Solve the following trigonometric equations, using the domain given?

Question

1. 2cos(x)+1=0 for 0 ≤ θ < 360° 
2. 2+4sin(x)=4 for 0 ≤ θ < 720° 
3. 4csc(x)+8=0 for 0 ≤ x< 2π 
4. 3tan(x)+√3=0 0 ≤ x < 4π

Answer 1

 1) The trigonometric equation is 2cos(x) + 1 = 0

2cos(x) = - 1

cos(x) = - 1/2

cos(x) is negative in second and third quadrant.

Find all solutions in 0 ≤ θ < 360°.

cos(120°) = cos(180° - 60°) = -1/2

cos(240°) = cos(180° + 60°) = -1/2

x = θ

The solutions are in the interval 0 ≤ θ < 360° are 120° and  240°.

Answer 2

 2) The trigonometric equation is 2 + 4 sin(x) = 4

4 sin(x) = 4 - 2

4 sin(x) = 2

sin(x) = 2/4

sin(x) = 1/2

sin(x) = sin(30°)

The general solution of sin(x) = sin(α) is x = 180°n+(-1)ⁿα, where n is an integer.

α = 30°

For n = 0, x = 180°(0)+(-1)⁰(30°) = 30°

For n = 1, x = 180°(1)+(-1)¹(30°) = 150°

For n = 2, x = 180°(2)+(-1)²(30°) = 390°

For n = 3, x = 180°(3)+(-1)³(30°) = 510°

For n = 4, x = 180°(4)+(-1)⁴(30°) = 750°

The solutions in the interval 0 ≤ θ < 720° are 30°, 150°, 390° and 510°.

Answer 3

 3) The trigonometric equation is 4csc(x) + 8 = 0

4csc(x) = - 8

csc(x) = - 8/4

csc(x) = - 2

csc(x) is negative in third and fourth quadrant.

Find all solutions in 0 ≤ θ < 2π.

csc(7π/6) = csc(π + π/6) = - 2

csc(11π/6) = cos(2π - π/6) = - 2

The solutions are in the interval 0 ≤ x < 2π are 7π/6 and  11π/6.

Answer 4

4) The trigonometric equation is 3tan(x) + √3 = 0

3tan(x) = - √3

tan(x) = (- √3)/3

tan(x) = (- √3)/(√3√3)

tan(x) = (- 1)/(√3)

tan(x) is negative in second and fourth quadrant.

Find all solutions in 0 ≤ x < 4π.

tan(5π/6) = csc(π - π/6) = -(1/√3)

tan(11π/6) = cos(2π - π/6) = -(1/√3)

tan(17π/6) = csc(3π - π/6) = -(1/√3)

tan(23π/6) = cos(4π - π/6) = -(1/√3)

The solutions are in the interval 0 ≤ x < 4π are 5π/6, 11π/6, 17π/6 and 23π/6.