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Difference between revisions of "1993 AHSME Problems/Problem 23"

(Created page with "== Problem == <asy> draw(circle((0,0),10),black+linewidth(.75)); draw((-10,0)--(10,0),black+linewidth(.75)); draw((-10,0)--(9,sqrt(19)),black+linewidth(.75)); draw((-10,0)--(9,-s...")
 
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<math>\text{(A) } cos(6^\circ)cos(12^\circ)sec(18^\circ)\quad\\
+
<math>\text{(A) } \cos(6^\circ)\cos(12^\circ)\sec(18^\circ)\quad\\
−
\text{(B) } cos(6^\circ)sin(12^\circ)csc(18^\circ)\quad\\
+
\text{(B) } \cos(6^\circ)\sin(12^\circ)\csc(18^\circ)\quad\\
−
\text{(C) } cos(6^\circ)sin(12^\circ)sec(18^\circ)\quad\\
+
\text{(C) } \cos(6^\circ)\sin(12^\circ)\sec(18^\circ)\quad\\
−
\text{(D) } sin(6^\circ)sin(12^\circ)csc(18^\circ)\quad\\
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\text{(D) } \sin(6^\circ)\sin(12^\circ)\csc(18^\circ)\quad\\
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\text{(E) } sin(6^\circ)sin(12^\circ)sec(18^\circ)</math>
+
\text{(E) } \sin(6^\circ)\sin(12^\circ)\sec(18^\circ)</math>
  
 
== Solution ==
 
== Solution ==
 +
 +
We have all the angles we need, but most obviously, we see that right angle in triangle <math>ABD</math>.
 +
 +
Note also that angle <math>BAD</math> is 6 degrees, so length <math>AB = cos(6)</math> because the diameter, <math>AD</math>, is 1.
 +
 +
Now, we can concentrate on triangle <math>ABX</math> (after all, now we can decipher all angles easily and use Law of Sines).
 +
 +
We get:
 +
 +
<math>\frac{AB}{\sin(\angle{AXB})} =\frac{AX}{\sin(\angle{ABX})}</math>
 +
 +
That's equal to
 +
 +
<math>\frac{\cos(6)}{\sin(180-18)} =\frac{AX}{\sin(12)}</math>
 +
 +
Therefore, our answer is equal to:
 
<math>\fbox{B}</math>
 
<math>\fbox{B}</math>
 +
 +
Note that <math>\sin(162) = \sin(18)</math>, and don't accidentally put <math>\text{C}</math> because you thought <math>\frac{1}{\sin}</math> was <math>\sec</math>!
  
 
== See also ==
 
== See also ==

Latest revision as of 14:38, 2 October 2024

Problem

[asy] draw(circle((0,0),10),black+linewidth(.75)); draw((-10,0)--(10,0),black+linewidth(.75)); draw((-10,0)--(9,sqrt(19)),black+linewidth(.75)); draw((-10,0)--(9,-sqrt(19)),black+linewidth(.75)); draw((2,0)--(9,sqrt(19)),black+linewidth(.75)); draw((2,0)--(9,-sqrt(19)),black+linewidth(.75)); MP("X",(2,0),N);MP("A",(-10,0),W);MP("D",(10,0),E);MP("B",(9,sqrt(19)),E);MP("C",(9,-sqrt(19)),E); [/asy]

Points $A,B,C$ and $D$ are on a circle of diameter $1$, and $X$ is on diameter $\overline{AD}.$

If $BX=CX$ and $3\angle{BAC}=\angle{BXC}=36^\circ$, then $AX=$


$\text{(A) } \cos(6^\circ)\cos(12^\circ)\sec(18^\circ)\quad\\ \text{(B) } \cos(6^\circ)\sin(12^\circ)\csc(18^\circ)\quad\\ \text{(C) } \cos(6^\circ)\sin(12^\circ)\sec(18^\circ)\quad\\ \text{(D) } \sin(6^\circ)\sin(12^\circ)\csc(18^\circ)\quad\\ \text{(E) } \sin(6^\circ)\sin(12^\circ)\sec(18^\circ)$

Solution

We have all the angles we need, but most obviously, we see that right angle in triangle $ABD$.

Note also that angle $BAD$ is 6 degrees, so length $AB = cos(6)$ because the diameter, $AD$, is 1.

Now, we can concentrate on triangle $ABX$ (after all, now we can decipher all angles easily and use Law of Sines).

We get:

$\frac{AB}{\sin(\angle{AXB})} =\frac{AX}{\sin(\angle{ABX})}$

That's equal to

$\frac{\cos(6)}{\sin(180-18)} =\frac{AX}{\sin(12)}$

Therefore, our answer is equal to: $\fbox{B}$

Note that $\sin(162) = \sin(18)$, and don't accidentally put $\text{C}$ because you thought $\frac{1}{\sin}$ was $\sec$!

See also

1993 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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