Students get through Maths 1A Important Questions Inter 1st Year Maths 1A Hyperbolic Functions Important Questions which are most likely to be asked in the exam.

## Intermediate 1st Year Maths 1A Hyperbolic Functions Important Questions

Question 1.
Prove that for any x ∈ R, sinh (3x) = 3 sinh x + 4sinh x
LHS = sinh (3x)
= sinh (2x + x)
= sinh (2x) . cosh(x) + cosh (2x) . sinh (x) = (2sinh x cosh x)cosh x (1 +2sinh2 x)sinh x
= 2 sinh x (cosh2 x) + (1 + 2 sinh2 x) sinh x
= 2 sinh x (1 + sinh2 x) + (1 + 2 sinh2 x) sinh x
∵ cosh2 x – sinh2 x = 1
= 3 sinh x + 4 sinh3 x
∴ sinh (3x) = 3 sinh x + 4 sinh3 x

Question 2.
Prove that for any x ∈ R, tanh 3x = $$\frac{3 \tanh x+\tanh ^{3} x}{1+3 \tanh ^{2} x}$$
tanh 3x = tan (2x + x)

Question 3.
If cosh x = $$\frac{5}{2}$$, find the values of
i) cosh (2x) and
ii) sinh (2x)
cosh (x) = $$\frac{5}{2}$$
(i) cosh (2x) = 2 cosh2 (x) – 1
= 2($$\frac{5}{2}$$)2 – 1 = $$\frac{25}{2}$$ – 1 = $$\frac{23}{2}$$

ii) sinh2 (2x) = cosh2 (2x) – 1
= ($$\frac{23}{2}$$)2 – 1 = $$\frac{529-4}{2}$$ = $$\frac{525}{4}$$
∴ sinh (2x) = ±$$\sqrt{\frac{525}{4}}$$ = ±$$\frac{5 \sqrt{21}}{2}$$

Question 4.
If coshx = sec θ then prove that tan h2$$\frac{x}{2}$$ = tan2$$\frac{\theta}{2}$$
tan h2$$\frac{x}{2}$$ = $$\frac{\cosh x-1}{\cosh x+1}$$
= $$\frac{\sec \theta-1}{\sec \theta+1}$$ = $$\frac{1-\cos \theta}{1+\cos \theta}$$ = tan2$$\frac{\theta}{2}$$

Question 5.
If θ ∈ (-$$\frac{\pi}{4}$$, $$\frac{\pi}{4}$$) and x = loge(cot($$\frac{\pi}{4}$$ + θ) then prove that
i) cosh x = sec 2θ and
ii) sinh x = -tan 2θ

Question 6.
If sinh x = 5, show that x = loge (5 + $$\sqrt{26}$$)
= loge (5 + $$\sqrt{5^{2}+1}$$)
= loge (5 + $$\sqrt{26}$$)
[sin-1 (x) = loge (x + $$\sqrt{x^{2}+1}$$) for all x ∈ R]
Show that tanh-1($$\frac{1}{2}$$) = $$\frac{1}{2}$$ loge3 (A.P) [Mar 15; May 07, 05; Mar 08, 05]
∵tanh-1 (x) = $$\frac{1}{2}$$loge($$\frac{1+x}{1-x}$$) for all x ∈ (-1, 1)
∵ tanh-1 ($$\frac{1}{2}$$) = $$\frac{1}{2}$$loge($$\frac{1+\frac{1}{2}}{1-\frac{1}{2}}$$)
= $$\frac{1}{2}$$loge(3)