Find the sum, in degrees, of all solutions to the trigonometric equation \sqrt{3}\tan(2x) … — JAMB UTME

Find the sum, in degrees, of all solutions to the trigonometric equation $\sqrt{3}\tan(2x) = 3$ in the interval $0^\circ \le x \le 180^\circ$ .

Cevap: 150 degrees

The sum of all solutions to the equation in the given interval is 150 degrees.

Isolating

\tan(2x)

gives

\sqrt{3}

. For

0^\circ \le 2x \le 360^\circ

\tan(2x) = \sqrt{3}

yields solutions at

2x = 60^\circ

and

2x = 240^\circ

. Dividing by 2 gives

x = 30^\circ

and

x = 120^\circ

. Adding these solutions yields

30^\circ + 120^\circ = 150^\circ

Isolate the trigonometric function

\tan(2x) = \frac{3}{\sqrt{3}} = \sqrt{3}

Dividing both sides by \sqrt{3} simplifies the expression to a standard special angle ratio.

Determine the domain for the argument

2x

Since

0^\circ \le x \le 180^\circ

, multiplying the inequality by 2 gives

0^\circ \le 2x \le 360^\circ

This establishes the range of angles to search for

2x

within one complete turn.

Find all values of

2x

where tangent equals

\sqrt{3}

2x = 60^\circ

(1st quadrant) and

2x = 180^\circ + 60^\circ = 240^\circ

(3rd quadrant)

The tangent function is positive in Quadrants I and III with a reference angle of

60^\circ

Solve for

x

x = \frac{60^\circ}{2} = 30^\circ

and

x = \frac{240^\circ}{2} = 120^\circ

Dividing each angle by 2 yields the values of

x

lying within the domain

0^\circ \le x \le 180^\circ

Calculate the sum of the solutions

30^\circ + 120^\circ = 150^\circ

The question specifically requests the sum of all valid solutions.

Solving trigonometric equations using reference angles and domain transformation

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