Find the Other Trig Values in Quadrant I cos(x)=4/9

Problem

cos(x)=4/9,0<x<π/2

Solution

1. Identify the given values for the right triangle. Since cos(x)=adjacent/hypotenuse we let the adjacent side a=4 and the hypotenuse c=9

2. Apply the Pythagorean theorem to find the opposite side b

a2+b2=c2

4+b2=9

16+b2=81

b2=65

b=√(,65)

3. Determine the sign of the trigonometric functions. Since x is in Quadrant I, all trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) are positive.

4. Calculate sine using the ratio opposite/hypotenuse

sin(x)=√(,65)/9

5. Calculate tangent using the ratio opposite/adjacent

tan(x)=√(,65)/4

6. Calculate cosecant by taking the reciprocal of sine.

csc(x)=9/√(,65)

csc(x)=(9√(,65))/65

7. Calculate secant by taking the reciprocal of cosine.

sec(x)=9/4

8. Calculate cotangent by taking the reciprocal of tangent.

cot(x)=4/√(,65)

cot(x)=(4√(,65))/65

Final Answer

sin(x)=√(,65)/9,tan(x)=√(,65)/4,csc(x)=(9√(,65))/65,sec(x)=9/4,cot(x)=(4√(,65))/65

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