Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin \left(\arcsin \frac{1}{2}+\arccos 0\right)\) (b) \(\cos \left[\arctan \left(-\frac{3}{4}\right)-\arcsin \frac{4}{5}\right]\) (c) \(\tan \left(\arctan \frac{4}{3}+\arccos \frac{8}{17}\right)\)

Step-by-Step Solution

Answer

(a) \(\frac{\sqrt{3}}{2}\), (b) 0, (c) \(-\frac{77}{36}\).

Step 1: Simplify \(\arcsin \left( \frac{1}{2} \right)\)

\(\arcsin \left( \frac{1}{2} \right)\) represents the angle whose sine is \(\frac{1}{2}\). This angle is \(\frac{\pi}{6}\) or 30 degrees.

Step 2: Simplify \(\arccos(0)\)

\(\arccos(0)\) represents the angle whose cosine is 0. This angle is \(\frac{\pi}{2}\) or 90 degrees.

Step 3: Evaluate \(\sin\left(\arcsin \frac{1}{2} + \arccos 0\right)\)

Add the angles: \(\frac{\pi}{6} + \frac{\pi}{2} = \frac{2\pi}{3}\). The sine of \(\frac{2\pi}{3}\) is \(\frac{\sqrt{3}}{2}\).

Step 4: Simplify \(\arctan \left( -\frac{3}{4}\right)\)

\(\arctan \left( -\frac{3}{4} \right)\) is the angle whose tangent is \(-\frac{3}{4}\). Denote this angle as \(\theta\).

Step 5: Simplify \(\arcsin \left( \frac{4}{5}\right)\)

\(\arcsin \left( \frac{4}{5} \right)\) is the angle whose sine is \(\frac{4}{5}\). Denote this angle as \(\phi\).

Step 6: Use cosine of angle difference formula

The cosine of the difference of two angles is \(\cos(\theta - \phi) = \cos \theta \cos \phi + \sin \theta \sin \phi\). For \(\theta\) and \(\phi\), first find their respective sine and cosine values.

Step 7: Calculate sine and cosine of \(\theta\) and \(\phi\)

For \(\theta\), use: \(\sin \theta = -\frac{3}{5}\), \(\cos \theta = \frac{4}{5}\). For \(\phi\), use: \(\sin \phi = \frac{4}{5}\), \(\cos \phi = \frac{3}{5}\).

Step 8: Calculate \(\cos(\theta - \phi)\)

\(\cos(\theta - \phi) = \frac{4}{5} \cdot \frac{3}{5} + \left(-\frac{3}{5}\right) \cdot \frac{4}{5} = \frac{12}{25} - \frac{12}{25} = 0\).

Step 9: Simplify \(\arctan \left( \frac{4}{3} \right)\)

\(\arctan \left( \frac{4}{3} \right)\) is the angle whose tangent is \(\frac{4}{3}\). Denote this angle as \(\psi\).

Step 10: Simplify \(\arccos \left( \frac{8}{17} \right)\)

\(\arccos \left( \frac{8}{17} \right)\) is the angle whose cosine is \(\frac{8}{17}\). Denote this angle as \(\chi\).

Step 11: Use tangent of angle sum formula

The tangent of the sum of two angles is \(\tan(\psi + \chi) = \frac{\tan \psi + \tan \chi}{1 - \tan \psi \tan \chi}\). First, find \(\tan \chi\).

Step 12: Calculate \(\tan \chi\)

Using \(\cos \chi = \frac{8}{17}\), find \(\sin \chi = \sqrt{1 - \left(\frac{8}{17}\right)^2} = \frac{15}{17}\). Thus, \(\tan \chi = \frac{15}{8}\).

Step 13: Calculate \(\tan(\psi + \chi)\) using the formula

\(\tan(\psi + \chi) = \frac{\frac{4}{3} + \frac{15}{8}}{1 - \frac{4}{3} \cdot \frac{15}{8}} = \frac{\frac{32}{24} + \frac{45}{24}}{1 - \frac{60}{24}} = \frac{\frac{77}{24}}{-1.5} = -\frac{77}{36}\).

Step 14: Answer the Question

Calculate the total expressions for parts (a), (b), and (c) for their respective trigonometric functions:

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find the angles associated with certain trigonometric ratios. When working with \( \arcsin, \arccos, \) and \( \arctan \), you're essentially finding an angle with a known sine, cosine, or tangent value. \( \arcsin(x) \) returns an angle whose sine is \( x \), while \( \arccos(x) \) provides the angle whose cosine is \( x \). Lastly, \( \arctan(x) \) finds an angle whose tangent is \( x \).
For example, \( \arcsin\left( \frac{1}{2} \right) \) gives us the angle 30 degrees (or \( \frac{\pi}{6} \) radians), because the sine of 30 degrees is \( \frac{1}{2} \). Similarly, \( \arccos(0) = \frac{\pi}{2} \) radians because the cosine of 90 degrees is zero.
These functions are crucial because they help determine angles in various mathematical and real-world applications, enabling calculations involving angles that are not "standard" angles such as 30, 45, or 60 degrees.

Angle Sum and Difference Identities

Trigonometric identities for angle sums and differences let you evaluate trig functions for sums or differences of two angles. The sine, cosine, and tangent of these combined angles are given by specific formulas.
The sine of a sum or difference is represented as:

• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
• \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)

For cosine, the formulas are:

• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

For tangent, the formulas are:

• \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)
• \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \)

This allows for solving more complex expressions like those found in the exercise, where angles determined by inverse functions are combined and evaluated using the above identities. It's vital to find each original angle and substitute into these formulas accurately, accounting for the sign and relative magnitude of sine, cosine, and tangent values.

Exact Values of Trigonometric Expressions

Finding the exact values of trigonometric expressions involves using known values of sine, cosine, and tangent for common angles. It often includes simplifying expressions by employing identities or evaluating inverse trigonometric results.
For example, knowing the exact value of sine and cosine for \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3} \), etc., is essential. In the exercise, for instance, the exact value of \( \sin\left(\frac{2\pi}{3}\right) \) involves understanding that it equals \( \frac{\sqrt{3}}{2} \), based on the unit circle properties and transformations.
These exact values are crucial for accurately solving math problems and understanding complex trigonometric relationships. Using a combination of \( \arcsin \, \arccos \, \arctan \, \sin \, \cos \, \) and other trigonometric identities helps in obtaining these values efficiently.