Find the Other Trig Values in Quadrant IV sec(t)=2

Problem

sec(t)=2, Quadrant IV

Solution

1. Identify the given trigonometric ratio and the quadrant. Since sec(t)=2 and sec(t)=1/cos(t) we find the value of the cosine function.

cos(t)=1/2

2. Determine the signs of the trigonometric functions in Quadrant IV. In this quadrant, cos(t) and sec(t) are positive, while sin(t) csc(t) tan(t) and cot(t) are negative.

3. Use the Pythagorean identity sin2(t)+cos2(t)=1 to find the sine value.

sin2(t)+(1/2)2=1

sin2(t)+1/4=1

sin2(t)=3/4

sin(t)=−√(,3/4)

sin(t)=−√(,3)/2

4. Calculate the cosecant value by taking the reciprocal of the sine value.

csc(t)=1/sin(t)

csc(t)=−2/√(,3)

csc(t)=−(2√(,3))/3

5. Calculate the tangent value using the identity tan(t)=sin(t)/cos(t)

tan(t)=(−√(,3)/2)/1/2

tan(t)=−√(,3)

6. Calculate the cotangent value by taking the reciprocal of the tangent value.

cot(t)=1/tan(t)

cot(t)=−1/√(,3)

cot(t)=−√(,3)/3

Final Answer

sin(t)=−√(,3)/2,cos(t)=1/2,tan(t)=−√(,3),csc(t)=−(2√(,3))/3,cot(t)=−√(,3)/3

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