1.1 Powers and Roots of Exponential and Trigonometric Functions
Exponential Functions: \(f(x) = a^x\), where \(a\) is a positive constant. When raising an exponential function to a power, use the property \(a^{x+y} = a^x \cdot a^y\). For roots of exponential functions, \(a^{1/n}\) represents the \(n\)th root of \(a\).
Trigonometric Functions: Trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions are periodic and have specific properties, e.g., \(\sin^2(\theta) + \cos^2(\theta) = 1\).
1.2 Expansion of \(\sin^n \theta\) and \(\cos^n \theta\)
To expand \(\sin^n \theta\) or \(\cos^n \theta\) in terms of sines and cosines of multiples of \(\theta\), you can use the binomial expansion. For \(\sin^n \theta\), use the binomial theorem to express it as a sum of \(\sin(k\theta)\) terms, where \(k\) ranges from 0 to \(n\).
Example: \(\sin^3 \theta = 3\sin\theta - 4\sin^3\theta\).
To expand \(\sin^n \theta\) or \(\cos^n \theta\) in powers of \(\sin \theta\) and \(\cos \theta\), you can use trigonometric identities and recursion.
Example: \(\cos^3 \theta = 4\cos^3 \theta - 3\cos \theta\).
1.3 Circular Functions of Complex Numbers and Hyperbolic Functions
Circular Functions of Complex Numbers: Complex numbers can be expressed in polar form as \(z = re^{i\theta}\), where \(r\) is the modulus and \(\theta\) is the argument. Circular functions in complex numbers include \(\sin(z)\) and \(\cos(z)\).
Example: \(\sin(z) = \sin(r)\cosh(i\theta) + i\cos(r)\sinh(i\theta)\).
Hyperbolic Functions: Hyperbolic functions include \(\sinh(x)\), \(\cosh(x)\), and \(\tanh(x)\). They are analogous to trigonometric functions but are defined using exponentials.
Inverse Circular and Inverse Hyperbolic Functions: Inverse circular functions like \(\arcsin(x)\), \(\arccos(x)\), and \(\arctan(x)\) find the angle corresponding to a given trigonometric value. Inverse hyperbolic functions like \(\sinh^{-1}x\), \(\cosh^{-1}(x)\), and \(\tanh^{-1}(x)\) do the same for hyperbolic functions.
Separation of Real and Imaginary Parts
To separate real and imaginary parts, apply trigonometric or hyperbolic functions to complex numbers, and then express the results in terms of real and imaginary parts.
Example: \(\sin(z) = \sinh(y)\cos(x) + i\cosh(y)\sin(x)\).
Sample Problems
- Expand \(\cos^4 \theta\) in terms of sines and cosines of multiples of \(\theta\).
- Compute the value of \(\sin(i\pi/3)\) using circular functions of complex numbers.
- Find the inverse hyperbolic function of \(\sinh(2)\).
- Separate the real and imaginary parts of \(\cos(3+4i)\).
