Evaluating Inverse Trigonometric Functions

Summary

This video script offers a comprehensive tutorial on evaluating inverse trigonometric functions, focusing on arcsine, arccosine, and arctangent. It explains the range and domain of these functions, emphasizing why certain angles are chosen over others based on their quadrant and the function's range. The script clarifies common confusions, such as why arcsine of 1/2 is π/6 and not 5π/6, and demonstrates how to calculate values for various inverse trigonometric expressions, including compositions of functions.

Takeaways

• 📏 The range of the arcsine function is from -π/2 to π/2, which means it only accounts for angles in Quadrants I and IV within that range.
• 🔍 For arcsine(1/2), the correct answer is π/6, not 5π/6, because 5π/6 is outside the range of the arcsine function.
• 🔺 When evaluating inverse trigonometric functions, consider the quadrant in which the function is defined and ensure the angle falls within the function's range.
• 📐 The arcsine function is positive in Quadrant I and negative in Quadrant IV, which is crucial for determining the correct angle for a given sine value.
• 🌀 For the arcsine of values like √3/2, the angle must be in Quadrant I (60 degrees or π/3 radians), not in Quadrant II (120 degrees or 2π/3 radians).
• ❌ Angles in Quadrants II and III are not valid for the arcsine function due to its range and definition.
• 📉 The arccosine function has a range from 0 to π and is defined in Quadrants I and II, meaning it only accounts for angles in those quadrants within the range.
• 🔄 When evaluating compositions of inverse trigonometric functions, such as arcsine of sine, the result simplifies to the original angle if it falls within the function's range.
• 🔢 For specific values like arcsine(-1/2), the correct angle is -30 degrees (or -π/6 radians), which is within the range and the appropriate quadrant.
• 📋 The key to solving inverse trigonometric problems is understanding the range and domain of each function and applying these constraints to find the correct angle.

Q & A

What is the value of the inverse sine of one half?

The value of the inverse sine of one half is pi over six, which is equivalent to 30 degrees. This is because the sine of pi over six (or 30 degrees) is equal to one half, and the range of the arcsine function is from negative pi over two to pi over two, which includes pi over six but excludes five pi over six.

Why is the arcsine of one half not equal to five pi over six?

The arcsine of one half is not equal to five pi over six because the range of the arcsine function is from negative pi over two to pi over two. Five pi over six is outside this range, as it falls outside the interval from -90 to 90 degrees.

In which quadrants does the arcsine function exist?

The arcsine function exists in quadrants one and four. This is because the sine function is positive in quadrant one and negative in quadrant four, and the arcsine function must return an angle within the range of -90 to 90 degrees.

How do you determine the quadrant for the arcsine function when given a positive value?

For a positive value, the arcsine function will return an angle in quadrant one because sine is positive in this quadrant. The angle will be within the range of 0 to 90 degrees.

What is the value of the arcsine of the square root of three over two?

The value of the arcsine of the square root of three over two is 60 degrees, which can also be written as pi over three. This is because the sine of 60 degrees (or pi over three radians) is equal to the square root of three over two, and 60 degrees is within the range of the arcsine function.

Why is the arcsine of negative one-half equal to negative 30 degrees?

The arcsine of negative one-half is equal to negative 30 degrees because the sine of negative 30 degrees (or negative pi over six radians) is equal to negative one-half, and negative 30 degrees is within the range of the arcsine function, which is from negative 90 to 90 degrees.

What is the range of the arccosine function?

The range of the arccosine function is from 0 to pi. This means that the arccosine of a value will return an angle within this range, typically in radians.

In which quadrants does the arccosine function exist?

The arccosine function exists in quadrants one and two. This is because the cosine function is positive in quadrant one and negative in quadrant two, and the arccosine function must return an angle within the range of 0 to 180 degrees.

What is the value of the arccosine of one half?

The value of the arccosine of one half is pi over three, which is equivalent to 60 degrees. This is because the cosine of 60 degrees (or pi over three radians) is equal to one half, and 60 degrees is within the range of the arccosine function.

How do you evaluate the composition of an inverse trig function with a trig function, such as arcsine of sine pi over three?

When evaluating a composition like arcsine of sine pi over three, you first find the value of the sine function, which is the square root of three over two. Then, since the arcsine function will return an angle in quadrants one or four that corresponds to this value, and pi over three is in quadrant one, the final answer is simply pi over three.