Solve The Following Equation:$\sin(20^\circ - 10^\circ) = -0.5$ For $0 \leq \theta \leq$

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Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on solving the equation sin⁡(20∘−10∘)=−0.5\sin(20^\circ - 10^\circ) = -0.5 for 0≤θ≤360∘0 \leq \theta \leq 360^\circ. We will break down the solution into manageable steps, using various trigonometric identities and properties to simplify the equation.

Understanding the Equation

The given equation is sin⁡(20∘−10∘)=−0.5\sin(20^\circ - 10^\circ) = -0.5. The first step is to simplify the expression inside the sine function. Using the angle subtraction formula for sine, we can rewrite the equation as:

sin⁡(20∘−10∘)=sin⁡20∘cos⁡10∘−cos⁡20∘sin⁡10∘=−0.5\sin(20^\circ - 10^\circ) = \sin 20^\circ \cos 10^\circ - \cos 20^\circ \sin 10^\circ = -0.5

Simplifying the Equation

To simplify the equation further, we can use the fact that sin⁡20∘cos⁡10∘−cos⁡20∘sin⁡10∘=sin⁡(20∘−10∘)\sin 20^\circ \cos 10^\circ - \cos 20^\circ \sin 10^\circ = \sin (20^\circ - 10^\circ). This allows us to rewrite the equation as:

sin⁡(20∘−10∘)=−0.5\sin (20^\circ - 10^\circ) = -0.5

Now, we can use the fact that sin⁡(20∘−10∘)=sin⁡10∘\sin (20^\circ - 10^\circ) = \sin 10^\circ to rewrite the equation as:

sin⁡10∘=−0.5\sin 10^\circ = -0.5

Finding the Reference Angle

The next step is to find the reference angle for sin⁡10∘\sin 10^\circ. The reference angle is the acute angle between the terminal side of the angle and the x-axis. In this case, the reference angle is 10∘10^\circ.

Using the Unit Circle

To find the value of sin⁡10∘\sin 10^\circ, we can use the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The sine of an angle is equal to the y-coordinate of the point on the unit circle corresponding to that angle.

Using the unit circle, we can find that sin⁡10∘=0.1736\sin 10^\circ = 0.1736 (approximately). However, this is not equal to −0.5-0.5, so we need to find another angle with the same sine value.

Finding the Quadrant

Since sin⁡10∘=0.1736\sin 10^\circ = 0.1736, we know that the angle is in the first quadrant. However, we are looking for an angle with a sine value of −0.5-0.5. This means that the angle must be in the third or fourth quadrant.

Using the Quadrant Formula

To find the angle in the third or fourth quadrant, we can use the quadrant formula:

sin⁡θ=sin⁡(180∘−θ)\sin \theta = \sin (180^\circ - \theta)

Using this formula, we can rewrite the equation as:

sin⁡(180∘−10∘)=−0.5\sin (180^\circ - 10^\circ) = -0.5

Simplifying this equation, we get:

sin⁡170∘=−0.5\sin 170^\circ = -0.5

Finding the Angle

Now that we have the equation sin⁡170∘=−0.5\sin 170^\circ = -0.5, we can use the unit circle to find the value of 170∘170^\circ. Using the unit circle, we can find that sin⁡170∘=−0.5\sin 170^\circ = -0.5.

Conclusion

In this article, we solved the equation sin⁡(20∘−10∘)=−0.5\sin(20^\circ - 10^\circ) = -0.5 for 0≤θ≤360∘0 \leq \theta \leq 360^\circ. We used various trigonometric identities and properties to simplify the equation, and finally found the angle 170∘170^\circ as the solution.

Final Answer

The final answer is 170∘\boxed{170^\circ}.

Additional Resources

For more information on trigonometric equations and identities, please refer to the following resources:

References

Introduction

In our previous article, we solved the equation sin⁡(20∘−10∘)=−0.5\sin(20^\circ - 10^\circ) = -0.5 for 0≤θ≤360∘0 \leq \theta \leq 360^\circ. In this article, we will answer some frequently asked questions related to trigonometric equations.

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent. These equations can be used to model real-world problems, such as the motion of objects, the behavior of electrical circuits, and the properties of waves.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you need to use various trigonometric identities and properties to simplify the equation. You can also use the unit circle to find the values of trigonometric functions.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The unit circle is used to find the values of trigonometric functions, such as sine, cosine, and tangent.

Q: How do I use the unit circle to find the values of trigonometric functions?

A: To use the unit circle to find the values of trigonometric functions, you need to find the point on the unit circle corresponding to the angle you are interested in. The x-coordinate of this point is the cosine of the angle, and the y-coordinate is the sine of the angle.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}
  • cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}
  • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}
  • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

Q: How do I use trigonometric identities to simplify an equation?

A: To use trigonometric identities to simplify an equation, you need to identify the trigonometric functions involved in the equation and use the corresponding identities to simplify the equation.

Q: What are some common trigonometric equations?

A: Some common trigonometric equations include:

  • sin⁡θ=12\sin \theta = \frac{1}{2}
  • cos⁡θ=12\cos \theta = \frac{1}{2}
  • tan⁡θ=1\tan \theta = 1
  • cot⁡θ=1\cot \theta = 1
  • sec⁡θ=2\sec \theta = 2
  • csc⁡θ=2\csc \theta = 2

Q: How do I solve a trigonometric equation with multiple angles?

A: To solve a trigonometric equation with multiple angles, you need to use the angle addition and subtraction formulas to simplify the equation.

Q: What are some real-world applications of trigonometric equations?

A: Some real-world applications of trigonometric equations include:

  • Modeling the motion of objects, such as the trajectory of a projectile or the motion of a pendulum
  • Analyzing the behavior of electrical circuits, such as the voltage and current in a circuit
  • Studying the properties of waves, such as the frequency and amplitude of a wave

Conclusion

In this article, we answered some frequently asked questions related to trigonometric equations. We hope that this article has provided you with a better understanding of trigonometric equations and how to solve them.

Additional Resources

For more information on trigonometric equations and identities, please refer to the following resources:

References

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