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 == |
Latest revision as of 20:45, 18 June 2021
Problem
If , , , and , then equals
Solution
Since ,
-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 |
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