Find the Other Trig Values in Quadrant I tan(x)=1/3

Problem

tan(x)=1/3, 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, such that the opposite side a=1 and the adjacent side b=3

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

1+3=c2

1+9=c2

c=√(,10)

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

sin(x)=1/√(,10)=√(,10)/10

cos(x)=3/√(,10)=(3√(,10))/10

5. Find the reciprocal functions by inverting the primary ratios.

cot(x)=1/tan(x)=3

sec(x)=1/cos(x)=√(,10)

csc(x)=1/sin(x)=√(,10)

6. Verify the signs based on the location in Quadrant I, where all trigonometric functions are positive.

Final Answer

sin(x)=√(,10)/10,cos(x)=(3√(,10))/10,cot(x)=3,sec(x)=√(,10),csc(x)=√(,10)

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