Find the Other Trig Values in Quadrant I sec(x)=13/12

Problem

sec(x)=13/12,0<x<π/2

Solution

1. Identify the relationship between the secant function and the sides of a right triangle.

sec(x)=hypotenuse/adjacent=13/12

2. Assign values to the hypotenuse r and the adjacent side (x_side)

r=13

(x_side)=12

3. Calculate the opposite side (y_side) using the Pythagorean theorem (x_side)2+(y_side)2=r2

12+(y_side)2=13

144+(y_side)2=169

(y_side)2=25

(y_side)=5

4. Determine the sine value using the ratio opposite/hypotenuse

sin(x)=5/13

5. Determine the cosine value using the ratio adjacent/hypotenuse or the reciprocal of secant.

cos(x)=12/13

6. Determine the tangent value using the ratio opposite/adjacent

tan(x)=5/12

7. Determine the cosecant value using the reciprocal of sine.

csc(x)=13/5

8. Determine the cotangent value using the reciprocal of tangent.

cot(x)=12/5

Final Answer

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

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