Solving Trigonometric Equation. Rewrite the equation 3cscx - sinx = 2 in terms of sine. Then, solve algebraically for 0 < x < 2pi?

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2 Answers
Martin C.
Apr 2, 2018

x_(1_1)=pi/2 or x_(1_2)=-7/4pi

Explanation:

3cscx - sinx = 2|*sin(x)
3-sin(x)^2=2sin(x)|+sin(x)^2|-3
0=sin(x)^2+2sin(x)+1-1-3
0=(sin(x)+1)^2-4|+4
4=(sin(x)+1)^2|sqrt()
+-2=sin(x)+1|-1
-1+-2=sin(x)

arcsin(-1+-2)=x
x_1=arcsin(1)+2pin, n∈ZZ
x_1=pi/2+2pin, n∈ZZ

x_2=arcsin(-3)+2pin
arcsin(-3) " can't be expressed with real numbers."

arcsin(-3) and x_2 ∈CC

"For " 0 < x < 2pi:
x_(1_1)=pi/2+2pi*0=pi/2 or x_(1_2)=pi/2-1*2pi=-7/4pi

Nghi N
Apr 3, 2018

pi/2

Explanation:

3csc x - sin x = 2
3/(sin x) - sin x = 2
3 - sin^2 x = 2sin x
Solve this quadratic equation for sin x
sin^2 x + 2sin x - 3 = 0
Since a + b + c = 0, use shortcut. There are 2 real roots:
sin x = 1 , and sin x = c/a = - 3 (rejected)
Answer -->
sin x = 1 --> x = pi/2

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