Which Equation Shows That The Pythagorean Identity Is True For Θ = 3 Π 2 \theta=\frac{3 \pi}{2} Θ = 2 3 Π ​ ?Select The Equation That Is In The Form Sin ⁡ 2 ( 3 Π 2 ) + Cos ⁡ 2 ( 3 Π 2 ) = 1 \sin^2\left(\frac{3 \pi}{2}\right) + \cos^2\left(\frac{3 \pi}{2}\right) = 1 Sin 2 ( 2 3 Π ​ ) + Cos 2 ( 2 3 Π ​ ) = 1 .A. 1 2 + 0 2 = 1 1^2 + 0^2 = 1 1 2 + 0 2 = 1

The Pythagorean identity is a fundamental concept in trigonometry that relates the sine and cosine functions. It states that for any angle $\theta$, the following equation holds:

$$\sin^2(\theta) + \cos^2(\theta) = 1$$

This identity is a cornerstone of trigonometric mathematics and has numerous applications in various fields, including physics, engineering, and mathematics.

Evaluating the Pythagorean Identity for $\theta=\frac{3 \pi}{2}$

To determine which equation shows that the Pythagorean identity is true for $\theta=\frac{3 \pi}{2}$, we need to evaluate the sine and cosine functions at this specific angle.

Recall the Unit Circle

The unit circle is a fundamental concept in trigonometry that helps us visualize the relationships between angles and their corresponding sine and cosine values. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane.

Sine and Cosine Values for $\theta=\frac{3 \pi}{2}$

Using the unit circle, we can determine the sine and cosine values for $\theta=\frac{3 \pi}{2}$.

• The sine function is positive in the second and third quadrants, and the cosine function is negative in the second and third quadrants.
• At $\theta=\frac{3 \pi}{2}$, the point on the unit circle has coordinates $(0, -1)$.
• Therefore, the sine value is $-1$, and the cosine value is $0$.

Evaluating the Pythagorean Identity

Now that we have the sine and cosine values for $\theta=\frac{3 \pi}{2}$, we can evaluate the Pythagorean identity:

$$\sin^2\left(\frac{3 \pi}{2}\right) + \cos^2\left(\frac{3 \pi}{2}\right) = (-1)^2 + 0^2 = 1$$

Selecting the Correct Equation

Based on our evaluation, the correct equation that shows the Pythagorean identity is true for $\theta=\frac{3 \pi}{2}$ is:

$$1^2 + 0^2 = 1$$

This equation matches the form $\sin^2\left(\frac{3 \pi}{2}\right) + \cos^2\left(\frac{3 \pi}{2}\right) = 1$.

Conclusion

In conclusion, the Pythagorean identity is a fundamental concept in trigonometry that relates the sine and cosine functions. By evaluating the sine and cosine values for $\theta=\frac{3 \pi}{2}$, we can determine that the correct equation that shows the Pythagorean identity is true for this angle is:

$$1^2 + 0^2 = 1$$

This equation matches the form $\sin^2\left(\frac{3 \pi}{2}\right) + \cos^2\left(\frac{3 \pi}{2}\right) = 1$.

Additional Resources

For further reading on the Pythagorean identity and trigonometry, we recommend the following resources:

• Khan Academy: Trigonometry
• MIT OpenCourseWare: 18.01 Single Variable Calculus
• Wolfram MathWorld: Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that relates the sine and cosine functions. In this article, we will answer some frequently asked questions about the Pythagorean identity.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a mathematical equation that relates the sine and cosine functions. It states that for any angle $\theta$, the following equation holds:

$$\sin^2(\theta) + \cos^2(\theta) = 1$$

Q: Why is the Pythagorean identity important?

A: The Pythagorean identity is important because it provides a fundamental relationship between the sine and cosine functions. It is used in a wide range of applications, including physics, engineering, and mathematics.

Q: How do I prove the Pythagorean identity?

A: There are several ways to prove the Pythagorean identity. One way is to use the unit circle and the definitions of the sine and cosine functions. Another way is to use the trigonometric identities for the sum and difference of angles.

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

A: The Pythagorean identity has numerous applications in various fields, including:

• Physics: The Pythagorean identity is used to describe the relationships between the sine and cosine functions in physics problems, such as the motion of objects and the behavior of waves.
• Engineering: The Pythagorean identity is used in engineering to describe the relationships between the sine and cosine functions in problems involving the design of electrical circuits and the analysis of mechanical systems.
• Mathematics: The Pythagorean identity is used in mathematics to prove other trigonometric identities and to solve problems involving the sine and cosine functions.

Q: Can I use the Pythagorean identity to solve problems involving the sine and cosine functions?

A: Yes, the Pythagorean identity can be used to solve problems involving the sine and cosine functions. By using the Pythagorean identity, you can simplify complex expressions and solve problems more easily.

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 trigonometric identities: Make sure to use the correct trigonometric identities when applying the Pythagorean identity.
• Not simplifying expressions: Make sure to simplify expressions when applying the Pythagorean identity.
• Not checking units: Make sure to check units when applying the Pythagorean identity.

Q: Can I use the Pythagorean identity to prove other trigonometric identities?

A: Yes, the Pythagorean identity can be used to prove other trigonometric identities. By using the Pythagorean identity, you can derive other trigonometric identities and simplify complex expressions.

Conclusion

In conclusion, the Pythagorean identity is a fundamental concept in trigonometry that relates the sine and cosine functions. By understanding the Pythagorean identity and its applications, you can gain a deeper appreciation for the beauty and power of trigonometry.

Additional Resources

For further reading on the Pythagorean identity and trigonometry, we recommend the following resources:

• Khan Academy: Trigonometry
• MIT OpenCourseWare: 18.01 Single Variable Calculus
• Wolfram MathWorld: Pythagorean Identity

By understanding the Pythagorean identity and its applications, you can gain a deeper appreciation for the beauty and power of trigonometry.