Help on Trigonometric sums. (Assorted type)

[surajkrishna]

I have some questions and doubts in trigonometry. I hope somebody can solve these questions.

Q1) If for real values of x, cos$$\theta = x +\frac{1}{x}$$, then

a) $$\theta$$ is acute angle b) $$\theta$$ is right angle c) $$\theta$$ is an obtuse angle d) no value of $$\theta$$ is possible

I will post the following questions soon.

 

[surajkrishna]

Q2) If sec$$\theta$$ + tan$$\theta$$ = x, then sin$$\theta$$ is equal to

a) x²+1/2x b) x²+1/x²-1 c) x²-1/x²+1 d) 2x/x²​-1

 

[chisigma]

I have some questions and doubts in trigonometry. I hope somebody can solve these questions.

Q1) If for real values of x, cos$$\theta = x +\frac{1}{x}$$, then

a) $$\theta$$ is acute angle b) $$\theta$$ is right angle c) $$\theta$$ is an obtuse angle d) no value of $$\theta$$ is possible

I will post the following questions soon.

The function $\displaystyle f(x)=x+\frac{1}{x}$...

a) for x>0 f(x) is greater than 0 and it has a minimum for x=1, where is f(x)=2...

b) for x<0 f(x) is less than 0 and it has a maximum for x=-1, where is f(x)=-2...

Consequence of a) and b) is that it doesn't exist any real $\theta$ for which is $\displaystyle \cos \theta= x+\frac{1}{x}$...

Kind regards

$\chi$ $\sigma$

 

[chisigma]

Q2) If sec$$\theta$$ + tan$$\theta$$ = x, then sin$$\theta$$ is equal to

a) x²+1/2x b) x²+1/x²-1 c) x²-1/x²+1 d) 2x/x²​-1

$\displaystyle \frac{1}{\cos \theta}+ \frac{\sin \theta}{\cos \theta}=x \implies \frac{1+ \sin \theta}{\sqrt {1 - \sin^{2} \theta}}+ x \implies \sqrt{\frac{1+\sin \theta}{1-\sin \theta}}=x \implies \sin \theta = \frac{x^{2}-1}{x^{2}+1}$

Kind regards

$\chi$ $\sigma$

 

[surajkrishna]

$\displaystyle \frac{1}{\cos \theta}+ \frac{\sin \theta}{\cos \theta}=x \implies \frac{1+ \sin \theta}{\sqrt {1 - \sin^{2} \theta}}+ x \implies \sqrt{\frac{1+\sin \theta}{1-\sin \theta}}=x \implies \sin \theta = \frac{x^{2}-1}{x^{2}+1}$

Kind regards

$\chi$ $\sigma$

$\displaystyle \frac{1+ \sin \theta}{\sqrt {1 - \sin^{2} \theta}}= x $

After this how do you get this:

$\sqrt{\frac{1+\sin \theta}{1-\sin \theta}}=x \implies \sin \theta = \frac{x^{2}-1}{x^{2}+1}$

 

[chisigma]

$\displaystyle \frac{1+ \sin \theta}{\sqrt {1 - \sin^{2} \theta}}= x $

After this how do you get this:

$\sqrt{\frac{1+\sin \theta}{1-\sin \theta}}=x \implies \sin \theta = \frac{x^{2}-1}{x^{2}+1}$

$\displaystyle \frac{1+ \sin \theta}{\sqrt {1 - \sin^{2} \theta}}= x \implies \sqrt{\frac{(1+\sin \theta)^{2}}{(1+\sin \theta)\ (1-\sin \theta)}} = x \implies \sqrt{\frac{1+ \sin \theta}{1-\sin \theta}}=x \implies 1 + \sin \theta= x^{2}\ (1-\sin \theta) \implies \sin \theta= \frac{x^{2}-1}{x^{2}+1} $

Kind regards

$\chi$ $\sigma$

 

[surajkrishna]

Thanks for the help.