Reducing Fractions to Lowest Form

Oct 13, 2025 by miodeya.com 133 views

Introduction

Trigonometric functions are a crucial part of mathematics, and understanding how to evaluate them is essential for solving various mathematical problems. In this article, we will focus on evaluating the expression $\arcsin(\frac{1}{2})$ in radians. We will also discuss the importance of reducing fractions to their lowest form and provide examples of how to do so.

What is the Arcsine Function?

The arcsine function, denoted by $\arcsin(x)$ , is the inverse of the sine function. It returns the angle whose sine is a given number. In other words, if $\sin(\theta) = x$ , then $\arcsin(x) = \theta$ . The arcsine function is defined only for values of $x$ between $-1$ and $1$ , inclusive.

Evaluating the Expression

To evaluate the expression $\arcsin(\frac{1}{2})$ , we need to find the angle whose sine is $\frac{1}{2}$ . We know that the sine function has a period of $2\pi$ , so we can restrict our search to the interval $[0, \pi]$ . Within this interval, the sine function is increasing, so we can use the inverse sine function to find the angle.

Using a calculator or a trigonometric table, we find that $\sin(\frac{\pi}{6}) = \frac{1}{2}$ . Therefore, $\arcsin(\frac{1}{2}) = \frac{\pi}{6}$ .

Reducing Fractions to Lowest Form

When evaluating trigonometric expressions, it is often necessary to reduce fractions to their lowest form. This is because fractions can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).

For example, consider the fraction $\frac{6}{8}$ . To reduce this fraction to its lowest form, we need to find the GCD of $6$ and $8$ . The GCD of $6$ and $8$ is $2$ , so we can divide both the numerator and the denominator by $2$ to get $\frac{3}{4}$ .

Examples of Reducing Fractions

Here are a few more examples of reducing fractions to their lowest form:

$\frac{4}{6} = \frac{2}{3}$
$\frac{8}{12} = \frac{2}{3}$
$\frac{9}{18} = \frac{1}{2}$

Importance of Reducing Fractions

Reducing fractions to their lowest form is important for several reasons:

It makes it easier to compare fractions and determine which one is larger or smaller.
It makes it easier to add and subtract fractions.
It makes it easier to multiply and divide fractions.

Conclusion

In conclusion, evaluating trigonometric expressions in radians is an essential part of mathematics. The arcsine function is the inverse of the sine function, and it returns the angle whose sine is a given number. When evaluating trigonometric expressions, it is often necessary to reduce fractions to their lowest form. This makes it easier to compare fractions, add and subtract fractions, and multiply and divide fractions.

Final Answer

The final answer to the expression $\arcsin(\frac{1}{2})$ is $\boxed{\frac{\pi}{6}}$ .

Additional Resources

For more information on trigonometric functions and how to evaluate them, check out the following resources:

Khan Academy: Trigonometry
Mathway: Trigonometry
Wolfram Alpha: Trigonometry

Frequently Asked Questions

Here are a few frequently asked questions about trigonometric functions:

Q: What is the arcsine function?
A: The arcsine function is the inverse of the sine function. It returns the angle whose sine is a given number.
Q: How do I evaluate the expression $\arcsin(\frac{1}{2})$ ?
A: To evaluate the expression $\arcsin(\frac{1}{2})$ , you need to find the angle whose sine is $\frac{1}{2}$ . You can use a calculator or a trigonometric table to find this angle.
Q: Why is it important to reduce fractions to their lowest form?
A: Reducing fractions to their lowest form makes it easier to compare fractions, add and subtract fractions, and multiply and divide fractions.
Trigonometric Functions Q&A =============================

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are two of the most basic trigonometric functions. The sine function returns the ratio of the length of the side opposite a given angle to the length of the hypotenuse, while the cosine function returns the ratio of the length of the side adjacent to a given angle to the length of the hypotenuse.

Q: How do I evaluate the expression $\sin(\frac{\pi}{4})$ ?

A: To evaluate the expression $\sin(\frac{\pi}{4})$ , you need to find the sine of the angle $\frac{\pi}{4}$ . Using a calculator or a trigonometric table, you can find that $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ .

Q: What is the tangent function?

A: The tangent function is the ratio of the sine and cosine functions. It is defined as $\tan(x) = \frac{\sin(x)}{\cos(x)}$ . The tangent function is often used to find the angle whose sine and cosine are given.

Q: How do I evaluate the expression $\tan(\frac{\pi}{3})$ ?

A: To evaluate the expression $\tan(\frac{\pi}{3})$ , you need to find the tangent of the angle $\frac{\pi}{3}$ . Using a calculator or a trigonometric table, you can find that $\tan(\frac{\pi}{3}) = \sqrt{3}$ .

Q: What is the cotangent function?

A: The cotangent function is the reciprocal of the tangent function. It is defined as $\cot(x) = \frac{1}{\tan(x)}$ . The cotangent function is often used to find the angle whose sine and cosine are given.

Q: How do I evaluate the expression $\cot(\frac{\pi}{4})$ ?

A: To evaluate the expression $\cot(\frac{\pi}{4})$ , you need to find the cotangent of the angle $\frac{\pi}{4}$ . Using a calculator or a trigonometric table, you can find that $\cot(\frac{\pi}{4}) = 1$ .

Q: What is the secant function?

A: The secant function is the reciprocal of the cosine function. It is defined as $\sec(x) = \frac{1}{\cos(x)}$ . The secant function is often used to find the angle whose sine and cosine are given.

Q: How do I evaluate the expression $\sec(\frac{\pi}{4})$ ?

A: To evaluate the expression $\sec(\frac{\pi}{4})$ , you need to find the secant of the angle $\frac{\pi}{4}$ . Using a calculator or a trigonometric table, you can find that $\sec(\frac{\pi}{4}) = \sqrt{2}$ .

Q: What is the cosecant function?

A: The cosecant function is the reciprocal of the sine function. It is defined as $\csc(x) = \frac{1}{\sin(x)}$ . The cosecant function is often used to find the angle whose sine and cosine are given.

Q: How do I evaluate the expression $\csc(\frac{\pi}{3})$ ?

A: To evaluate the expression $\csc(\frac{\pi}{3})$ , you need to find the cosecant of the angle $\frac{\pi}{3}$ . Using a calculator or a trigonometric table, you can find that $\csc(\frac{\pi}{3}) = 2$ .

Q: What is the difference between the trigonometric functions and the inverse trigonometric functions?

A: The trigonometric functions are used to find the ratio of the sides of a triangle, while the inverse trigonometric functions are used to find the angle whose sine, cosine, or tangent is given.

Q: How do I evaluate the expression $\sin^{-1}(\frac{1}{2})$ ?

A: To evaluate the expression $\sin^{-1}(\frac{1}{2})$ , you need to find the angle whose sine is $\frac{1}{2}$ . Using a calculator or a trigonometric table, you can find that $\sin^{-1}(\frac{1}{2}) = \frac{\pi}{6}$ .

Q: What is the difference between the trigonometric identities and the trigonometric formulas?

A: The trigonometric identities are used to simplify trigonometric expressions, while the trigonometric formulas are used to find the values of trigonometric functions.

Q: How do I use the trigonometric identities to simplify the expression $\sin^2(x) + \cos^2(x)$ ?

A: To simplify the expression $\sin^2(x) + \cos^2(x)$ , you can use the trigonometric identity $\sin^2(x) + \cos^2(x) = 1$ . This identity is known as the Pythagorean identity.

Q: What is the difference between the trigonometric functions and the hyperbolic functions?

A: The trigonometric functions are used to find the ratio of the sides of a triangle, while the hyperbolic functions are used to find the ratio of the sides of a hyperbola.

Q: How do I evaluate the expression $\sinh(x)$ ?

A: To evaluate the expression $\sinh(x)$ , you need to find the hyperbolic sine of the angle $x$ . Using a calculator or a hyperbolic table, you can find that $\sinh(x) = \frac{e^x - e^{-x}}{2}$ .

Q: What is the difference between the trigonometric functions and the exponential functions?

A: The trigonometric functions are used to find the ratio of the sides of a triangle, while the exponential functions are used to find the values of exponential expressions.

Q: How do I evaluate the expression $e^x$ ?

A: To evaluate the expression $e^x$ , you need to find the value of the exponential function at the angle $x$ . Using a calculator or an exponential table, you can find that $e^x = e^x$ .

Q: What is the difference between the trigonometric functions and the logarithmic functions?

A: The trigonometric functions are used to find the ratio of the sides of a triangle, while the logarithmic functions are used to find the values of logarithmic expressions.

Q: How do I evaluate the expression $\log(x)$ ?

A: To evaluate the expression $\log(x)$ , you need to find the value of the logarithmic function at the angle $x$ . Using a calculator or a logarithmic table, you can find that $\log(x) = \log(x)$ .