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Magazine Chapter 8: Problem 18
Find the exact value of the expression, whenever it is defined. a) \(\sin \left[\sin ^{-1} \frac{5}{13}-\cos ^{-1}\left(-\frac{3}{5}\right)\right]\) (b) \(\cos \left(\sin ^{-1} \frac{4}{5}+\tan ^{-1} \frac{3}{4}\right)\) c) \(\tan \left[\cos ^{-1} \frac{1}{2}-\sin ^{-1}\left(-\frac{1}{2}\right)\right]\)
Short Answer
Expert verified
(a) \(-\frac{63}{65}\), (b) 0, (c) 2.
Step by step solution
01
Understand the Problem
We need to evaluate trigonometric expressions involving inverse trigonometric functions and basic trigonometric identities.
02
Solve Expression (a)
For expression \( \sin \left[\sin ^{-1} \frac{5}{13}-\cos^{-1}\left(-\frac{3}{5}\right)\right] \), use properties of inverse trig functions. Let \( A = \sin^{-1} \frac{5}{13} \) and \( B = \cos^{-1} (-\frac{3}{5}) \). Then, \( \sin(A) = \frac{5}{13} \) and \( \cos(B) = -\frac{3}{5} \). The angle \( B \) also implies \( \sin(B) = \sqrt{1 - \cos^2(B)} = \frac{4}{5} \). Use the identity \( \sin(X - Y) = \sin X \cos Y - \cos X \sin Y \). Substitute values: \( \sin(A) = \frac{5}{13} \), \( \sin(B) = \frac{4}{5} \), and \( \cos(B) = -\frac{3}{5} \). Find \( \cos(A) = \sqrt{1 - \left( \frac{5}{13} \right)^2} = \frac{12}{13} \). Now, \( \sin \left( A - B \right) = \frac{5}{13} \cdot (-\frac{3}{5}) - \frac{12}{13} \cdot \frac{4}{5} = \frac{-15}{65} - \frac{48}{65} = -\frac{63}{65} \).
03
Solve Expression (b)
For expression \( \cos \left(\sin ^{-1} \frac{4}{5}+\tan ^{-1} \frac{3}{4}\right) \), let \( C = \sin^{-1} \frac{4}{5} \) and \( D = \tan^{-1} \frac{3}{4} \). This implies \( \sin(C) = \frac{4}{5}\) and \( \cos(C) = \sqrt{1 - \left( \frac{4}{5} \right)^2} = \frac{3}{5} \). For \( D \), \( \tan(D) = \frac{3}{4} \) indicates a right triangle with opposite \( 3 \), adjacent \( 4 \), hence \( \sec^2(D) = 1 + \tan^2(D) = \frac{25}{16} \) so \( \cos(D) = \frac{4}{5}\), \( \sin(D) = \frac{3}{5} \). Apply the identity \( \cos(X + Y) = \cos X \cos Y - \sin X \sin Y \). Substitute: \( \cos(C) = \frac{3}{5} \), \( \sin(C) = \frac{4}{5} \), \( \cos(D) = \frac{4}{5} \), \( \sin(D) = \frac{3}{5} \). \( \cos(C+D) = \frac{3}{5} \cdot \frac{4}{5} - \frac{4}{5} \cdot \frac{3}{5} = \frac{12}{25} - \frac{12}{25} = 0 \).
04
Solve Expression (c)
For expression \( \tan \left[\cos^{-1} \frac{1}{2}-\sin^{-1}\left(-\frac{1}{2}\right)\right] \), set \( E = \cos^{-1} \frac{1}{2} \) and \( F = \sin^{-1} \left(-\frac{1}{2}\right) \). Then \( \cos(E) = \frac{1}{2} \), \( E = \frac{\pi}{3} \) and for \( F \), \( \sin(F) = -\frac{1}{2} \) implies \( F = -\frac{\pi}{6} \). Use the identity \( \tan(X - Y) = \frac{\tan X - \tan Y}{1 + \tan X \tan Y} \). Calculate \( \tan(E) = \sqrt{3} \) and \( \tan(F) = -\frac{1}{\sqrt{3}} \). Find \( \tan(E - F) = \frac{\sqrt{3} + \frac{1}{\sqrt{3}}}{1 - \sqrt{3} \cdot -\frac{1}{\sqrt{3}}} = \frac{\frac{3\sqrt{3} + 1}{\sqrt{3}}}{2} = 2 \).
05
Conclusion
Combine all solutions: - Part (a) evaluates to \(-\frac{63}{65}\). - Part (b) evaluates to \(0\). - Part (c) evaluates to \(2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Identities
Trigonometric identities are fundamental tools in trigonometry that express relationships between different trigonometric functions. They are essential for simplifying expressions and solving equations. Here are some key identities crucial for solving problems involving inverse trigonometric functions:
- Pythagorean identity: This is probably one of the most important ones. It shows the intrinsic relationship between sine and cosine in terms of squares: \( \sin^2 \theta + \cos^2 \theta = 1 \).
- Angle subtraction and addition: For sine and cosine, these identities help in determining the sine or cosine of two angles summed or subtracted: \( \sin(X - Y) = \sin X \cos Y - \cos X \sin Y \) and \( \cos(X + Y) = \cos X \cos Y - \sin X \sin Y \).
- Tangent identity for subtraction: Used for calculating the tangent of an angle that is the difference of two angles: \( \tan(X - Y) = \frac{\tan X - \tan Y}{1 + \tan X \tan Y} \).
Sine and Cosine Functions
The sine and cosine functions are the two primary trigonometric functions that describe the ratios of sides of right-angled triangles. Here's a closer look at them:
- Sine function (\( \sin \theta \)): This function gives the ratio of the opposite side to the hypotenuse in a right triangle. It ranges from -1 to 1. For a unit circle, \( \sin \theta \) is the y-coordinate of a point on the circle.
- Cosine function (\( \cos \theta \)): Like sine, the cosine is the ratio of the adjacent side to the hypotenuse. In the unit circle framework, \( \cos \theta \) represents the x-coordinate of the point on the circle. It also ranges between -1 and 1.
- Inverse Sine \( (\sin^{-1} x) \) and Inverse Cosine \( (\cos^{-1} x) \): These functions return angles whose sine or cosine is the given value. The angle range is important in these functions, with \( \sin^{-1}(x) \) yielding values in \([-\frac{\pi}{2}, \frac{\pi}{2}] \) and \( \cos^{-1}(x) \) yielding values in \([0, \pi] \).
Tangent Function
The tangent function provides the ratio of the sine of an angle to its cosine and is particularly handy in expressing certain trigonometric solutions.
- Tangent function (\( \tan \theta \)): It represents the ratio of the opposite side to the adjacent side in a right triangle and can also be expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Unlike sine and cosine, the tangent function's range is all real numbers, though it has vertical asymptotes wherever the cosine of the angle is zero.
- Inverse Tangent \( (\tan^{-1} x) \): This function returns angles whose tangent is the given value, and the output is typically within \((-\frac{\pi}{2}, \frac{\pi}{2}) \).
- Tangent subtraction identity: When subtracting angles, the identity \( \tan(X - Y) = \frac{\tan X - \tan Y}{1 + \tan X \tan Y} \) is used, as featured in complex expressions involving the tangent of angle differences.
