-1, \,1] \quad y \in [0, \, \pi] math

and

math \\ . \qquad \qquad y=Tan^{-1}(x), \; x \in R \quad y \in \left( -\dfrac{\pi}{2}, \; \dfrac{\pi}{2} \right) \\. \\ . \qquad \qquad \text {Asymptotes } \;\; y=-\dfrac{\pi}{2}, \;\; y=\dfrac{\pi}{2} math


 * ** Notice that the domains and ranges have been reversed from Sin, Cos and Tan **

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