Solve the trigonometric equation from 0¡ã¡Üx¡Ü360

tan⁡x+¡Ì3=0

To solve the trigonometric equation tan(x) + √3 = 0 for 0° ≤ x ≤ 360°, we can follow these steps:

Step 1: Recall the trigonometric identity for tangent:
tan(x) = sin(x) / cos(x)

Step 2: Substitute the values of sin(x) and cos(x) into the equation:
sin(x) / cos(x) + √3 = 0

Step 3: Multiply the entire equation by cos(x) to eliminate the denominator:
sin(x) + √3 * cos(x) = 0

Step 4: Rearrange the equation:
√3 * cos(x) = -sin(x)

Step 5: Square both sides of the equation to get rid of the square root:
3 * cos^2(x) = sin^2(x)

Step 6: Use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to rewrite the equation:
3 * cos^2(x) = 1 - cos^2(x)

Step 7: Simplify the equation:
4 * cos^2(x) = 1

Step 8: Divide both sides of the equation by 4:
cos^2(x) = 1/4

Step 9: Take the square root of both sides to solve for cos(x):
cos(x) = ± √(1/4) = ± 1/2

Step 10: Find the corresponding angles in the interval [0°, 360°] where cos(x) = 1/2 or cos(x) = -1/2.

For cos(x) = 1/2:
x = arccos(1/2) = 60° or x = 360° - arccos(1/2) = 300°

For cos(x) = -1/2:
x = arccos(-1/2) = 120° or x = 360° - arccos(-1/2) = 240°

So the solutions to the equation tan(x) + √3 = 0 in the range 0° ≤ x ≤ 360° are:
x = 60°, 120°, 240°, and 300°.