What is the Difference between sin,sin^-1 and sinh^-1?

Converting fractions to decimals is an important foundational math skill. It demonstrates understanding of fractions as division and reveals decimal equivalents of common fractions. Changing a fraction to decimal form requires dividing the numerator by the denominator.

In this post, we’ll convert the fraction 1/3 to a decimal step-by-step and explain the underlying reasoning. We’ll also look at strategies for converting fractions and discuss insights into the repeating decimal nature of thirds.

What is the Difference between sin,sin^-1 and sinh^-1?

• sin(x) is the trigonometric function representing the sine of an angle x.
• sin^-1(x) is the inverse sine function, also known as arcsin, returning an angle whose sine is x.
• sinh^-1(x) is the inverse hyperbolic sine function, arcsinh, yielding a value such that the hyperbolic sine of that value equals x.

Here’s a table summarizing the differences between sin(x), sin^-1(x), and sinh^-1(x):

Function	Notation	Input Range	Output Range	Domain Restrictions
sin(x)	Sine	-∞ to ∞	-1 to 1	None
sin^-1(x)	Arcsine	-1 to 1	-π/2 to π/2	-1 ≤ x ≤ 1
sinh^-1(x)	Hyperbolic	-∞ to ∞	-∞ to ∞	None

Note: The domain restrictions for sin^-1(x) indicate that x must be within the range of -1 to 1 for the function to be defined.

Reviewing Fractions as Division

Let’s first review the concept that fractions represent division:

• The numerator is the dividend (number being divided).
• The denominator is the divisor (number dividing by).
• Any fraction a/b can be written as a ÷ b.
• To convert to decimal, divide numerator by denominator.

Keeping this fraction-as-division perspective in mind, let’s convert our fraction 1/3.

Converting 1/3 to a Decimal

To convert 1/3 to a decimal:

1. Set up long division: 1/3 = 1 ÷ 3
2. Calculate: 1 ÷ 3 = 0.333…

Therefore, 1/3 as a decimal equals 0.333… with the 3 repeating infinitely.

Explaining the Repeating Decimal

• 1/3 means dividing 1 by 3.
• 1 cannot be divided by 3 evenly.
• The remainder 1 is carried over repeatedly.
• This makes the 3 repeat endlessly in the decimal.

Understanding Fractions like Thirds

Fractions with denominators like 3, 6, 7, 11 etc will produce repeating decimals when converted to decimals. The repeats help identify fraction equivalents. Recognizing these patterns aids in conceptualizing fractions fundamentally as divisions.

Strategies for Converting Fractions to Decimals

Some helpful strategies:

• Double check long division setup for numerator ÷ denominator.
• Note any repeats and identify the equivalent fraction.
• Use a calculator to convert if unsure. Look for repeats.
• Memorize common ones like halves, quarters, tenths.
• Visualize fractions as division on a number line.#

FAQs

1. Is sin inverse and sinh the same?

• No, sin inverse (often denoted as sin⁻¹ or arcsin) and sinh are not the same. Sin inverse refers to the inverse trigonometric function of sine, while sinh is the hyperbolic sine function, a completely different mathematical function.

2. What is the difference between sin and sinh?

• The main difference between sin and sinh is that sin is a trigonometric function used to calculate the sine of an angle in a right triangle, while sinh is a hyperbolic function used to calculate the hyperbolic sine of a real number.

3. What is the difference between sin 1 and sin 1?

• “sin 1” typically represents the sine of an angle of 1 radian, while “sin 1” (with a space) doesn’t have a specific mathematical meaning. If you meant “sin(1)” in standard notation, it represents the sine of 1 degree.

4. Is Sin 1 the same as Asin?

• “Sin 1” and “Asin” are not the same. “Sin 1” usually refers to the sine of 1 degree or 1 radian, while “Asin” typically represents the inverse sine function, which is denoted as sin⁻¹ or arcsin.

5. What is sinh inverse?

• The inverse hyperbolic sine function is denoted as sinh⁻¹ or arcsinh. It is used to find the value of a real number for which sinh(x) equals a given value.

6. What is sinh equivalent to?

• Sinh(x) is equivalent to (e^x – e^(-x))/2, where “e” is the base of the natural logarithm (approximately 2.71828).

7. How do you convert sin to sinh?

• There is no direct conversion between sin and sinh because they represent different mathematical concepts. Sin is a trigonometric function, while sinh is a hyperbolic function. You cannot convert one into the other.

8. What is the relation between Sinx and Sinhx?

• Sinx and Sinhx are related in the context of complex numbers and Euler’s formula. Specifically, Euler’s formula states that e^(ix) = cos(x) + i*sin(x), and e^(x) = cosh(x) + sinh(x), where “i” is the imaginary unit. This relates the trigonometric functions (sin and cos) to their hyperbolic counterparts (sinh and cosh).

9. What does sinh differentiate to?

• The derivative of sinh(x) with respect to x is cosh(x).

10. Why do you use sin 1? – “Sin 1” is often used to represent the sine of a specific angle, which could be 1 degree or 1 radian, depending on the context of the problem.

11. What does sin 1 differentiate to? – The derivative of sin(1) with respect to x is 0 because sin(1) is a constant value.

12. What is sin 1 equal to? – Sin(1) is approximately equal to 0.01745240643 if 1 is measured in radians or approximately 0.8414709848 if 1 is measured in degrees.

13. Why is arcsin written as sin⁻¹? – Arcsin is often written as sin⁻¹ to denote the inverse sine function. This notation helps distinguish it from the reciprocal of sine (1/sin), which is not the same as the inverse function.

14. Is Arcsin the same as Sinx 1? – No, Arcsin and Sinx 1 are not the same. Arcsin represents the inverse sine function (denoted as sin⁻¹), while Sinx 1 is not a standard mathematical notation.

15. Is sin⁻¹ the reciprocal of sin? – No, sin⁻¹ is not the reciprocal of sin. Sin⁻¹ represents the inverse sine function, whereas the reciprocal of sin(x) is 1/sin(x).

16. What is the trig identity of sinh? – A trigonometric identity involving sinh is sinh^2(x) – cosh^2(x) = 1.

17. Is sinh opposite over hypotenuse? – No, the concept of opposite over hypotenuse is specific to right triangles and the trigonometric functions sine, cosine, and tangent. Sinh is not related to right triangles and does not have an opposite or hypotenuse.

18. Why are sinh called hyperbolic functions? – Hyperbolic functions, including sinh (hyperbolic sine) and cosh (hyperbolic cosine), are named after the hyperbola because they have mathematical properties similar to those of the trigonometric functions sine and cosine. They are used in various mathematical and scientific contexts, especially in areas involving exponential growth and decay.

Conclusion

In this post, we showed step-by-step how to convert the fraction 1/3 to its decimal equivalent 0.333… by treating it as long division. We also noted patterns in repeating decimals that reveal fraction identities.

Explaining the thought process reinforces the connections between equivalent fraction and decimal representations. Building fluency in interconverting fractions and decimals provides a key foundation for more advanced math and analytical applications. Mastering the basics prepares learners for success.