Find the Other Trig Values in Quadrant II sec(theta)=-4

Problem

sec(θ)=−4,θ∈Quadrant II

Solution

1. Identify the given trigonometric ratio and the quadrant. Since sec(θ)=−4 the reciprocal is cos(θ)=−1/4 In Quadrant II, x is negative and y is positive.

2. Determine the values of x and r using the definition cos(θ)=x/r Let x=−1 and r=4

3. Calculate the value of y using the Pythagorean identity x2+y2=r2

(−1)2+y2=4

1+y2=16

y2=15

y=√(,15)

Note that y is positive because θ is in Quadrant II.

4. Find the sine and cosecant values using y and r

sin(θ)=y/r=√(,15)/4

csc(θ)=r/y=4/√(,15)=(4√(,15))/15

5. Find the tangent and cotangent values using x and y

tan(θ)=y/x=√(,15)/(−1)=−√(,15)

cot(θ)=x/y=(−1)/√(,15)=−√(,15)/15

Final Answer

sin(θ)=√(,15)/4,cos(θ)=−1/4,tan(θ)=−√(,15),csc(θ)=(4√(,15))/15,sec(θ)=−4,cot(θ)=−√(,15)/15

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