Converting the Angle to a Standard Angle

Oct 13, 2025 by miodeya.com 62 views

Introduction

Trigonometric equations are an essential part of mathematics, and solving them requires a deep understanding of the subject. In this article, we will focus on finding the exact value of $\sin \left(157.5^{\circ}\right)$ . This problem involves using trigonometric identities and formulas to simplify the expression and arrive at the correct answer.

Understanding the Problem

The problem asks us to find the exact value of $\sin \left(157.5^{\circ}\right)$ . To solve this problem, we need to use trigonometric identities and formulas to simplify the expression. We can start by converting the angle from degrees to radians, but in this case, we will work with degrees.

Converting the Angle to a Standard Angle

To solve this problem, we can use the fact that the sine function has a period of $360^{\circ}$ . This means that we can add or subtract multiples of $360^{\circ}$ from the angle without changing the value of the sine function. We can also use the fact that the sine function is an odd function, which means that $\sin (-x) = -\sin x$ .

Using Trigonometric Identities

We can use the trigonometric identity $\sin (x) = \sin (180^{\circ} - x)$ to rewrite the expression as $\sin (157.5^{\circ}) = \sin (22.5^{\circ})$ . This is because $157.5^{\circ} = 180^{\circ} - 22.5^{\circ}$ .

Finding the Exact Value of $\sin (22.5^{\circ})$

To find the exact value of $\sin (22.5^{\circ})$ , we can use the fact that $\sin (22.5^{\circ}) = \frac{\sqrt{2-\sqrt{2}}}{2}$ . This is because $22.5^{\circ}$ is a special angle that can be expressed in terms of the square root of 2.

Verifying the Answer

To verify the answer, we can use a calculator to find the approximate value of $\sin (157.5^{\circ})$ . We can also use the fact that the sine function is an odd function to check if the answer is correct.

Conclusion

In this article, we have shown how to find the exact value of $\sin \left(157.5^{\circ}\right)$ . We used trigonometric identities and formulas to simplify the expression and arrive at the correct answer. We also verified the answer using a calculator and the fact that the sine function is an odd function.

The Final Answer

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

Discussion

This problem is a great example of how to use trigonometric identities and formulas to solve trigonometric equations. It also shows how to verify the answer using a calculator and the fact that the sine function is an odd function. If you have any questions or comments, please feel free to ask.

Solving Trigonometric Equations: Tips and Tricks

Here are some tips and tricks for solving trigonometric equations:

Use trigonometric identities and formulas to simplify the expression.
Use the fact that the sine function is an odd function to check if the answer is correct.
Use a calculator to find the approximate value of the trigonometric function.
Use the fact that the trigonometric functions have a period of $360^{\circ}$ to simplify the expression.

Conclusion

Solving trigonometric equations requires a deep understanding of the subject. By using trigonometric identities and formulas, we can simplify the expression and arrive at the correct answer. We also verified the answer using a calculator and the fact that the sine function is an odd function. If you have any questions or comments, please feel free to ask.

Introduction

In our previous article, we discussed how to find the exact value of $\sin \left(157.5^{\circ}\right)$ . We used trigonometric identities and formulas to simplify the expression and arrive at the correct answer. In this article, we will answer some frequently asked questions about solving trigonometric equations.

Q: What are some common trigonometric identities that I should know?

A: There are several common trigonometric identities that you should know when solving trigonometric equations. Some of the most common ones include:

$\sin^2 x + \cos^2 x = 1$
$\tan x = \frac{\sin x}{\cos x}$
$\cot x = \frac{\cos x}{\sin x}$
$\sec x = \frac{1}{\cos x}$
$\csc x = \frac{1}{\sin x}$

Q: How do I simplify a trigonometric expression?

A: To simplify a trigonometric expression, you can use the following steps:

Use the trigonometric identities to rewrite the expression in a simpler form.
Use the fact that the sine function is an odd function to simplify the expression.
Use the fact that the cosine function is an even function to simplify the expression.
Use the fact that the trigonometric functions have a period of $360^{\circ}$ to simplify the expression.

Q: How do I find the exact value of a trigonometric function?

A: To find the exact value of a trigonometric function, you can use the following steps:

Use the trigonometric identities to rewrite the expression in a simpler form.
Use the fact that the sine function is an odd function to simplify the expression.
Use the fact that the cosine function is an even function to simplify the expression.
Use the fact that the trigonometric functions have a period of $360^{\circ}$ to simplify the expression.
Use a calculator to find the approximate value of the trigonometric function.

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

A: There are several common trigonometric functions that you should know when solving trigonometric equations. Some of the most common ones include:

$\sin x$
$\cos x$
$\tan x$
$\cot x$
$\sec x$
$\csc x$

Q: How do I use a calculator to find the approximate value of a trigonometric function?

A: To use a calculator to find the approximate value of a trigonometric function, you can follow these steps:

Enter the angle in degrees or radians.
Select the trigonometric function that you want to evaluate.
Press the "Enter" or "Calculate" button to get the approximate value of the trigonometric function.

Q: What are some common mistakes that I should avoid when solving trigonometric equations?

A: There are several common mistakes that you should avoid when solving trigonometric equations. Some of the most common ones include:

Not using the correct trigonometric identities.
Not simplifying the expression correctly.
Not using the fact that the sine function is an odd function.
Not using the fact that the cosine function is an even function.
Not using the fact that the trigonometric functions have a period of $360^{\circ}$ .

Conclusion

If you are interested in learning more about solving trigonometric equations, here are some related articles:

Solving Trigonometric Equations: Finding the Exact Value of $\sin \left(157.5^{\circ}\right)$
Solving Trigonometric Equations: Tips and Tricks
Solving Trigonometric Equations: Common Mistakes to Avoid

Discussion

This article is a great resource for anyone who wants to learn more about solving trigonometric equations. If you have any questions or comments, please feel free to ask. We would be happy to help you with any questions or concerns that you may have.