Problem

Question 11 of 29
Find the exact value of each of the remaining trigonometric functions of θ.
sinθ=12,cotθ<0
cosθ=
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
cotθ=
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
tanθ=
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
cscθ=
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
secθ=
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Answer

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Answer

So, the exact values of the remaining trigonometric functions of θ are cosθ=32, tanθ=13, cotθ=3, cscθ=2, and secθ=233.

Steps

Step 1 :Given that sinθ=12 and cotθ<0, we can determine the quadrant in which θ lies. Since sinθ>0 and cotθ<0, θ must be in the second quadrant.

Step 2 :Next, we can find the value of cosθ using the Pythagorean identity sin2θ+cos2θ=1. Solving for cosθ, we get cosθ=1sin2θ=1(12)2=114=34=32. However, since θ is in the second quadrant, cosθ is negative. Therefore, cosθ=32.

Step 3 :Then, we can find the value of tanθ using the identity tanθ=sinθcosθ. Substituting the given values, we get tanθ=1232=13.

Step 4 :Next, we can find the value of cotθ using the identity cotθ=1tanθ. Substituting the value of tanθ we found, we get cotθ=113=3.

Step 5 :Then, we can find the value of cscθ using the identity cscθ=1sinθ. Substituting the given value of sinθ, we get cscθ=112=2.

Step 6 :Finally, we can find the value of secθ using the identity secθ=1cosθ. Substituting the value of cosθ we found, we get secθ=132=23=233.

Step 7 :So, the exact values of the remaining trigonometric functions of θ are cosθ=32, tanθ=13, cotθ=3, cscθ=2, and secθ=233.

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