Find the Other Trig Values in Quadrant I tan(x)=1/8

Problem

tan(x)=1/8, Quadrant I

Solution

1. Identify the definitions of the trigonometric functions in terms of a right triangle where tan(x)=opposite/adjacent

2. Assign values to the sides of the triangle based on the given ratio: opposite=1 and adjacent=8

3. Calculate the hypotenuse r using the Pythagorean theorem r=√(,opposite2+adjacent2)

r=√(,1+8)

r=√(,1+64)

r=√(,65)

4. Determine the sine and cosine values using the ratios sin(x)=opposite/hypotenuse and cos(x)=adjacent/hypotenuse Since x is in Quadrant I, all values are positive.

sin(x)=1/√(,65)

cos(x)=8/√(,65)

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

sin(x)=√(,65)/65

cos(x)=(8√(,65))/65

6. Find the reciprocal functions: cot(x)=1/tan(x) sec(x)=1/cos(x) and csc(x)=1/sin(x)

cot(x)=8

sec(x)=√(,65)/8

csc(x)=√(,65)

Final Answer

sin(x)=√(,65)/65,cos(x)=(8√(,65))/65,cot(x)=8,sec(x)=√(,65)/8,csc(x)=√(,65)

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