Difference between revisions of "User:Temperal/sandbox"

Latest revision as of 12:28, 24 March 2020

Problem 1

Evaluate the following expressions:

(a) $\tan(45^\circ)$

(b) $\cos\left(\frac {7\pi}{4}\right)$

(c) $\sin\left(\frac {5\pi}{3}\right)$

(d) $\csc(135^\circ)$

(e) $\cot(945^\circ)$

(f) $\sin(\pi \sin(\pi/6))$

(g) $\tan(21\pi)$

(h) $\sec( - 585^\circ)$

Problem 2

Using the unit circle, find $\sin \left( x + \frac {\pi}{2} \right)$ and $\cos \left( x + \frac {\pi}{2} \right)$ in terms of $\sin x$ and $\cos x$ .

Problem 3

In triangle $ABC$ , $\angle B = 90^\circ$ , $\sin A = 7/9$ , and $BC = 21$ . What is $AB$ ?

Problem 4

What does the graph of $\sin 4x$ look like compared to the graphs of $\sin x$ and $\cos x$ ? What about the graph of $2\sin \left( 3x + \frac {\pi}{4} \right) - 1$ ?

Problem 5

Find the value of $\tan(\pi/12) \cdot \tan(2\pi/12) \cdot \tan(3\pi/12) \cdots \tan(5\pi/12).$

Problem 6

Suppose that parallelogram $ABCD$ has $\angle A = \angle C = 30^\circ$ , $\angle B = \angle D = 150^\circ$ , and the shorter diagonal $BD$ has length 2. If the height of the parallelogram is $a$ , find the perimeter of $ABCD$ in terms of $a$ .

Problem 7

Given a positive number $n$ and a number $c$ satisfying $- 1 < c < 1$ , for how many values of $q$ with $0 \leq q < 2\pi$ is $\sin nq = c$ ? What if $c = 1$ or $c = - 1$ ?

Problem 8

How many solutions are there to the equation $\cos x = \frac {x^2}{1000}$ , where $x$ is in radians?

Problem 9

Determine all $\theta$ such that $0 \le \theta \le \frac {\pi}{2}$ and $\sin^5\theta + \cos^5\theta = 1$ .

Problem 10

Find the value of $\sin(15^\circ)$ . Hint: Draw an isosceles triangle with vertex angle $30^\circ$ .

@@ Line 1: / Line 1: @@
-== Problem ==
+==Problem 1==
-A bunny is nice. Why?
-<math>\mathrm{(A)}\ {{{answera}}} \qquad \mathrm{(B)}\ {{{answerb}}} \qquad \mathrm{(C)}\ {{{answerc}}} \qquad \mathrm{(D)}\ {{{answerd}}} \qquad \mathrm{(E)}\ {{{answere}}}</math>
+Evaluate the following expressions:
-== Solution ==
+(a) <math>\tan(45^\circ)</math>
-Because he is nice. yay, bunnies!
-== See also ==
+(b) <math>\cos\left(\frac {7\pi}{4}\right)</math>
-{{AIME box|year=2007|ab=|num-b=2|num-a=4|n=I}}
-[[Category:Intermediate Combinatorics Problems]]
+(c) <math>\sin\left(\frac {5\pi}{3}\right)</math>
+(d) <math>\csc(135^\circ)</math>
+(e) <math>\cot(945^\circ)</math>
+(f) <math>\sin(\pi \sin(\pi/6))</math>
+(g) <math>\tan(21\pi)</math>
+(h) <math>\sec( - 585^\circ)</math>
+==Problem 2==
+Using the unit circle, find <math>\sin \left( x + \frac {\pi}{2} \right)</math> and <math>\cos \left( x + \frac {\pi}{2} \right)</math> in terms of <math>\sin x</math> and <math>\cos x</math>.
+==Problem 3==
+In triangle <math>ABC</math>, <math>\angle B = 90^\circ</math>, <math>\sin A = 7/9</math>, and <math>BC = 21</math>. What is <math>AB</math>?
+==Problem 4==
+What does the graph of <math>\sin 4x</math> look like compared to the graphs of <math>\sin x</math> and <math>\cos x</math>? What about the graph of <math>2\sin \left( 3x + \frac {\pi}{4} \right) - 1</math>?
+==Problem 5==
+Find the value of <math>\tan(\pi/12) \cdot \tan(2\pi/12) \cdot \tan(3\pi/12) \cdots \tan(5\pi/12).</math>
+==Problem 6==
+Suppose that parallelogram <math>ABCD</math> has <math>\angle A = \angle C = 30^\circ</math>, <math>\angle B = \angle D = 150^\circ</math>, and the shorter diagonal <math>BD</math> has length 2. If the height of the parallelogram is <math>a</math>, find the perimeter of <math>ABCD</math> in terms of <math>a</math>.
+==Problem 7==
+Given a positive number <math>n</math> and a number <math>c</math> satisfying <math>- 1 < c < 1</math>, for how many values of <math>q</math> with <math>0 \leq q < 2\pi</math> is <math>\sin nq = c</math>? What if <math>c = 1</math> or <math>c = - 1</math>?
+==Problem 8==
+How many solutions are there to the equation <math>\cos x = \frac {x^2}{1000}</math>, where <math>x</math> is in radians?
+==Problem 9==
+Determine all <math>\theta</math> such that <math>0 \le \theta \le \frac {\pi}{2}</math> and <math>\sin^5\theta + \cos^5\theta = 1</math>.
+==Problem 10==
+Find the value of <math>\sin(15^\circ)</math>. Hint: Draw an isosceles triangle with vertex angle <math>30^\circ</math>.

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