1984 AHSME Problems/Problem 15
Problem 15
If , then one value for is
Solution
We divide both sides of the equation by to get , or .
This looks a lot like the formula relating the slopes of two perpendicular lines, which is , where and are the slopes. It's made even more relatable by the fact that the tangent of an angle can be defined by the slope of the line that makes that angle with the x-axis.
We can make this look even more like the slope formula by multiplying both sides by :
, and using the trigonometric identity , we have .
Now it's time for a diagram:
Since the product of the two slopes, and , is , the lines are perpendicular, and the angle between them is . The angle between them is also , so and , or .
NOTE: To show that is not the only solution, we can also set equal to another angle measure congruent to , such as , yielding another solution as , which clearly is a solution to the equation.
Solution 2
We can simply try the answers. We quickly see that works, since by the identity
Solution 3
Start by subtracting from both sides to get . We recognize that this is of the form , so . , so .
~purplepenguin2
See Also
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Followed by Problem 16 | |
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