2cos (3t) = -1

Question

Help!!

Find all solutions.

Please show steps.

 

Thank you!

Answer 1

cos(3t) = -1/2

 

cosine is negative in quadrants 2 and 3

 

3t = 2π/3 + 2kπ or 4π/3 + 2kπ, k = 0, ±1, ±2, ...

 

t = 2π/9 + (2/3)kπ, or 4π/9 + (2/3)kπ, k = 0, ±1, ±2, ...

 

If k = 0, we get t = 2π/9 or 4π/9

If k = 1, we get t = 8π/9 or 10π/9

If k = 2, we get t = 14π/9 or 16π/9

 

Other choices of k give values of t that lie outside of the interval [0, 2π].  

 

So, there are 6 solutions between 0 and 2π. 

 

 

Answer 2

Hi Anna,

 

First, you have to divide by 2 on both sides to make

 

cos (3t) = -1/2

 

From here, you change 3t to u (to make equation simpler).  In other words, u = 3t.

 

And then, find where cos values are (-).  In this case, it will be in 2nd & 3rd quadrants of unit circle.

 

And then, find when cos is -1/2  (x value from x,y) : 2π/3 and 4π/3.

 

 

 

 

So, u = 2π/3 and 4π/3.

 

u = 3t.  To find t, you divide by 3 (u/3 = t) and to its values (2π/3 and 4π/3).  At the end, you will get:

 

t = 2π/9 and 4π/9.

 

I hope you got it.  Good luck to you.

 

 

Yohan Chu   (LV  NV)