Difference between revisions of "1991 AHSME Problems/Problem 21"
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== Problem == | == Problem == | ||
| − | For all real numbers <math>x</math> except <math>x=0</math> and <math>x=1</math> the function <math>f(x)</math> is defined by <math>f(x/(1 | + | For all real numbers <math>x</math> except <math>x=0</math> and <math>x=1</math> the function <math>f(x)</math> is defined by <math>f(x/(x-1))=1/x</math>. Suppose <math>0\leq t\leq \pi/2</math>. What is the value of <math>f(\sec^2t)</math>? |
| + | |||
| + | <math>\text{(A) } sin^2\theta\quad | ||
| + | \text{(B) } cos^2\theta\quad | ||
| + | \text{(C) } tan^2\theta\quad | ||
| + | \text{(D) } cot^2\theta\quad | ||
| + | \text{(E) } csc^2\theta</math> | ||
== Solution == | == Solution == | ||
| + | Let <math>y=\frac{x}{x-1} \Rightarrow xy-y=x \Rightarrow x=\frac{y}{y-1}</math> | ||
| + | |||
| + | <math>f(y)=\frac{1}{x}=\frac{y-1}{y}=1-\frac{1}{y}</math> | ||
| + | |||
| + | <math>f(sec^2t)=sin^2t</math> | ||
| + | |||
<math>\fbox{A}</math> | <math>\fbox{A}</math> | ||
Latest revision as of 20:56, 17 October 2016
Problem
For all real numbers 
except 
and 
the function 
is defined by 
. Suppose 
. What is the value of 
?
Solution
Let 
See also
| 1991 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 20 |
Followed by Problem 22 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
