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xDAN-Agentic-Math-Level5

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Find the ordered pair $(a,b)$ of integers such that \[\sqrt{9 - 8 \sin 50^\circ} = a + b \csc 50^\circ.\]

Level 5

We write \[9 - 8 \sin 50^\circ = \frac{9 \sin^2 50^\circ - 8 \sin^3 50^\circ}{\sin^2 50^\circ} = \frac{9 \sin^2 50^\circ - 6 \sin 50^\circ + 6 \sin 50^\circ - 8 \sin^3 50^\circ}{\sin^2 50^\circ}.\]By the triple angle identity, \begin{align*} 6 \sin 50^\circ - 8 \sin^3 50^\circ &= 2 \sin (3 \cdot 50^\circ) \\ &= 2 \sin 150^\circ \\ &= 1, \end{align*}so \[9 - 8 \sin 50^\circ = \frac{9 \sin^2 50^\circ - 6 \sin 50^\circ + 1}{\sin^2 50^\circ} = \left( \frac{3 \sin 50^\circ - 1}{\sin 50^\circ} \right)^2.\]Since $3 \sin 50^\circ > 3 \sin 30^\circ = \frac{3}{2} > 1,$ $3 \sin 50^\circ - 1 > 0.$ Therefore, \[\sqrt{9 - 8 \sin 50^\circ} = \frac{3 \sin 50^\circ - 1}{\sin 50^\circ} = 3 - \csc 50^\circ,\]so $(a,b) = \boxed{(3,-1)}.$

Precalculus

Find the smallest positive integer $k$ such that $ z^{10} + z^9 + z^6+z^5+z^4+z+1 $ divides $z^k-1$.

Level 5

First, we factor the given polynomial. The polynomial has almost all the powers of $z$ from 1 to $z^6,$ which we can fill in by adding and subtracting $z^2$ and $z^3.$ This allows us to factor as follows: \begin{align*} z^{10} + z^9 + z^6 + z^5 + z^4 + z + 1 &= (z^{10} - z^3) + (z^9 - z^2) + (z^6 + z^5 + z^4 + z^3 + z^2 + z + 1) \\ &= z^3 (z^7 - 1) + z^2 (z^7 - 1) + (z^6 + z^5 + z^4 + z^3 + z^2 + z + 1) \\ &= z^3 (z - 1)(z^6 + z^5 + z^4 + z^3 + z^2 + z + 1) \\ &\quad + z^2 (z - 1)(z^6 + z^5 + z^4 + z^3 + z^2 + z + 1) \\ &\quad + (z^6 + z^5 + z^4 + z^3 + z^2 + z + 1) \\ &= (z^4 - z^2 + 1)(z^6 + z^5 + z^4 + z^3 + z^2 + z + 1). \end{align*}Viewing $z^4 - z^2 + 1 = 0$ as a quadratic in $z^2,$ we can solve to get \[z^2 = \frac{1 \pm i \sqrt{3}}{2},\]or $\operatorname{cis} \frac{\pi}{3}$ and $\operatorname{cis} \frac{5 \pi}{3}.$ Therefore, the roots of $z^4 - z^2 + 1 = 0$ are \[\operatorname{cis} \frac{\pi}{6}, \ \operatorname{cis} \frac{7 \pi}{6}, \ \operatorname{cis} \frac{5 \pi}{6}, \ \operatorname{cis} \frac{11 \pi}{6}.\]We write these as \[\operatorname{cis} \frac{2 \pi}{12}, \ \operatorname{cis} \frac{14 \pi}{12}, \ \operatorname{cis} \frac{10 \pi}{12}, \ \operatorname{cis} \frac{22 \pi}{12}.\]If $z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 = 0,$ then \[(z - 1)(z^6 + z^5 + z^4 + z^3 + z^2 + z + 1) = 0,\]which simplifies to $z^7 = 1.$ Thus, the roots of $z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 = 0$ are of the form \[\operatorname{cis} \frac{2 \pi j}{7},\]where $1 \le j \le 6.$ The roots of $z^k - 1 = 0$ are of the form \[\operatorname{cis} \frac{2 \pi j}{k}.\]Thus, we need $k$ to be a multiple of both 12 and 7. The smallest such $k$ is $\boxed{84}.$

Precalculus

If \[\sin x + \cos x + \tan x + \cot x + \sec x + \csc x = 7,\]then find $\sin 2x.$

Level 5

Expressing everything in terms of $\sin x$ and $\cos x,$ we get \[\sin x + \cos x + \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} + \frac{1}{\sin x} + \frac{1}{\cos x} = 7.\]Then \[\sin x + \cos x + \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} + \frac{\sin x + \cos x}{\sin x \cos x} = 7,\]which becomes \[\sin x + \cos x + \frac{\sin x + \cos x}{\sin x \cos x} = 7 - \frac{1}{\sin x \cos x}.\]We can factor the left-hand side, and replace $\sin x \cos x$ with $\frac{1}{2} \sin 2x$: \[(\sin x + \cos x) \left( 1 + \frac{2}{\sin 2x} \right) = 7 - \frac{2}{\sin 2x}.\]Hence, \[(\sin x + \cos x)(\sin 2x + 2) = 7 \sin 2x - 2.\]Squaring both sides, we get \[(\sin^2 x + 2 \sin x \cos + \cos^2 x)(\sin^2 2x + 4 \sin 2x + 4) = 49 \sin^2 x - 28 \sin x + 4.\]We can write this as \[(\sin 2x + 1)(\sin^2 2x + 4 \sin 2x + 4) = 49 \sin^2 x - 28 \sin x + 4.\]This simplifies to \[\sin^3 2x - 44 \sin^2 2x + 36 \sin 2x = 0,\]so $\sin 2x (\sin^2 2x - 44 \sin 2x + 36) = 0.$ If $\sin 2x = 2 \sin x \cos x = 0,$ then the expression in the problem becomes undefined. Otherwise, \[\sin^2 2x - 44 \sin 2x + 36 = 0.\]By the quadratic formula, \[\sin 2x = 22 \pm 8 \sqrt{7}.\]Since $22 + 8 \sqrt{7} > 1,$ we must have $\sin 2x = \boxed{22 - 8 \sqrt{7}}.$

Precalculus

Define the sequence $a_1, a_2, a_3, \ldots$ by $a_n = \sum\limits_{k=1}^n \sin{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$.

Level 5

By the product-to-sum formula, \[\sin \frac{1}{2} \sin k = \frac{1}{2} \left[ \cos \left( k - \frac{1}{2} \right) - \cos \left( k + \frac{1}{2} \right) \right].\]Thus, we can make the sum in the problem telescope: \begin{align*} a_n &= \sum_{k = 1}^n \sin k \\ &= \sum_{k = 1}^n \frac{\sin \frac{1}{2} \sin k}{\sin \frac{1}{2}} \\ &= \sum_{k = 1}^n \frac{\cos (k - \frac{1}{2}) - \cos (k + \frac{1}{2})}{2 \sin \frac{1}{2}} \\ &= \frac{(\cos \frac{1}{2} - \cos \frac{3}{2}) + (\cos \frac{3}{2} - \cos \frac{5}{2}) + \dots + (\cos \frac{2n - 1}{2} - \cos \frac{2n + 1}{2})}{2 \sin \frac{1}{2}} \\ &= \frac{\cos \frac{1}{2} - \cos \frac{2n + 1}{2}}{2 \sin \frac{1}{2}}. \end{align*}Then $a_n < 0$ when $\cos \frac{1}{2} < \cos \frac{2n + 1}{2}.$ This occurs if and only if \[2 \pi k - \frac{1}{2} < \frac{2n + 1}{2} < 2 \pi k + \frac{1}{2}\]for some integer $k.$ Equivalently, \[2 \pi k - 1 < n < 2 \pi k.\]In other words, $n = \lfloor 2 \pi k \rfloor.$ The 100th index of this form is then $\lfloor 2 \pi \cdot 100 \rfloor = \boxed{628}.$

Precalculus