How do you use the inverse functions where needed to find all solutions of the equation sec^2x+tanx-3=0 in the interval [0,2pi)?

1 Answer
Nghi N.
Jun 3, 2018

45^@; 116^@57;225^@; 296^@57

Explanation:

1/cos^2 x + tan x - 3 = 0 (1)
Reminder of trig identity:
cos^2 x = 1/(1 + tan^2 x) --> 1/cos^2 x = 1 + tan^2 x
We get from equation (1):
1 + tan^2 x + tan x - 3 = 0
tan^2 x + tan x - 2 = 0.
Solve this quadratic equation for tan x.
Since a + b + c = 0, use shortcut. The 2 real roots are:
tan x = 1 and tan x = c/a = - 2
a. tan x = 1
Trig table and Unit circle give -->
x = pi/4 and x = pi + pi/4 = (5pi)/4
b. tan x = - 2
Calculator and unit circle give -->
x = - 63^@43, and x = -63.43 + 180 = 116^@57
Note. x = -63.43 is co-terminal to x = 360 - 63.43 = 296.57.
Answers for (0, 360):
45^@; 116^@57; 225^@; 296^@57

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