Question

Can you solve `sin^2theta+sintheta-1=0` by factoring or quadratic formula?

Answer 1

`38^@17 + k360^@`
`141^@83 + k360^@`

Explanation:

`sin^2 t + sin t - 1 = 0`
Solve this quadratic equation for sin t by using the improved quadratic formula (Socratic Search).
`D = d^2 = b^ - 4ac = 1 + 4 = 5` --> `d = +- sqrt5`
There are 2 real roots:
`sin t = -b/(2a) +- d/(2a) = - 1/2 +- sqrt5/2 = (-1 +- sqrt5)/2`
a. `sin t = (- 1 - sqrt5)/2 = -3.236/2 = - 1.62` (rejected as < - 1)
b. `sin t = (-1 + sqrt5)/2 = (1.236)/2 = 0.618`
Calculator gives -->
sin t = 0.618 --> `t = 38^@ 17 + k360^@`
Unit circle gives another t that has the same sine value (0.618):
`x = 180 - 38.17 = 141^@83 + k360^@`