Finding arctan(1/sqrt3): Solving the Puzzle

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SUMMARY

The discussion centers on finding arctan(1/sqrt3), where the user initially believes the answer to be pi/6 based on right triangle properties. The coordinates of pi/6 yield an opposite side of 1/2 and an adjacent side of sqrt(3)/2, resulting in a ratio of y/x equal to 1/sqrt(3). However, the user mistakenly associates the angle 7pi/6 with the same ratio, leading to confusion. The correct interpretation is that arctan(1/sqrt3) corresponds to pi/6, as it represents the angle whose tangent is 1/sqrt3.

PREREQUISITES
  • Understanding of trigonometric functions, specifically tangent and arctangent.
  • Familiarity with the unit circle and angle measurements in radians.
  • Basic knowledge of right triangle properties and ratios.
  • Ability to interpret coordinates in the context of trigonometric functions.
NEXT STEPS
  • Study the properties of the unit circle and how angles correspond to coordinates.
  • Learn about the relationships between angles and their tangent values, focusing on common angles.
  • Explore the derivation of arctan values for various ratios, particularly for special triangles.
  • Practice solving problems involving right triangles and their associated trigonometric ratios.
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Students studying trigonometry, educators teaching angle relationships, and anyone seeking to deepen their understanding of tangent and arctangent functions.

fk378
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This is a general question.
I was given a problem where you need to find arctan(1/sqrt3). Referring to a pie chart, I see that if you compute y/x, you can see that the coordinates of 7pi/6 equal 1/sqrt3.

However, I found the answer is pi/6 and I know the proof is in drawing a right triangle. But if you do y/x for the coordinates of pi/6, you get sqrt3 instead of 1/sqrt3. Can anyone tell me why?
 
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If I draw a right triangle with an angle of pi/6, I see an opposite side (y) with length 1/2 and and adjacent side (x) with length sqrt(3)/2. y/x=1/sqrt(3). What's the problem?
 

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