Problem 11 - Precalculus: Functions and Graphs

Question

Find the exact value of the expression whenever It is defined. (a) \(\arcsin \left(\sin \frac{5 \pi}{4}\right)\) (b) \(\arccos \left(\cos \frac{5 \pi}{4}\right)\) (c) arctan \(\left(\tan \frac{7 \pi}{4}\right)\)

Step-by-Step Solution

Answer

(a) \(-\frac{\pi}{4}\), (b) \(\frac{3\pi}{4}\), (c) \(-\frac{\pi}{4}\).

Step 1: Simplify the Sine Expression

The given expression is \( \arcsin(\sin(\frac{5\pi}{4})) \). First, simplify \( \sin(\frac{5\pi}{4}) \). The angle \( \frac{5\pi}{4} \) is located in the third quadrant (180° to 270°) where sine is negative. The reference angle for \( \frac{5\pi}{4} \) is \( \frac{\pi}{4} \), thus: \( \sin(\frac{5\pi}{4}) = -\sin(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2} \).

Step 2: Evaluate Arcsin

Next, evaluate \( \arcsin(-\frac{\sqrt{2}}{2}) \). The arcsine function \( \arcsin(x) \) provides an angle in the range \([ -\frac{\pi}{2}, \frac{\pi}{2}] \). The value for which sine is \(-\frac{\sqrt{2}}{2}\) in this range is \(-\frac{\pi}{4}\).

Step 3: Simplify the Cosine Expression

The given expression is \( \arccos(\cos(\frac{5\pi}{4})) \). First, simplify \( \cos(\frac{5\pi}{4}) \). The angle \( \frac{5\pi}{4} \) is in the third quadrant, where cosine is negative. So, \( \cos(\frac{5\pi}{4}) = -\cos(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2} \).

Step 4: Evaluate Arccos

Next, evaluate \( \arccos(-\frac{\sqrt{2}}{2}) \). The arccosine function \( \arccos(x) \) provides an angle in the range \([0, \pi] \). The angle for which cosine is \(-\frac{\sqrt{2}}{2}\) in this range is \(\frac{3\pi}{4}\).

Step 5: Simplify the Tangent Expression

The given expression is \( \arctan(\tan(\frac{7\pi}{4})) \). First, simplify \( \tan(\frac{7\pi}{4}) \). The angle \( \frac{7\pi}{4} \) is located in the fourth quadrant where tangent is negative. Furthermore, it is coterminal with \( -\frac{\pi}{4} \), since \( \tan(\frac{7\pi}{4}) = \tan(-\frac{\pi}{4}) \). So, \( \tan(\frac{7\pi}{4}) = -1 \).

Step 6: Evaluate Arctan

Finally, evaluate \( \arctan(-1) \). The arctangent function \( \arctan(x) \) provides an angle in the range \(( -\frac{\pi}{2}, \frac{\pi}{2}) \). For \( \arctan(-1) \), the angle is \(-\frac{\pi}{4}\).

Key Concepts

Understanding Arcsin

The arcsine function, denoted as `\( \arcsin \)` or sometimes `\( \sin^{-1} \)`, is the inverse of the sine function. It allows us to find the angle whose sine value is a specific number within the range of \( [-1, 1] \). The output is restricted to \( \left[-\frac{\pi}{2},\frac{\pi}{2}\right] \) or \([-90^\circ, 90^\circ]\) in degrees.
### Key Points about Arcsin- **Range**: The result from `\( \arcsin \)` is always an angle between \( \left[-\frac{\pi}{2},\frac{\pi}{2}\right] \). This means the angle is always located in the first or fourth quadrants.- **Usage**: Useful in trigonometry problems where you have to backtrack to find an angle when given the sine result.- **Example**: If \( \sin(\theta) = -\frac{\sqrt{2}}{2} \), then \( \arcsin\left(-\frac{\sqrt{2}}{2}\right) = -\frac{\pi}{4} \) because \(-\frac{\pi}{4}\) is in the range \( \left[-\frac{\pi}{2},\frac{\pi}{2}\right] \) where sine has this result.
Understanding these properties is crucial for solving problems where we need to find angles using the sine values in \([ -1, 1] \).

Exploring Arccos

The arccosine function, denoted as `\( \arccos \)` or occasionally `\( \cos^{-1} \)`, is the inverse of the cosine function. It determines the angle whose cosine value equals a specific number within the range of \([ -1, 1] \). The output values are restricted to \([0, \pi]\) or \([0^\circ, 180^\circ]\) in degrees.
### Key Points about Arccos- **Range**: This function will give an angle always between \([0, \pi]\), meaning it covers the first and second quadrants.- **Usage**: In trigonometry, it helps us find exact angles when the cosine result is given.- **Example**: If \( \cos(\theta) = -\frac{\sqrt{2}}{2} \), then \( \arccos\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4} \) which is in the allowed range for `\( \arccos \)`.
By understanding this, one can solve trigonometric equations where cosine values need interpreting through angles.

Investigating Arctan

The arctangent function, denoted as `\( \arctan \)`, or sometimes `\( \tan^{-1} \)`, is the inverse of the tangent function. It indicates the angle whose tangent value matches a specific number, usually falling within the entire set of real numbers. The resulting angles are restricted to the range \(( -\frac{\pi}{2}, \frac{\pi}{2})\) or \((-90^\circ, 90^\circ)\) in degrees.
### Key Points about Arctan- **Range**: The angle resulting from `\( \arctan \)` is always found in the intervals \((-\frac{\pi}{2}, \frac{\pi}{2})\), pertinent to the first and fourth quadrants.- **Usage**: It is incredibly useful in situations involving slopes and angle measurements given a specific tangent.- **Example**: For \( \tan(\theta) = -1 \), \( \arctan(-1) = -\frac{\pi}{4} \) since \(-\frac{\pi}{4}\) lies in the defined range of `\( \arctan \)`.
Arctan caters to problems that necessitate determining angles from tangent values, making it essential for various applications in trigonometry.