Operator notations Gradient: In the three-dimensional Cartesian coordinate variables, the gradient of some function f(x,y,z) is given by: grad(f)=∇f=(∂∂x,∂∂y,∂∂z)f=∂f∂xi+∂f∂yj+∂f∂zk where i, j, k are the standard unit vectors for the…
Negative angle identities 1. sin(−t)=−sin(t) 2. cos(−t)=cos(t) 3. tan(−t)=−tan(t) 4. cot(−t)=−cot(t) 5. sec(−t)=sec(t) 6. cosec(−t)=−cosec(t) Example 1: What is \(\cos \left(-\frac {3\pi }{4}\right) \)? Solution: cos-3π4=cos3π4=-22 Example 2: What is \(\sin \left(-\frac {\pi }{2}\right) \)? Solution:…
Allied angles table \(\alpha\) \(\sin \alpha\) \(\cos \alpha\) \(\tan \alpha\) \(\cot \alpha\) \(\sec \alpha\) \(\text{cosec} \alpha\) \(-\theta\) \(-\sin \theta\) \(+\cos \theta\) \(-\tan \theta\) \(-\cot \theta\) \(+\sec \theta\) \(-\text{cosec} \theta\)…
Euler's formula $$ 1. \ \sin \theta = \frac {1}{2i} \left( e^{i \theta } - e^{-i \theta } \right) $$ $$ 2. \ \cos \theta = \frac {1}{2} \left( e^{i…
Powers of functions $$ 1. \ \sin^2 α= \frac {(1 − \cos 2α)}{2} $$ $$ 2. \ \sin^3 α= \frac {(3 \sin α – \sin 3α)}{4} $$ $$ 3. \…
Product-to-Sum Identities 1. sinα·sinβ=12cos(α–β)–cos(α+β) 2. cosα·cosβ=12cos(α–β)+cos(α+β) 3. sinα·cosβ=12sin(α+β)+sin(α–β) 4. cosα·sinβ=12sin(α+β)–sin(α–β) 5. tanα·tanβ=(tanα+tanβ)(cotα+cotβ) 6. cotα·cotβ=(cotα+cotβ)(tanα+tanβ) 7. tanα·cotβ=(tanα+cotβ)(cotα+tanβ) Example: Write cos 3x cos 2x as a sum Solution: cosα·cosβ=12[cos(α–β)+cos(α+β)] cos3x·cos2x=12[cos(3x–2x)+cos(3x+2x)] cos3x·cos2x=12[cosx+cos5x] cos3x·cos2x=cosx2+cos5x2
Sum-to-Product Identities 1. Sum of sines sinα+sinβ=2sinα+β2cosα-β2 \sin \alpha + \sin \beta = 2 \sin \frac {\alpha + \beta}{2} \cos \frac {\alpha - \beta}{2} 2. Difference of sines sinα-sinβ=2cosα+β2sinα-β2 \sin…
Half/Double/Multiple angle formula Half angle formula: $$ 1.\ \sin \frac A2 = \pm \sqrt \frac{(1- \cos A)}{2} $$ $$ 2.\ \cos \frac A2 = \pm \sqrt \frac{(1+ \cos A)}{2} $$…
Angle transformation 1. sin(A+B)=sinAcosB+cosAsinB 1. \ \sin(A + B) = \sin A \cos B + \cos A \sin B 2. sin(A−B)=sinAcosB−cosAsinB 2. \ \sin(A − B) = \sin A \cos B −…
Table of Angle Angle (in Degrees) 0° 30° 45° 60° 90° 180° 270° 360° Angle (in Radians) 0 π/6 π/4 π/3 π/2 π 3π/2 2π Sin 0 1/2 1/√2 √3/2…