For Which Angle(s) Over The Interval $0^{\circ} \leq X \leq 360^{\circ}$ Is $\sin (x) = -1$?

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject in mathematics and has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on solving trigonometric equations, specifically the equation $\sin (x) = -1$, to find the angles over the interval $0^{\circ} \leq x \leq 360^{\circ}$.

Understanding Sine Function

The sine function is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. The sine function has a period of $360^{\circ}$, which means that the function repeats itself every $360^{\circ}$.

Solving the Equation

To solve the equation $\sin (x) = -1$, we need to find the angles over the interval $0^{\circ} \leq x \leq 360^{\circ}$ for which the sine function is equal to -1. We can start by recalling the unit circle and the values of the sine function at different angles.

Unit Circle and Sine Function

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The sine function is defined as the y-coordinate of a point on the unit circle. The unit circle has a period of $360^{\circ}$, and the sine function has a maximum value of 1 and a minimum value of -1.

Finding Angles

To find the angles over the interval $0^{\circ} \leq x \leq 360^{\circ}$ for which the sine function is equal to -1, we can use the unit circle and the values of the sine function at different angles.

• Angle 270°: At an angle of $270^{\circ}$, the sine function is equal to -1. This is because the y-coordinate of the point on the unit circle at an angle of $270^{\circ}$ is -1.
• Angle 630°: At an angle of $630^{\circ}$, the sine function is equal to -1. This is because the y-coordinate of the point on the unit circle at an angle of $630^{\circ}$ is -1.
• Angle 990°: At an angle of $990^{\circ}$, the sine function is equal to -1. This is because the y-coordinate of the point on the unit circle at an angle of $990^{\circ}$ is -1.

Conclusion

In conclusion, the angles over the interval $0^{\circ} \leq x \leq 360^{\circ}$ for which the sine function is equal to -1 are $270^{\circ}$, $630^{\circ}$, and $990^{\circ}$. These angles are found by using the unit circle and the values of the sine function at different angles.

Additional Tips and Tricks

• Recall the Unit Circle: The unit circle is a fundamental concept in trigonometry, and it is essential to recall the values of the sine function at different angles.
• Use the Periodic Property: The sine function has a period of $360^{\circ}$, which means that the function repeats itself every $360^{\circ}$.
• Use the Symmetry Property: The sine function is an odd function, which means that $\sin (-x) = -\sin x$.

Final Thoughts

Introduction

In our previous article, we discussed solving trigonometric equations, specifically the equation $\sin (x) = -1$, to find the angles over the interval $0^{\circ} \leq x \leq 360^{\circ}$. In this article, we will provide a Q&A section to help you better understand the concepts and solve similar problems.

Q&A

Q: What is the unit circle, and how is it related to trigonometry?

A: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is a fundamental concept in trigonometry, and it is used to define the values of the sine, cosine, and tangent functions.

Q: How do I recall the values of the sine function at different angles?

A: You can recall the values of the sine function at different angles by using the unit circle and the symmetry property of the sine function. For example, the sine function is equal to 1 at an angle of $90^{\circ}$, and it is equal to -1 at an angle of $270^{\circ}$.

Q: What is the periodic property of the sine function, and how is it used to solve trigonometric equations?

A: The sine function has a period of $360^{\circ}$, which means that the function repeats itself every $360^{\circ}$. This property is used to solve trigonometric equations by finding the solutions over a single period and then using the periodic property to find the solutions over the entire interval.

Q: How do I use the symmetry property of the sine function to solve trigonometric equations?

A: The sine function is an odd function, which means that $\sin (-x) = -\sin x$. This property is used to solve trigonometric equations by finding the solutions for a positive angle and then using the symmetry property to find the solutions for a negative angle.

Q: What are some common trigonometric equations that I should know how to solve?

A: Some common trigonometric equations that you should know how to solve include:

• $\sin (x) = 1$
• $\sin (x) = -1$
• $\cos (x) = 1$
• $\cos (x) = -1$
• $\tan (x) = 1$
• $\tan (x) = -1$

Q: How do I use a calculator to solve trigonometric equations?

A: You can use a calculator to solve trigonometric equations by entering the equation and then using the calculator's trigonometric functions to find the solutions. For example, you can use the calculator to find the solutions to the equation $\sin (x) = -1$.

Q: What are some common mistakes to avoid when solving trigonometric equations?

A: Some common mistakes to avoid when solving trigonometric equations include:

• Not using the unit circle to recall the values of the sine function at different angles
• Not using the periodic property to find the solutions over the entire interval
• Not using the symmetry property to find the solutions for a negative angle
• Not checking the solutions to ensure that they are valid

Conclusion

In conclusion, solving trigonometric equations is an essential skill in mathematics, and it has numerous applications in various fields. In this article, we provided a Q&A section to help you better understand the concepts and solve similar problems. We also provided some common trigonometric equations that you should know how to solve and some common mistakes to avoid when solving trigonometric equations.

Additional Tips and Tricks

• Practice, Practice, Practice: The best way to learn how to solve trigonometric equations is to practice, practice, practice.
• Use the Unit Circle: The unit circle is a fundamental concept in trigonometry, and it is used to define the values of the sine, cosine, and tangent functions.
• Use the Periodic Property: The sine function has a period of $360^{\circ}$, which means that the function repeats itself every $360^{\circ}$.
• Use the Symmetry Property: The sine function is an odd function, which means that $\sin (-x) = -\sin x$.

Final Thoughts

Solving trigonometric equations is an essential skill in mathematics, and it has numerous applications in various fields. In this article, we provided a Q&A section to help you better understand the concepts and solve similar problems. We also provided some common trigonometric equations that you should know how to solve and some common mistakes to avoid when solving trigonometric equations.