Problem 8 - Algebra 2 and Trigonometry

Question

Is arctan \(1=220^{\circ}\) a true statement? Justify your answer. \(y=\arccos 0\)

Step-by-Step Solution

Answer

No, \( \arctan(1) \neq 220^{\circ} \). \( y = \frac{\pi}{2} \).

Step 1: Understanding Arctan

The function \arctan(x)\ gives the angle \( \theta \) in radians such that \( \tan(\theta) = x \). The principal value of \( \arctan(x) \) is in the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Step 2: Convert 220 Degrees to Radians

Degrees can be converted to radians by multiplying by \( \frac{\pi}{180} \). Thus, \( 220^{\circ} = 220 \times \frac{\pi}{180} = \frac{11\pi}{9} \) radians.

Step 3: Comparing Radian Values

Compare if \( \arctan(1) \) equals \( \frac{11\pi}{9} \). Since \( \arctan(1) = \frac{\pi}{4} \) because \( \tan(\frac{\pi}{4}) = 1 \), and \( \frac{11\pi}{9} \) is not equal to \( \frac{\pi}{4} \), the statement is not true.

Step 4: Finding Arccos

The function \( \arccos(0) \) is the angle \( y \) such that \( \cos(y) = 0 \). The principal value of \( \arccos(0) \) is \( \frac{\pi}{2} \), as cosine is zero at \( \frac{\pi}{2} \) in the domain \( [0, \pi] \).

Key Concepts

Arctan

The inverse trigonometric function, known as arctan or \( \arctan(x) \), helps us find angles. More specifically, it provides the angle \( \theta \) for which \( \tan(\theta) = x \). This function is crucial when you need to reverse-engineer the tangent function to figure out an angle from a tangent value.

Remember that the principal value of \( \arctan(x) \) is confined within the interval \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), meaning we only consider angles in this range as possible outputs for the arctan function.

For instance, when you have \( \arctan(1) \), you are looking for an angle whose tangent is 1. The answer you find is \( \frac{\pi}{4} \), because the tangent of \( \frac{\pi}{4} \) is indeed 1. This is an essential check-point for solving inverse trigonometric problems.

Radians and Degrees

Radians and degrees are units of measurement for angles. Understanding how to convert between these units is important for solving trigonometry problems that mix them.

Degrees are a more familiar unit for many practical applications, commonly used in many settings from basic education to daily life. One full circle is 360 degrees. Radians, on the other hand, use the circumference of a circle for measurement, with one full circle being \( 2\pi \) radians.

To convert degrees to radians, multiply the degree measure by \( \frac{\pi}{180} \). For example, converting 220 degrees to radians involves \( 220 \times \frac{\pi}{180} = \frac{11\pi}{9} \) radians. Ensuring you can convert back and forth between these units is essential for solving mathematic problems where both are used.

Arccos

The inverse of the cosine function, known as arccos or \( \arccos(x) \), helps find the angle \( y \) for which \( \cos(y) = x \).

With all inverse cosine problems, we're looking at the interval \( [0, \pi] \) according to convention. This is because the cosine function is uniquely invertible over this range.

For example, \( \arccos(0) \) will return \( \frac{\pi}{2} \). This is because cosine reaches zero at an angle of \( \frac{\pi}{2} \). Understanding which angles correspond to specific cosine values is key for tackling inverse trigonometric function exercises.

Trigonometric Identities

Trigonometric identities are valuable equations involving trigonometric functions that are universally true for any angle. They can simplify complex trigonometric expressions and make it easier to solve equations.

Here are some of the most commonly used trigonometric identities:

• Pythagorean Identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Angle Sum and Difference Identities: These describe how to handle sums and differences of angles, such as \( \sin(a + b) \) or \( \cos(a - b) \).
• Double Angle Formulas: Such as \( \sin(2x) = 2\sin(x)\cos(x) \) and others, which relate trigonometric functions of double angles back to the original angle.
• Reciprocal Identities: Like \( \sec(x) = \frac{1}{\cos(x)} \), which relate trigonometric functions to each other.

These identities are instrumental in reducing the complexity of trigonometric expressions, proving mathematical equality, or solving trigonometric equations. They form an essential toolkit for anyone working with trigonometric functions.