How do you find the remaining trigonometric functions of #theta# given #tantheta=-1/2# and #sintheta>0#?

Answer 1
#tan t = - 1/2 --> cot t = - 2# #sin^2 t = 1/(1 + cot^2 t) = 1/(1 + 4) = 1/5# #sin t = +- 1/sqrt5 = = +- sqrt5/5# Take the positive value, because sin t > 0 #cos^2 t = 1 - sin^2 t = 1 - 1/5 = 4/5# #cos t = +- 2/sqrt5# Because tan t < 0, and sin t > o, then cos t < 0, take the negative value. #sec t = 1/(cos) = - sqrt5/2# #csc t = 1/(sin) = sqrt5#
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Answer 2

Given ( \tan(\theta) = -\frac{1}{2} ) and ( \sin(\theta) > 0 ), you can find the remaining trigonometric functions using the following steps:

  1. Since ( \tan(\theta) = -\frac{1}{2} ), we can use the definition of tangent to find that the opposite side is 1 and the adjacent side is 2.

  2. Using the Pythagorean theorem, we find the hypotenuse to be ( \sqrt{1^2 + 2^2} = \sqrt{5} ).

  3. Since ( \sin(\theta) > 0 ), the sine function is positive in the first and second quadrants. In the first quadrant, ( \sin(\theta) = \frac{1}{\sqrt{5}} ).

  4. Now, you can find the cosine function using the identity ( \cos(\theta) = \frac{1}{\sec(\theta)} ). Since ( \sec(\theta) = \frac{1}{\cos(\theta)} ), ( \cos(\theta) = \frac{2}{\sqrt{5}} ).

  5. Finally, you can find the cosecant, secant, and cotangent functions using the reciprocal identities:

  • ( \csc(\theta) = \frac{1}{\sin(\theta)} = \sqrt{5} )
  • ( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\sqrt{5}}{2} )
  • ( \cot(\theta) = \frac{1}{\tan(\theta)} = -2 ).
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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