Trigonometry sin cos tan Question

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SUMMARY

The discussion focuses on solving for tan(theta) given that Sin(theta) = 1/3 and cos(theta) < 0. Participants emphasize using the identity sin²(theta) + cos²(theta) = 1 to derive cos(theta) from the known sine value. The angle must be in the second quadrant due to the conditions provided, leading to the conclusion that tan(theta) can be calculated using the definition of tangent as sin(theta)/cos(theta). The correct quadrant determination is crucial for accurate results.

PREREQUISITES
  • Understanding of trigonometric functions: sine, cosine, and tangent
  • Familiarity with the unit circle and its quadrants
  • Knowledge of trigonometric identities, specifically sin²(theta) + cos²(theta) = 1
  • Ability to calculate inverse trigonometric functions, such as arcsin
NEXT STEPS
  • Study the unit circle and its application in trigonometry
  • Learn how to derive trigonometric values from known sine or cosine values
  • Explore the properties of angles in different quadrants
  • Practice solving trigonometric equations involving multiple identities
USEFUL FOR

Students studying trigonometry, educators teaching trigonometric concepts, and anyone seeking to deepen their understanding of angle relationships and trigonometric identities.

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If Sin(theta) = 1/3 and cos(theta) < 0, find the value(s) for tan(theta)

How do I do this? Thanks.
 
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recall the emanings, with a picture, of these functions sin, cos, tan, using a unit circle. you may be able to just see it.
 
Do you know that sin2(&theta;)+ cos2(&theta;)= 1? Since you know sin(&theta;)= 1/3, you can use that to find cos(&theta;). Now, how is tan(&theta;) defined?
 
404 said:
How do I do this? Thanks.

Take the arc sin of 1/3 - this will give you the angle, but it will be the principle angle (first quadrant). But, since Cos theta <0, you know it can't be in the first quadrant or fourth quadrant, so it must be in either the second or third quadrant. But, since sin theta is >0, it must be in the second quadrant. You should be able to get the correct angle from this info, and therefore the correct cosine and then - as was hinted before - remember the definition of tangent theta.
 
I got it now, Thanks guys :)
 

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