Not using the correct values for sine and cosine

Oct 13, 2025 by talynt.org 1 views

The Pythagorean identity is a fundamental concept in trigonometry, which states that the sum of the squares of the sine and cosine of an angle is equal to 1. This identity is expressed as $\sin^2 \theta + \cos^2 \theta = 1$ , where $\theta$ is the angle in question. In this article, we will explore which equation shows that the Pythagorean identity is true for $\theta=270^{\circ}$ .

The Pythagorean Identity: A Brief Overview

The Pythagorean identity is a fundamental concept in trigonometry, which states that the sum of the squares of the sine and cosine of an angle is equal to 1. This identity is expressed as $\sin^2 \theta + \cos^2 \theta = 1$ , where $\theta$ is the angle in question. The Pythagorean identity is a fundamental concept in trigonometry, and it has numerous applications in various fields, including physics, engineering, and mathematics.

**Understanding the Angle $\theta=270^{\circ}$

In this article, we will explore which equation shows that the Pythagorean identity is true for $\theta=270^{\circ}$ . To understand this, we need to recall the values of sine and cosine for the angle $270^{\circ}$ . The sine of $270^{\circ}$ is equal to -1, and the cosine of $270^{\circ}$ is equal to 0.

Evaluating the Options

Now that we have understood the values of sine and cosine for the angle $270^{\circ}$ , we can evaluate the options given in the problem. The options are:

A. $(-1)^2 + 0^2 = 1$ B. $0^2 + (-1)^2 = 1$ C. $0^2 + 0^2 = 1$

Option A: $(-1)^2 + 0^2 = 1$

Let's evaluate option A. We know that the sine of $270^{\circ}$ is equal to -1, and the cosine of $270^{\circ}$ is equal to 0. Therefore, we can substitute these values into the equation:

$(-1)^2 + 0^2 = 1$

Simplifying the equation, we get:

$1 + 0 = 1$

This equation is true, and it shows that the Pythagorean identity is true for $\theta=270^{\circ}$ .

Option B: $0^2 + (-1)^2 = 1$

Let's evaluate option B. We know that the sine of $270^{\circ}$ is equal to -1, and the cosine of $270^{\circ}$ is equal to 0. Therefore, we can substitute these values into the equation:

$0^2 + (-1)^2 = 1$

Simplifying the equation, we get:

$0 + 1 = 1$

This equation is also true, and it shows that the Pythagorean identity is true for $\theta=270^{\circ}$ .

Option C: $0^2 + 0^2 = 1$

Let's evaluate option C. We know that the sine of $270^{\circ}$ is equal to -1, and the cosine of $270^{\circ}$ is equal to 0. Therefore, we can substitute these values into the equation:

$0^2 + 0^2 = 1$

Simplifying the equation, we get:

$0 + 0 = 1$

This equation is not true, and it does not show that the Pythagorean identity is true for $\theta=270^{\circ}$ .

Conclusion

The Pythagorean identity is a fundamental concept in trigonometry, and it has numerous applications in various fields, including physics, engineering, and mathematics. In this article, we will answer some frequently asked questions (FAQs) about the Pythagorean identity.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry, which states that the sum of the squares of the sine and cosine of an angle is equal to 1. This identity is expressed as $\sin^2 \theta + \cos^2 \theta = 1$ , where $\theta$ is the angle in question.

Q: What is the significance of the Pythagorean identity?

A: The Pythagorean identity is significant because it provides a relationship between the sine and cosine of an angle. This relationship is essential in various fields, including physics, engineering, and mathematics. The Pythagorean identity is used to solve problems involving right triangles, and it is also used in the derivation of other trigonometric identities.

Q: How is the Pythagorean identity used in real-world applications?

A: The Pythagorean identity is used in various real-world applications, including:

Physics: The Pythagorean identity is used to calculate the distance and velocity of objects in motion.
Engineering: The Pythagorean identity is used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
Mathematics: The Pythagorean identity is used to prove other trigonometric identities and to solve problems involving right triangles.

Q: What are some common mistakes to avoid when using the Pythagorean identity?

A: Some common mistakes to avoid when using the Pythagorean identity include:

Not using the correct values for sine and cosine: Make sure to use the correct values for sine and cosine for the angle in question.
Not simplifying the equation: Make sure to simplify the equation to ensure that it is in the correct form.
Not checking the units: Make sure to check the units of the equation to ensure that they are correct.

Q: How can I prove the Pythagorean identity?

A: There are several ways to prove the Pythagorean identity, including:

Using the definition of sine and cosine: Use the definition of sine and cosine to prove the Pythagorean identity.
Using the Pythagorean theorem: Use the Pythagorean theorem to prove the Pythagorean identity.
Using other trigonometric identities: Use other trigonometric identities to prove the Pythagorean identity.

Q: What are some common applications of the Pythagorean identity?

A: Some common applications of the Pythagorean identity include:

Right triangle problems: The Pythagorean identity is used to solve problems involving right triangles.
Trigonometric identities: The Pythagorean identity is used to prove other trigonometric identities.
Engineering and physics problems: The Pythagorean identity is used to solve problems involving electrical circuits, mechanical systems, and other engineering systems.

Q: Can I use the Pythagorean identity to solve problems involving non-right triangles?

A: Yes, you can use the Pythagorean identity to solve problems involving non-right triangles. However, you will need to use other trigonometric identities to solve the problem.

Conclusion

In conclusion, the Pythagorean identity is a fundamental concept in trigonometry, and it has numerous applications in various fields, including physics, engineering, and mathematics. By understanding the Pythagorean identity and its applications, you can solve problems involving right triangles and other trigonometric identities.