Solve over the Interval sin(2x)-sin(x)=0 , [0,2pi)

Solution:

Step:1

Utilize the double-angle formula for sine: $sin(2x) = 2sin(x)cos(x)$. Thus, $2sin(x)cos(x) - sin(x) = 0$.

Step:2

Extract the common factor $sin(x)$ from the expression $2sin(x)cos(x) - sin(x)$.

Step:2.1

Take out $sin(x)$ from $2sin(x)cos(x)$: $sin(x)(2cos(x)) - sin(x) = 0$.

Step:2.2

Remove $sin(x)$ from $-sin(x)$: $sin(x)(2cos(x)) - sin(x)(-1) = 0$.

Step:2.3

Extract $sin(x)$ from the entire expression: $sin(x)(2cos(x) - 1) = 0$.

Step:3

Recognize that if either factor equals zero, the equation is satisfied: $sin(x) = 0$ or $2cos(x) - 1 = 0$.

Step:4

Isolate $sin(x)$ and determine the values of $x$.

Step:4.1

Set $sin(x)$ to zero: $sin(x) = 0$.

Step:4.2

Solve for $x$: $sin(x) = 0$.

Step:4.2.1

Apply the inverse sine function: $x = \arcsin(0)$.

Step:4.2.2

Evaluate the inverse sine: $x = 0$.

Step:4.2.3

Use the symmetry of the sine function to find additional solutions: $x = \pi - 0$.

Step:4.2.4

Calculate the second solution: $x = \pi$.

Step:4.2.5

Determine the period of $sin(x)$.

Step:4.2.5.1

The period of sine is given by $2\pi$.

Step:4.2.6

List all solutions within the interval using the period: $x = 0, \pi, 2\pi n, \pi + 2\pi n$ for integers $n$.

Step:5

Address the second factor: $2cos(x) - 1 = 0$.

Step:5.1

Set the equation: $2cos(x) - 1 = 0$.

Step:5.2

Solve for $x$: $2cos(x) - 1 = 0$.

Step:5.2.1

Add 1 to both sides: $2cos(x) = 1$.

Step:5.2.2

Divide by 2: $cos(x) = \frac{1}{2}$.

Step:5.2.3

Apply the inverse cosine function: $x = \arccos(\frac{1}{2})$.

Step:5.2.4

Find the exact values: $x = \frac{\pi}{3}$.

Step:5.2.5

Use the symmetry of the cosine function to find additional solutions: $x = 2\pi - \frac{\pi}{3}$.

Step:5.2.6

Calculate the second solution: $x = \frac{5\pi}{3}$.

Step:5.2.7

Determine the period of $cos(x)$.

Step:5.2.7.1

The period of cosine is given by $2\pi$.

Step:5.2.8

List all solutions within the interval using the period: $x = \frac{\pi}{3} + 2\pi n, \frac{5\pi}{3} + 2\pi n$ for integers $n$.

Step:6

Combine all solutions that satisfy $sin(x)(2cos(x) - 1) = 0$: $x = 0, \pi, \frac{\pi}{3}, \frac{5\pi}{3}$.

Step:7

Simplify the solution set considering the interval $[0, 2\pi)$.

Step:8

Verify which solutions fall within the given interval.

Step:8.1

Test $n = 0$: $x = 0$ is in the interval.

Step:8.2

Test $n = 0$ for the second solution: $x = \frac{\pi}{3}$ is in the interval.

Step:8.3

Test $n = 1$: $x = \pi$ is in the interval.

Step:8.4

Test $n = 0$ for the fourth solution: $x = \frac{5\pi}{3}$ is in the interval.

The final solution set within the interval $[0, 2\pi)$ is $x = 0, \frac{\pi}{3}, \pi, \frac{5\pi}{3}$.

Knowledge Notes:

The problem involves solving a trigonometric equation over a specific interval. The process includes the following knowledge points:

1. Double-Angle Formula: The double-angle formula for sine, $sin(2x) = 2sin(x)cos(x)$, is used to transform the original equation into a more solvable form.

2. Factoring: The ability to factor out common terms, such as $sin(x)$ in this case, simplifies the equation into a product of factors.

3. Zero Product Property: If a product of factors equals zero, at least one of the factors must be zero. This property allows us to set each factor equal to zero and solve for $x$ separately.

4. Inverse Trigonometric Functions: The use of inverse functions, such as $\arcsin$ and $\arccos$, helps to find the angle whose sine or cosine is a given value.

5. Symmetry and Periodicity of Trigonometric Functions: Understanding that sine and cosine functions are periodic and symmetric allows us to find additional solutions within the given interval.

6. Interval Consideration: The solutions must be within the interval $[0, 2\pi)$. This requires checking each solution to ensure it falls within the specified range.

7. Simplification: The process involves simplifying expressions and combining like terms to find the most concise form of the solution set.

By applying these concepts, the problem-solving process systematically finds all solutions to the trigonometric equation within the given interval.