Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1973 AHSME Problems/Problem 17"

(Solution to Problem 17)
 
(→‎See Also)
 
Line 24: Line 24:
  
 
==See Also==
 
==See Also==
−
{{AHSME 35p box|year=1973|num-b=16|num-a=18}}
+
{{AHSME 30p box|year=1973|num-b=16|num-a=18}}
  
 
[[Category:Introductory Trigonometry Problems]]
 
[[Category:Introductory Trigonometry Problems]]

Latest revision as of 13:00, 20 February 2020

Problem

If $\theta$ is an acute angle and $\sin \frac12 \theta = \sqrt{\frac{x-1}{2x}}$, then $\tan \theta$ equals

$\textbf{(A)}\ x \qquad \textbf{(B)}\ \frac1{x} \qquad \textbf{(C)}\ \frac{\sqrt{x-1}}{x+1} \qquad \textbf{(D)}\ \frac{\sqrt{x^2-1}}{x} \qquad \textbf{(E)}\ \sqrt{x^2-1}$

Solution

Since $\tfrac{\theta}{2}$ is acute and $\cos \tfrac{\theta}{2} = \sqrt{1 - \sin (\tfrac{\theta}{2})^2}$, \[\cos \frac{\theta}{2} = \sqrt{1 - (\frac{x-1}{2x})^2}\] \[\cos \frac{\theta}{2} = \sqrt{\frac{x+1}{2x}}\] Using the definition of tangent, \[\tan \frac{\theta}{2} = \sqrt{\frac{x-1}{2x}} \div \sqrt{\frac{x+1}{2x}}\] \[\tan \frac{\theta}{2} = \sqrt{\frac{x-1}{x+1}}\] Finally, by using the Double Angle Identity for Tangent, \[\tan \theta = \frac{2 \cdot \sqrt{\frac{x-1}{x+1}}}{1 - \frac{x-1}{x+1}}\] \[\tan \theta = \frac{\frac{2\sqrt{x^2-1}}{x+1}}{\frac{2}{x+1}}\] \[\tan \theta = \sqrt{x^2-1}\] The answer is $\boxed{\textbf{(E)}}$.

See Also

1973 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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 • 31 • 32 • 33 • 34 • 35
All AHSME Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1973_AHSME_Problems/Problem_17&oldid=118198"