Find the Other Trig Values in Quadrant I tan(x)=9

Problem

tan(x)=9, Quadrant I

Solution

1. Identify the given information and the quadrant. Since x is in Quadrant I, all trigonometric functions (sin(x) cos(x) tan(x) csc(x) sec(x) cot(x) will be positive.

2. Represent the tangent ratio as a fraction. Since tan(x)=opposite/adjacent we can let the opposite side be 9 and the adjacent side be 1

tan(x)=9/1

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

1+9=c2

1+81=c2

c=√(,82)

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

sin(x)=9/√(,82)

cos(x)=1/√(,82)

5. Rationalize the denominators for the sine and cosine functions.

sin(x)=(9√(,82))/82

cos(x)=√(,82)/82

6. Find the reciprocal functions csc(x) sec(x) and cot(x) by flipping the ratios of sin(x) cos(x) and tan(x)

csc(x)=√(,82)/9

sec(x)=√(,82)

cot(x)=1/9

Final Answer

sin(x)=(9√(,82))/82,cos(x)=√(,82)/82,csc(x)=√(,82)/9,sec(x)=√(,82),cot(x)=1/9

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