Find the Other Trig Values in Quadrant I tan(x)=5/12

Problem

tan(x)=5/12, Quadrant I

Solution

1. Identify the given trigonometric ratio and the quadrant. Since tan(x)=opposite/adjacent=5/12 and x is in Quadrant I, both the opposite side a and the adjacent side b are positive.

a=5

b=12

2. Calculate the hypotenuse c using the Pythagorean theorem a2+b2=c2

5+12=c2

25+144=c2

169=c2

c=13

3. Determine the sine and cosine values using the definitions sin(x)=opposite/hypotenuse and cos(x)=adjacent/hypotenuse

sin(x)=5/13

cos(x)=12/13

4. Find the reciprocal trigonometric values by inverting the sine, cosine, and tangent ratios.

csc(x)=1/sin(x)=13/5

sec(x)=1/cos(x)=13/12

cot(x)=1/tan(x)=12/5

Final Answer

sin(x)=5/13,cos(x)=12/13,csc(x)=13/5,sec(x)=13/12,cot(x)=12/5

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