Find Trig Functions Using Identities cot(theta)=4/3 , sin(theta)<0

Math
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The function is negative in the and quadrants. The function is positive in the and quadrants. The set of solutions for are limited to the since that is the only quadrant found in both sets.
Solution is in third quadrant.
Use the definition of cotangent to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.
Find the hypotenuse of the unit circle triangle. Since the opposite and adjacent sides are known, use the Pythagorean theorem to find the remaining side.
Replace the known values in the equation.
Simplify inside the radical.
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Raise to the power of .
Hypotenuse
Raise to the power of .
Hypotenuse
Add and .
Hypotenuse
Rewrite as .
Hypotenuse
Pull terms out from under the radical, assuming positive real numbers.
Hypotenuse
Hypotenuse
Find the value of sine.
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Use the definition of sine to find the value of .
Substitute in the known values.
Find the value of cosine.
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Use the definition of cosine to find the value of .
Substitute in the known values.
Find the value of tangent.
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Use the definition of tangent to find the value of .
Substitute in the known values.
Find the value of secant.
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Use the definition of secant to find the value of .
Substitute in the known values.
Find the value of cosecant.
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Use the definition of cosecant to find the value of .
Substitute in the known values.
This is the solution to each trig value.
Find Trig Functions Using Identities cot(theta)=4/3 , sin(theta)<0

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