Evaluate Sin ⁡ 135 ∘ \sin 135^{\circ} Sin 13 5 ∘ Without Using A Calculator.A. − 2 2 -\frac{\sqrt{2}}{2} − 2 2 ​ ​ B. 3 2 \frac{\sqrt{3}}{2} 2 3 ​ ​ C. − 1 2 -\frac{1}{2} − 2 1 ​ D. 2 2 \frac{\sqrt{2}}{2} 2 2 ​ ​

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Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the fundamental concepts in trigonometry is the sine function, which is defined as the ratio of the length of the side opposite a given angle to the length of the hypotenuse in a right-angled triangle. In this article, we will evaluate the sine of $135^{\circ}$ without using a calculator.

Understanding the Angle

The angle $135^{\circ}$ is in the second quadrant of the unit circle. In the unit circle, the angle is measured counterclockwise from the positive x-axis. Since the angle is in the second quadrant, the sine function will be negative.

Using the Unit Circle

To evaluate the sine of $135^{\circ}$, 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 in the unit circle is equal to the y-coordinate of the point where the terminal side of the angle intersects the unit circle.

Converting the Angle to a Reference Angle

To evaluate the sine of $135^{\circ}$, we need to convert it to a reference angle. The reference angle is the acute angle between the terminal side of the angle and the x-axis. Since $135^{\circ}$ is in the second quadrant, the reference angle is $180^{\circ} - 135^{\circ} = 45^{\circ}$.

Evaluating the Sine Function

Now that we have the reference angle, we can evaluate the sine function. The sine of $45^{\circ}$ is equal to $\frac{\sqrt{2}}{2}$. Since the angle $135^{\circ}$ is in the second quadrant, the sine function will be negative. Therefore, the sine of $135^{\circ}$ is equal to $-\frac{\sqrt{2}}{2}$.

Conclusion

In this article, we evaluated the sine of $135^{\circ}$ without using a calculator. We used the unit circle and the reference angle to find the sine function. The sine of $135^{\circ}$ is equal to $-\frac{\sqrt{2}}{2}$. This is the correct answer among the options provided.

Final Answer

The final answer is $-\frac{\sqrt{2}}{2}$.

Additional Information

The sine function is a periodic function, which means that it repeats itself after a certain interval. The sine function has a period of $360^{\circ}$, which means that the sine of an angle is equal to the sine of the angle plus or minus any multiple of $360^{\circ}$. This property can be used to evaluate the sine of angles in different quadrants.

Common Mistakes

One common mistake when evaluating the sine of angles in the second quadrant is to assume that the sine function is positive. However, since the angle is in the second quadrant, the sine function will be negative.

Real-World Applications

The sine function has many real-world applications, including physics, engineering, and navigation. In physics, the sine function is used to describe the motion of objects in circular motion. In engineering, the sine function is used to design and analyze electrical circuits. In navigation, the sine function is used to calculate distances and directions.

Summary

In this article, we evaluated the sine of $135^{\circ}$ without using a calculator. We used the unit circle and the reference angle to find the sine function. The sine of $135^{\circ}$ is equal to $-\frac{\sqrt{2}}{2}$. This is the correct answer among the options provided. The sine function has many real-world applications, including physics, engineering, and navigation.

Introduction

In our previous article, we evaluated the sine of 135° without using a calculator. We used the unit circle and the reference angle to find the sine function. In this article, we will answer some frequently asked questions related to evaluating the sine of 135°.

Q: What is the reference angle for 135°?

A: The reference angle for 135° is 45°. This is because 135° is in the second quadrant, and the reference angle is the acute angle between the terminal side of the angle and the x-axis.

Q: Why is the sine of 135° negative?

A: The sine of 135° is negative because the angle is in the second quadrant. In the second quadrant, the sine function is negative.

Q: How do I find the sine of 135° without using a calculator?

A: To find the sine of 135° without using a calculator, you can use the unit circle and the reference angle. The sine of 135° is equal to the sine of the reference angle (45°) multiplied by -1.

Q: What is the sine of 45°?

A: The sine of 45° is equal to √2/2.

Q: Why is the sine of 45° equal to √2/2?

A: The sine of 45° is equal to √2/2 because the point where the terminal side of the angle intersects the unit circle has a y-coordinate of √2/2.

Q: Can I use a calculator to evaluate the sine of 135°?

A: Yes, you can use a calculator to evaluate the sine of 135°. However, the question asks you to evaluate the sine of 135° without using a calculator.

Q: What is the period of the sine function?

A: The period of the sine function is 360°. This means that the sine of an angle is equal to the sine of the angle plus or minus any multiple of 360°.

Q: How do I use the period of the sine function to evaluate the sine of 135°?

A: You can use the period of the sine function to evaluate the sine of 135° by finding the sine of an angle that is equal to 135° plus or minus any multiple of 360°.

Q: What are some real-world applications of the sine function?

A: The sine function has many real-world applications, including physics, engineering, and navigation. In physics, the sine function is used to describe the motion of objects in circular motion. In engineering, the sine function is used to design and analyze electrical circuits. In navigation, the sine function is used to calculate distances and directions.

Q: Why is the sine function important in real-world applications?

A: The sine function is important in real-world applications because it helps us to describe and analyze complex phenomena. In physics, the sine function helps us to understand the motion of objects in circular motion. In engineering, the sine function helps us to design and analyze electrical circuits. In navigation, the sine function helps us to calculate distances and directions.

Q: Can I use the sine function to solve problems in other areas of mathematics?

A: Yes, you can use the sine function to solve problems in other areas of mathematics. The sine function is a fundamental concept in trigonometry, and it has many applications in other areas of mathematics, including algebra and geometry.

Q: What are some common mistakes when evaluating the sine of angles in the second quadrant?

A: One common mistake when evaluating the sine of angles in the second quadrant is to assume that the sine function is positive. However, since the angle is in the second quadrant, the sine function will be negative.

Q: How can I avoid making this mistake?

A: To avoid making this mistake, you can use the unit circle and the reference angle to find the sine function. You can also use the fact that the sine function is negative in the second quadrant.

Q: What are some other trigonometric functions that I should know?

A: In addition to the sine function, you should also know the cosine function and the tangent function. The cosine function is defined as the ratio of the length of the side adjacent to a given angle to the length of the hypotenuse. The tangent function is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to the angle.

Q: How do I use the cosine and tangent functions to solve problems?

A: You can use the cosine and tangent functions to solve problems in a similar way to the sine function. You can use the unit circle and the reference angle to find the cosine and tangent functions. You can also use the fact that the cosine function is positive in the first and fourth quadrants, and the tangent function is positive in the first and third quadrants.

Q: What are some real-world applications of the cosine and tangent functions?

A: The cosine and tangent functions have many real-world applications, including physics, engineering, and navigation. In physics, the cosine and tangent functions are used to describe the motion of objects in circular motion. In engineering, the cosine and tangent functions are used to design and analyze electrical circuits. In navigation, the cosine and tangent functions are used to calculate distances and directions.

Q: Why are the cosine and tangent functions important in real-world applications?

A: The cosine and tangent functions are important in real-world applications because they help us to describe and analyze complex phenomena. In physics, the cosine and tangent functions help us to understand the motion of objects in circular motion. In engineering, the cosine and tangent functions help us to design and analyze electrical circuits. In navigation, the cosine and tangent functions help us to calculate distances and directions.