Problem 35 - Precalculus: Functions and Graphs

Question

Find the solutions of the equation that are in the interval \([0,2 \pi).\) \(\sin 2 t+\sin t=0\)

Step-by-Step Solution

Answer

The solutions are \( t = 0, \pi, \frac{2\pi}{3}, \frac{4\pi}{3} \).

Step 1: Recognize Trigonometric Identity

Identify the double angle identity for sine, which states that \( \sin 2t = 2 \sin t \cos t \). Use this identity to rewrite the given equation.

Step 2: Substitute the Identity

Substitute the identity into the equation: \( 2 \sin t \cos t + \sin t = 0 \).

Step 3: Factor the Equation

Notice that \( \sin t \) is common in both terms of the equation. Factor it out: \( \sin t (2 \cos t + 1) = 0 \).

Step 4: Solve Each Factor

The product of two factors equals zero implies at least one of the factors must be zero. Solve for each factor separately. \( \sin t = 0 \) and \( 2\cos t + 1 = 0 \).

Step 5: Solve \( \sin t = 0 \)

\( \sin t = 0 \) at \( t = 0, \pi, 2\pi \). However, since the interval is \([0, 2\pi)\), we take \( t = 0, \pi \).

Step 6: Solve \( 2\cos t + 1 = 0 \)

Solve for \( \cos t \) by isolating it: \( 2\cos t = -1 \). Therefore, \( \cos t = -\frac{1}{2} \). This occurs at \( t = \frac{2\pi}{3}, \frac{4\pi}{3} \) in the given interval.

Step 7: Combine the Solutions

Combine the solutions from Steps 5 and 6 to get all solutions in the interval \([0, 2\pi)\).

Key Concepts

Double Angle Identity

The double angle identity is a vital tool in trigonometry. It allows you to express functions of double angles in terms of single angles, simplifying equations and making it easier to solve them. One of the most common forms is for sine, specifically \( \sin 2t = 2 \sin t \cos t \). This identity is useful because it transforms a single nonlinear equation into a more manageable form.

• In the context of our problem, using the double angle identity helps replace \( \sin 2t \) with \( 2 \sin t \cos t \).
• This substitution simplifies the initial equation, making it easier to solve by using algebraic methods like factoring.

Recognizing and applying the correct identity is crucial, as it affects the potential solutions you can find. Always check if a double angle identity fits into your equation, especially in equations involving trigonometric functions multiplied together.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another. They are extremely helpful in various computations, including simplifying expressions and solving equations. Some well-known identities include Pythagorean identities, sum and difference formulas, and of course, double angle identities.

• Using these identities, trigonometric terms can be rewritten in different forms. This can help to simplify or solve equations more efficiently.
• In our exercise, we used the double angle identity for sine function to make the equation solvable by familiar algebraic methods like factoring.

Trigonometric identities are fundamental tools in solving complex trigonometric problems. They allow us to manipulate expressions and find solutions that are otherwise difficult to identify or solve.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angles that satisfy the given equation within a specified interval. It often requires using trigonometric identities and algebraic techniques like factoring or using quadratic formulas.

• First, transform the equation using relevant identities. This makes the equation easier to manage.
• Next, use algebraic methods such as factoring to split the equation into simpler components.
• Lastly, solve each component separately to find all possible solutions, considering the given interval.

In our case, we used these steps: applying double angle identity, factoring, and solving the factors to find all solutions \( t = 0, \pi, \frac{2\pi}{3}, \frac{4\pi}{3} \).
By understanding and effectively using these techniques, solving trigonometric equations becomes much more approachable and less intimidating.