TS Inter 1st Year Maths 1A Hyperbolic Functions Formulas

Learning these TS Inter 1st Year Maths 1A Formulas Chapter 9 Hyperbolic Functions will help students to solve mathematical problems quickly.

TS Inter 1st Year Maths 1A Hyperbolic Functions Formulas

→ Hyperbolic Functions:

Hyperbolic Function	Definition	Domain	Range
1. sin hx	\(\frac{e^x-e^{-x}}{2}\)	R	R
2. cos hx	\(\frac{\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{x}}}{2}\)	R	[1, ∞)
3. tan hx	\(\frac{e^x-e^{-x}}{e^x+e^{-x}}\)	R	(-1, 1)
4. cot hx	\(\frac{e^x+e^{-x}}{e^x-e^{-x}}\)	R – {0}	(-∞, -1] ∪ [1, ∞)
5. sec hx	\(\frac{2}{e^x+e^{-x}}\)	R	[0, 1]
6. cosec hx	\(\frac{2}{e^x-e^{-x}}\)	R – {0}	R – {0}

→ Inverse Hyperbolic Functions:

Inverse Hyperbolic Function	Definition	Domain	Range
1. sin h-1x	loge(x + \(\sqrt{x^2+1} \))	R	R
2. cos h-1x	loge(x + \(\sqrt{x^2-1} \))	[1, ∞)	[0, ∞)
3. tan h-1x	\(\frac{1}{2}\)loge\( \left(\frac{1+x}{1-x}\right) \)	(-1, 1)	R
4. cot h-1x	\(\frac{1}{2}\)loge\( \left(\frac{x+1}{x-1}\right) \)	R – [-1, 1]	R – {0}
5. sec h-1x	loge\( \left(\frac{1+\sqrt{1-x^2}}{x}\right) \)	(0, 1]	[0, ∞)
6. cosec h-1x	loge\( \left(\frac{1 \pm \sqrt{1+x^2}}{x}\right) \)	R – {0}	R – {0}

→ Hyperbolic Identities:

• cosh²x – slnh²x = 1
• sech²x – tanh²x = 1
• coth²x – cosech²x = 1
• sinh(2x) = 2sinhx coshx
• cosh(2x) = cosh²x + sinh²x = 1 + 2sinh²x = 2cosh²x – 1

→ sinh (- x) = – sin hx

→ cosh (- x) = cosh x

→ tanh(-x)= -tanhx

→ coth(-x) = -coth x

→ cosech(-x) = -cosech x

→ sech(-x) = sech x

→ sinh (x + y) = shih x . cosh y + cosh x sin h y

→ cosh (x + y) = cosh x. cosh y + sin h x sinb y

→ sinh(x – y) sinhx.cosh y – cosh x sinh y

→ cosh(x – y)=coshx.coshy – sinhxsinhy

→ tanh (x + y) = \(\frac{\tanh x+\tanh y}{1+\tanh x \tanh y}\)

→ tanh (x – y) = \(\frac{\tanh x-\tanh y}{1-\tanh x \tanh y}\)

→ sinh3x = 3sinhx + 4sinh³x

→ cosh3x = 4cosh³x – 3coshx

→ tanh3x = \(\frac{3 \tanh x+\tanh ^3 x}{1+3 \tanh ^2 x}\)