Show that ##2 {\cos \theta} +(1- \tan \theta)^2≈ 3 - 2\theta##

[chwala]

Homework Statement

Show that $2 \cos \theta +(1- \tan \theta)^2≈ 3 - 2\theta$

Relevant Equations

trigonometry

This is a past paper question...

find the solution here from ms

...wawawawawa...it took a little bit of my time because i was only thinking of taking limits...then realized that i was wrong as we have to remain with $2θ$...i later realized that for small approximations, $\cos θ≈1-\frac {1}{2}θ^2, \tan θ≈θ$.

Is there a different way of solving this problem?

 

[anuttarasammyak]

Term order of 1 comes as 2+1=3
Coefficient of $\theta$ comes as 0-2=-2
Coefficient of $\theta^2$ comes as -1+1=0
Coefficient of $\theta^3$ comes as 0+0=0
and so on.

When we write the formula
[2\ cos\theta+\sec^2\theta]+[-2\tan\theta]
=[even function of $\theta$]+[odd function of $\theta$]
All the even power terms come from the first term. All the odd power terms come from the second term and it is easy to get.

 

[chwala]

This is the continuation of the question...part (b)... i really am not getting how they are solving this...the angles are in radians...approximation is their approach...

solution;

any better way?

 

[Mark44]

any better way?

I doubt it. In this part, as in the first part, they're assuming that $\theta$ is reasonably small, so $\sin(\theta) \approx \theta$.

Using the result of the first part of the problem, we have $3 - 2 \theta = 28\sin(\theta) \approx 28\theta$. The result follows almost immediately.

 

[chwala]

I doubt it. In this part, as in the first part, they're assuming that $\theta$ is reasonably small, so $\sin(\theta) \approx \theta$.

Using the result of the first part of the problem, we have $3 - 2 \theta = 28\sin(\theta) \approx 28\theta$. The result follows almost immediately.

Noted ...cheers Mark.

 

[SammyS]

This is the continuation of the question...part (b)... i really am not getting how they are solving this...the angles are in radians...approximation is their approach...

View attachment 297267

It would have been helpful to include, in Post #1, a statement about θ being a small angle.