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Magazine Problem 10
Evaluate \(\tan ^{-1}\left(\tan \frac{17 \pi}{7}\right)\)
Problem 10
Emphasize the importance of understanding inverse notation as well as the importance of parentheses in determining the order of operations. For \(x=3,\) evaluate each of the following: (a) \(\tan ^{-1} x\) (c) \(\tan \left(x^{-1}\right)\) (b) \((\tan x)^{-1}\) (d) \(\left(\tan ^{-1} x\right)^{-1}\)
Problem 10
Given that \(\sin \frac{3 \pi}{10}=\frac{\sqrt{5}+1}{4},\) find an exact expression for \(\cos \frac{3 \pi}{5}\) [Problem 73 asks you to explain how the value for \(\sin \frac{3 \pi}{10}\) used here follows from the solution to Exercise 9.]
Problem 11
Use the following figure. Find the value of \(\theta\) (in degrees) if \(b=3, c=6,\) the area of the triangle equals \(5,\) and \(\theta>90^{\circ}\).
Problem 11
Evaluate \(\sin ^{-1}\left(\sin \frac{6 \pi}{7}\right)\)
Problem 11
Emphasize the importance of understanding inverse notation as well as the importance of parentheses in determining the order of operations. For \(x=4\), evaluate each of the following: (a) \(\left(\cos \left(x^{-1}\right)\right)^{-1}\) (c) \(\left(\cos ^{-1}\left(x^{-1}\right)\right)^{-1}\) (b) \(\cos ^{-1}\left(x^{-1}\right)\)
Problem 12
For Exercises 11-26, evaluate the given quantities assuming that \(u\) and \(v\) are both in the interval \(\left(0, \frac{\pi}{2}\right)\) and $$ \cos u=\frac{1}{3} \quad \text { and } \quad \sin v=\frac{1}{4} $$ \(\cos v\)
Problem 12
Evaluate \(\cos ^{-1}\left(\cos \frac{10 \pi}{9}\right)\)
Problem 12
Emphasize the importance of understanding inverse notation as well as the importance of parentheses in determining the order of operations. For \(x=5\), evaluate each of the following: (a) \(\left(\cos \left(x^{-1}\right)\right)^{-1}\) (c) \(\left(\cos ^{-1}\left(x^{-1}\right)\right)^{-1}\) (b) \(\cos ^{-1}\left(x^{-1}\right)\)
Problem 13
Emphasize the importance of understanding inverse notation as well as the importance of parentheses in determining the order of operations. For \(x=6\), evaluate each of the following: (a) \(\left(\sin \left(x^{-1}\right)\right)^{-1}\) (c) \(\left(\sin ^{-1}\left(x^{-1}\right)\right)^{-1}\) (b) \(\sin ^{-1}\left(x^{-1}\right)\)
