Find the Other Trig Values in Quadrant I cos(x)=3/7

Problem

cos(x)=3/7,0<x<π/2

Solution

1. Identify the given information and the quadrant. We are given cos(x)=3/7 in Quadrant I, where all trigonometric functions are positive.

2. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to find the value of sin(x)

sin2(x)+(3/7)2=1

3. Solve for sin(x) by isolating the term and taking the square root.

sin2(x)=1−9/49

sin2(x)=40/49

sin(x)=√(,40/49)

sin(x)=(2√(,10))/7

4. Calculate tan(x) using the identity tan(x)=sin(x)/cos(x)

tan(x)=(2√(,10))/7/3/7

tan(x)=(2√(,10))/3

5. Determine the reciprocal functions sec(x) csc(x) and cot(x) by inverting the primary values.

sec(x)=1/cos(x)=7/3

csc(x)=1/sin(x)=7/(2√(,10))=(7√(,10))/20

cot(x)=1/tan(x)=3/(2√(,10))=(3√(,10))/20

Final Answer

sin(x)=(2√(,10))/7,tan(x)=(2√(,10))/3,sec(x)=7/3,csc(x)=(7√(,10))/20,cot(x)=(3√(,10))/20

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