Use The Steps To Determine The Exact Value Of Sin ⁡ ( − 135 ) ∘ \sin (-135)^{\circ} Sin ( − 135 ) ∘ .1. Identify The Reference Angle, Θ \theta Θ . - Θ = 45 ∘ \theta = 45^{\circ} Θ = 4 5 ∘ ✓2. Determine The Sign Of The Value Based On The Quadrant Containing The Terminal Side.

Understanding the Problem

In trigonometry, we often encounter problems that involve finding the exact value of a trigonometric function for a given angle. In this article, we will focus on determining the exact value of $\sin (-135)^{\circ}$. To do this, we will follow a step-by-step approach that involves identifying the reference angle and determining the sign of the value based on the quadrant containing the terminal side.

Step 1: Identify the Reference Angle, $\theta$

The first step in determining the exact value of $\sin (-135)^{\circ}$ is to identify the reference angle, $\theta$. The reference angle is the acute angle between the terminal side of the given angle and the nearest x-axis. In this case, the given angle is $-135^{\circ}$.

To find the reference angle, we can use the fact that the reference angle is equal to the absolute value of the given angle. Therefore, the reference angle, $\theta$, is equal to $135^{\circ}$.

However, since the given angle is negative, we need to find the reference angle in the second quadrant. The reference angle in the second quadrant is equal to $180^{\circ} - 135^{\circ} = 45^{\circ}$.

Step 2: Determine the Sign of the Value Based on the Quadrant Containing the Terminal Side

Now that we have identified the reference angle, $\theta$, we need to determine the sign of the value based on the quadrant containing the terminal side. The terminal side of the given angle is in the third quadrant.

In the third quadrant, the sine function is negative. Therefore, the value of $\sin (-135)^{\circ}$ is negative.

Using the Reference Angle to Find the Exact Value

Now that we have identified the reference angle, $\theta$, and determined the sign of the value, we can use the reference angle to find the exact value of $\sin (-135)^{\circ}$.

The sine function is negative in the third quadrant, and the reference angle is $45^{\circ}$. Therefore, the exact value of $\sin (-135)^{\circ}$ is equal to $-\sin 45^{\circ}$.

We know that $\sin 45^{\circ} = \frac{\sqrt{2}}{2}$. Therefore, the exact value of $\sin (-135)^{\circ}$ is equal to $-\frac{\sqrt{2}}{2}$.

Conclusion

In this article, we have followed a step-by-step approach to determine the exact value of $\sin (-135)^{\circ}$. We identified the reference angle, $\theta$, and determined the sign of the value based on the quadrant containing the terminal side. We then used the reference angle to find the exact value of $\sin (-135)^{\circ}$.

The exact value of $\sin (-135)^{\circ}$ is equal to $-\frac{\sqrt{2}}{2}$. This value can be used in a variety of mathematical applications, including trigonometry and calculus.

Common Trigonometric Identities

In trigonometry, there are several common identities that can be used to simplify and evaluate trigonometric expressions. Some of the most common trigonometric identities include:

• Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$
• Quotient Identity: $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• Reciprocal Identity: $\csc \theta = \frac{1}{\sin \theta}$
• Cofunction Identity: $\sin \theta = \cos (90^{\circ} - \theta)$

These identities can be used to simplify and evaluate trigonometric expressions, and can be used to find the exact value of a trigonometric function for a given angle.

Real-World Applications of Trigonometry

Trigonometry has a wide range of real-world applications, including:

• Navigation: Trigonometry is used in navigation to determine the position and direction of an object.
• Physics: Trigonometry is used in physics to describe the motion of objects and to calculate the trajectory of projectiles.
• Engineering: Trigonometry is used in engineering to design and build structures, such as bridges and buildings.
• Computer Science: Trigonometry is used in computer science to create 3D models and to simulate the motion of objects.

These are just a few examples of the many real-world applications of trigonometry. Trigonometry is a fundamental subject that has a wide range of applications in many different fields.

Conclusion

Q: What is the reference angle for $\sin (-135)^{\circ}$?

A: The reference angle for $\sin (-135)^{\circ}$ is $45^{\circ}$.

Q: Why is the reference angle $45^{\circ}$?

A: The reference angle is $45^{\circ}$ because the given angle is $-135^{\circ}$, and the reference angle is the acute angle between the terminal side of the given angle and the nearest x-axis. Since the given angle is negative, we need to find the reference angle in the second quadrant, which is $180^{\circ} - 135^{\circ} = 45^{\circ}$.

Q: What is the sign of the value of $\sin (-135)^{\circ}$?

A: The value of $\sin (-135)^{\circ}$ is negative.

Q: Why is the value of $\sin (-135)^{\circ}$ negative?

A: The value of $\sin (-135)^{\circ}$ is negative because the terminal side of the given angle is in the third quadrant, and the sine function is negative in the third quadrant.

Q: How do I find the exact value of $\sin (-135)^{\circ}$?

A: To find the exact value of $\sin (-135)^{\circ}$, you can use the reference angle to find the value of $\sin 45^{\circ}$, and then multiply it by $-1$ to get the value of $\sin (-135)^{\circ}$.

Q: What is the exact value of $\sin (-135)^{\circ}$?

A: The exact value of $\sin (-135)^{\circ}$ is $-\frac{\sqrt{2}}{2}$.

Q: Can I use the Pythagorean identity to find the exact value of $\sin (-135)^{\circ}$?

A: Yes, you can use the Pythagorean identity to find the exact value of $\sin (-135)^{\circ}$. The Pythagorean identity is $\sin^2 \theta + \cos^2 \theta = 1$, and you can use it to find the value of $\sin 45^{\circ}$ and then multiply it by $-1$ to get the value of $\sin (-135)^{\circ}$.

Q: What are some common trigonometric identities that I can use to simplify and evaluate trigonometric expressions?

A: Some common trigonometric identities that you can use to simplify and evaluate trigonometric expressions include:

• Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$
• Quotient Identity: $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• Reciprocal Identity: $\csc \theta = \frac{1}{\sin \theta}$
• Cofunction Identity: $\sin \theta = \cos (90^{\circ} - \theta)$

Q: What are some real-world applications of trigonometry?

A: Some real-world applications of trigonometry include:

• Navigation: Trigonometry is used in navigation to determine the position and direction of an object.
• Physics: Trigonometry is used in physics to describe the motion of objects and to calculate the trajectory of projectiles.
• Engineering: Trigonometry is used in engineering to design and build structures, such as bridges and buildings.
• Computer Science: Trigonometry is used in computer science to create 3D models and to simulate the motion of objects.

Q: How can I practice determining the exact value of trigonometric functions?

A: You can practice determining the exact value of trigonometric functions by working through examples and exercises, such as finding the exact value of $\sin (-135)^{\circ}$. You can also use online resources, such as trigonometry calculators and worksheets, to help you practice and improve your skills.