Question

sin theta 1+ sin theta 2+ sin theta 3 + sin theta 4=4Then Cos theta 1+ cos theta 2+ cos theta 3+ cos theta 4=

Answer 1

ANSWER:

Given:

• $\sin\theta_1+\sin\theta_2+\sin\theta_3+\sin\theta_4=4$

To Find:

• $\cos\theta_1+\cos\theta_2+\cos\theta_3+\cos\theta_4$

Solution:

We are given that,

$\implies\sin\theta_1+\sin\theta_2+\sin\theta_3+\sin\theta_4=4$

We know that, the range of sin function, is from [-1, 1], i.e., all values from -1 to 1, including -1 and 1.

So, if we take the maximum range, possible, for sine function, in this question then only we will get the answer as 4.

Because, the maximum result possible by adding 4 sin functions is 4 only, as all other results will give a value less than 4.

That is,

$\implies\sin\theta_1+\sin\theta_2+\sin\theta_3+\sin\theta_4=4$

$\implies1+1+1+1=4$

$\implies4=4$

Hence,

$\implies\sin\theta_1=\sin\theta_2=\sin\theta_3=\sin\theta_4=1$

But, we know that,

$\implies\sin90^{\circ}=1$

Hence,

$\implies\theta_1=\theta_2=\theta_3=\theta_4=90^{\circ}$

Now, we need to find the value of,

$\implies\cos\theta_1+\cos\theta_2+\cos\theta_3+\cos\theta_4$

Substituting the value of all angles,

$\implies\cos90^{\circ}+\cos90^{\circ}+\cos90^{\circ}+\cos90^{\circ}$

$\implies4(\cos90^{\circ})$

We know that,

$\implies\cos90^{\circ}=0$

So,

$\implies4(\cos90^{\circ})$

$\implies4(0)$

$\implies0$

Therefore,

$\implies\bf cos\theta_1+cos\theta_2+cos\theta_3+cos\theta_4=0$