Solve over the Interval tan(theta)=-(2 square root of 3)/3sin(theta) , 0< =theta< 2pi

Solution:

Step 1: Simplify the equation.

Step 1.1: Rewrite the given equation.

Simplify $\tan (\theta )=-\frac{2\sqrt{3}}{3}\sin (\theta )$.

Step 1.1.1: Combine terms.

$\tan (\theta )=-\frac{2\sqrt{3}\sin (\theta )}{3}$.

Step 1.1.2: Rearrange the equation.

$\tan (\theta )=-\frac{2\sin (\theta )\sqrt{3}}{3}$.

Step 2: Isolate the variable.

Step 2.1: Divide by $\tan (\theta )$.

$\frac{\tan (\theta )}{\tan (\theta )}=\frac{-\frac{2\sin (\theta )\sqrt{3}}{3}}{\tan (\theta )}$.

Step 2.2: Simplify the left side.

Step 2.2.1: Cancel $\tan (\theta )$.

$1=\frac{-\frac{2\sin (\theta )\sqrt{3}}{3}}{\tan (\theta )}$.

Step 2.3: Simplify the right side.

Step 2.3.1: Multiply by the reciprocal.

$1=-\frac{2\sin (\theta )\sqrt{3}}{3}\cdot \frac{1}{\tan (\theta )}$.

Step 2.3.2: Express $\tan (\theta )$ as $\frac{\sin (\theta )}{\cos (\theta )}$.

$1=-\frac{2\sin (\theta )\sqrt{3}}{3}\cdot \frac{\cos (\theta )}{\sin (\theta )}$.

Step 2.3.3: Simplify the fraction.

$1=-\frac{2\sqrt{3}}{3}\cos (\theta )$.

Step 3: Solve for $\cos (\theta )$.

Step 3.1: Isolate $\cos (\theta )$.

$\cos (\theta )=-\frac{3}{2\sqrt{3}}$.

Step 4: Find the angles.

Step 4.1: Take the inverse cosine.

$\theta =\arccos (-\frac{\sqrt{3}}{2})$.

Step 4.2: Determine the angles in the specified interval.

$\theta =\frac{5\pi }{6},\frac{7\pi }{6}$.

Knowledge Notes:

1. Trigonometric identities: The tangent of an angle is the ratio of the sine to the cosine of that angle, $\tan (\theta )=\frac{\sin (\theta )}{\cos (\theta )}$.

2. Inverse trigonometric functions: To find an angle given a trigonometric value, we use the inverse trigonometric functions, such as $\arccos (x)$, which returns the angle whose cosine is $x$.

3. The cosine function is negative in the second and third quadrants of the unit circle.

4. Periodicity of trigonometric functions: The trigonometric functions repeat their values in regular intervals called periods. For the cosine function, the period is $2\pi$ radians.

5. Interval notation: When solving trigonometric equations, it is important to consider the interval in which the solution is sought. For example, $0\le \theta <2\pi$ means that the angle $\theta$ must be between $0$ and $2\pi$, including $0$ but not $2\pi$.

6. Simplifying expressions: When simplifying expressions, it is often useful to cancel common factors, combine like terms, and use algebraic properties to rewrite the expressions in a simpler form.