Difference between revisions of "1980 AHSME Problems/Problem 18"
(Created page with "== Problem == If <math>b>1</math>, <math>\sin x>0</math>, <math>\cos x>0</math>, and <math>\log_b \sin x = a</math>, then <math>\log_b \cos x</math> equals <math>\text{(A)} \ 2...") |
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== Solution == | == Solution == | ||
| − | <math>\ | + | <cmath>\log_b \sin x = a</cmath> |
| + | <cmath>b^a=\sin x</cmath> | ||
| + | <cmath>\log_b \cos x=c</cmath> | ||
| + | <cmath>b^c=\cos x</cmath> | ||
| + | Since <math>\sin^2x+\cos^2x=1</math>, | ||
| + | <cmath>(b^c)^2+(b^a)^2=1</cmath> | ||
| + | <cmath>b^{2c}+b^{2a}=1</cmath> | ||
| + | <cmath>b^{2c}=1-b^{2a}</cmath> | ||
| + | <cmath>\log_b 1-b^{2a} = 2c</cmath> | ||
| + | <cmath>c=\boxed{\text{(D)} \ \frac 12 \log_b(1-b^{2a})}</cmath> | ||
| + | |||
| + | -aopspandy | ||
== See also == | == See also == | ||
Revision as of 19:44, 18 June 2021
Problem
If 
, 
, 
, and 
, then 
equals
Solution
![\[\log_b \sin x = a\]](http://latex.artofproblemsolving.com/2/c/8/2c8cf7b5955094b815758760f5734edd34f0767d.png)
![\[b^a=\sin x\]](http://latex.artofproblemsolving.com/b/c/0/bc0847801f645a62c5bec436ce13eeb8acb84328.png)
![\[\log_b \cos x=c\]](http://latex.artofproblemsolving.com/2/5/b/25b98959f04b6bf0926aa330820f47739564ff88.png)
![\[b^c=\cos x\]](http://latex.artofproblemsolving.com/c/a/0/ca0bc27e245b3b643dc2bbbb4bc79225f4105307.png)
Since 
,
![\[(b^c)^2+(b^a)^2=1\]](http://latex.artofproblemsolving.com/6/0/5/605e9e508bc763c5ea257396727557b6681839fa.png)
![\[b^{2c}+b^{2a}=1\]](http://latex.artofproblemsolving.com/f/7/b/f7b34ab01d51fdafb59c1d1b2e10f8c3238ea83f.png)
![\[b^{2c}=1-b^{2a}\]](http://latex.artofproblemsolving.com/3/3/6/3366d9834575c0c37048035fc716f6c8118c0802.png)
![\[\log_b 1-b^{2a} = 2c\]](http://latex.artofproblemsolving.com/d/e/a/deaa367e9d70c7264d3ea45391aaa35c3234948a.png)
![\[c=\boxed{\text{(D)} \ \frac 12 \log_b(1-b^{2a})}\]](http://latex.artofproblemsolving.com/8/4/d/84d3a8a7ab992821e408c2a80509791fc3c4389f.png)
-aopspandy
See also
| 1980 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 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. 
