Find the Other Trig Values in Quadrant III cos(x)=-( square root of 3)/2

Problem

cos(x)=−√(,3)/2,π<x<(3*π)/2

Solution

1. Identify the quadrant and the signs of the trigonometric functions. Since x is in Quadrant III, sin(x) and csc(x) are negative, cos(x) and sec(x) are negative, and tan(x) and cot(x) are positive.

2. Find the sine using the Pythagorean identity sin2(x)+cos2(x)=1

sin2(x)+(−√(,3)/2)2=1

sin2(x)+3/4=1

sin2(x)=1/4

sin(x)=−1/2

3. Find the tangent using the identity tan(x)=sin(x)/cos(x)

tan(x)=(−1/2)/(−√(,3)/2)

tan(x)=1/√(,3)

tan(x)=√(,3)/3

4. Find the cosecant by taking the reciprocal of the sine.

csc(x)=1/sin(x)

csc(x)=−2

5. Find the secant by taking the reciprocal of the cosine.

sec(x)=1/cos(x)

sec(x)=−2/√(,3)

sec(x)=−(2√(,3))/3

6. Find the cotangent by taking the reciprocal of the tangent.

cot(x)=1/tan(x)

cot(x)=√(,3)

Final Answer

sin(x)=−1/2,tan(x)=√(,3)/3,csc(x)=−2,sec(x)=−(2√(,3))/3,cot(x)=√(,3)

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