Question

Find all solutions of the equation in the interval [0,2π) sec(theta) +2=0

Answer 1

I will first reveal the most obvious secret of trig: Most of the problems are 30/60/90 or 45/45/90 triangles of some sort.


`\sec \theta + 2 = 0`


`(1 )/( \cos \theta) = -2`


`\cos \theta = - \frac 1 2`

The rule to remember is
`\cos x = \cos a` has solutions


`x = \pm a + 2 \pi k \quad` integer
`k`

Continuing where we left off, a cosine of -1/2 is a 30/60/90 triangle in the second or third quadrant; we pick second, and a little thought tells us the angle is
`120^\circ`.


`\cos \theta = - \frac 1 2`


`\cos \theta = \cos \frac {2\pi} 3`


`\theta = \pm \frac {2\pi} 3 + 2 \pi k \quad` integer
`k`

In the range we want, that's
`k=0` with the plus sign and
`k=1` with the minus, so


`\theta = \frac {2\pi} 3` or
`-\frac {2\pi} 3 + 2 \pi = \frac{4 \pi} 3`


`\theta = \frac {2\pi} 3` or
`\frac{4 \pi} 3`