Solving Trigonometric Equations: Find Θ For -2sin(θ) = -√2

Hey guys! Let's dive into solving a trigonometric equation. We're going to find all the exact solutions for the equation $-2 ewline ewline sin(θ) = -

√2$ within the interval $0 ≤ θ < 2π$. It sounds a bit intimidating, but trust me, we'll break it down step by step so it's super easy to follow. Whether you're brushing up on your math skills or tackling homework, this guide is here to help you out!

Step-by-Step Solution

1. Isolate the Trigonometric Function

First things first, we need to isolate the sine function. Right now, we have $-2sin(θ) = -√2$. To get $sin(θ)$ by itself, we'll divide both sides of the equation by -2. This gives us:

$sin(θ) =

√2 / 2$

So, now we have a much simpler equation to work with: $sin(θ) =

√2 / 2$. This means we're looking for angles $θ$ where the sine function equals $

√2 / 2$.

2. Identify Reference Angles

Next, we need to figure out what angles have a sine value of $

√2 / 2$. Think back to your unit circle! The sine function corresponds to the y-coordinate on the unit circle. We know that $sin(π/4) =

√2 / 2$. So, $π/4$ is our reference angle. A reference angle is the acute angle made by the terminal side of the angle and the x-axis. Identifying the reference angle helps us find all angles with the same sine value.

3. Determine Quadrants

Since $sin(θ)$ is positive ($

√2 / 2$ is positive), we need to find the quadrants where sine is positive. Remember that sine is positive in the first and second quadrants. Here's a quick recap:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive.
• Quadrant III: Tangent is positive.
• Quadrant IV: Cosine is positive.

So, we'll be looking for solutions in Quadrants I and II.

4. Find Solutions in the Specified Interval

Now that we know the reference angle and the relevant quadrants, we can find our solutions. We're looking for solutions in the interval $0 ≤ θ < 2π$.

• Quadrant I Solution:

  In Quadrant I, the angle is the same as the reference angle. So, one solution is:

  $θ = π/4$

• Quadrant II Solution:

  In Quadrant II, the angle is $π$ minus the reference angle. So, the second solution is:

  $θ = π - π/4 = 3π/4$

Therefore, our solutions are $θ = π/4$ and $θ = 3π/4$.

5. Verify Solutions

Finally, it's always a good idea to check our solutions to make sure they're correct. Let's plug them back into the original equation:

• For $θ = π/4$:

  $-2sin(π/4) = -2(

√2 / 2) = -

√2$

This checks out!

• For $θ = 3π/4$:

  $-2sin(3π/4) = -2(

√2 / 2) = -

√2$

This also checks out!

Both solutions satisfy the original equation, so we're good to go!

Final Answer

The exact solutions for the equation $-2sin(θ) = -

√2$ on the interval $0 ≤ θ < 2π$ are:

$θ = π/4, 3π/4$

And that's it! We've successfully solved the trigonometric equation. Remember, the key steps are isolating the trigonometric function, identifying the reference angle, determining the quadrants, and finding the solutions within the given interval. Practice makes perfect, so keep at it, and you'll become a pro at solving these types of problems!

Additional Tips for Solving Trigonometric Equations

To become even more proficient in solving trigonometric equations, consider these additional tips. Understanding the underlying concepts and employing various techniques will enhance your problem-solving skills.

Mastering the Unit Circle

The unit circle is your best friend when it comes to trigonometry. Knowing the sine, cosine, and tangent values for common angles (like 0, π/6, π/4, π/3, π/2, etc.) will save you a lot of time and effort. Make sure you can quickly recall these values or visualize them on the unit circle. This will significantly speed up the process of finding reference angles and solutions. The unit circle provides a visual representation of trigonometric functions, making it easier to understand their values at different angles.

Understanding Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables. They are powerful tools for simplifying equations and expressing them in different forms. Some common identities include:

• Pythagorean identities: $sin^2(θ) + cos^2(θ) = 1$, $1 + tan^2(θ) = sec^2(θ)$, $1 + cot^2(θ) = csc^2(θ)$
• Double-angle identities: $sin(2θ) = 2sin(θ)cos(θ)$, $cos(2θ) = cos^2(θ) - sin^2(θ)$
• Sum and difference identities: $sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)$, $cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)$

Knowing these identities and how to use them can help you simplify complex trigonometric equations and make them easier to solve. It's essential to recognize when and how to apply these identities to transform equations into a solvable form.

Factoring Trigonometric Equations

Sometimes, trigonometric equations can be solved by factoring. Look for opportunities to factor out common trigonometric functions or use algebraic techniques to factor the equation. For example, consider the equation $sin^2(θ) - sin(θ) = 0$. You can factor out $sin(θ)$ to get $sin(θ)(sin(θ) - 1) = 0$. This gives you two separate equations to solve: $sin(θ) = 0$ and $sin(θ) = 1$.

Using Inverse Trigonometric Functions

Inverse trigonometric functions (arcsin, arccos, arctan) can be used to find angles when you know the value of the trigonometric function. For example, if $sin(θ) = x$, then $θ = arcsin(x)$. However, be careful when using inverse trigonometric functions because they only give you one solution within a specific range. You may need to find additional solutions based on the quadrant and periodicity of the trigonometric function. Always consider the range of the inverse function and the possible quadrants for additional solutions.

Checking for Extraneous Solutions

When solving trigonometric equations, it's important to check for extraneous solutions. These are solutions that satisfy the transformed equation but not the original equation. Extraneous solutions can arise when you square both sides of an equation or perform other operations that introduce new solutions. Always plug your solutions back into the original equation to make sure they are valid. Checking for extraneous solutions ensures that your final answer is accurate and reliable.

Utilizing Graphs of Trigonometric Functions

Visualizing trigonometric functions using graphs can provide valuable insights into the nature of the solutions. The graphs of sine, cosine, and tangent functions show their periodic behavior and the intervals where they are positive or negative. By sketching the graph of the trigonometric function and the line representing the value you're solving for, you can visually identify the solutions and their approximate locations. This method can be particularly helpful for understanding the multiple solutions within a given interval.

Common Mistakes to Avoid

Even with a solid understanding of the concepts, it's easy to make mistakes when solving trigonometric equations. Here are some common pitfalls to watch out for:

• Forgetting the Periodicity: Trigonometric functions are periodic, meaning they repeat their values at regular intervals. When finding solutions, make sure to consider all possible solutions within the given interval by adding or subtracting multiples of the period.
• Ignoring the Quadrant: The sign of the trigonometric function depends on the quadrant in which the angle lies. Make sure to identify the correct quadrants for your solutions based on the sign of the trigonometric function.
• Incorrectly Applying Identities: Using trigonometric identities incorrectly can lead to incorrect solutions. Double-check that you are using the correct identity and applying it properly.
• Not Checking Solutions: Failing to check your solutions can result in including extraneous solutions in your final answer. Always plug your solutions back into the original equation to verify their validity.

By avoiding these common mistakes and consistently applying the steps outlined above, you'll be well-equipped to tackle a wide range of trigonometric equations. Keep practicing, and you'll become a confident and skilled problem solver in trigonometry.