Find the Other Trig Values in Quadrant I tan(x)=3/4

Problem

tan(x)=3/4,x∈Quadrant I

Solution

1. Identify the given trigonometric ratio and the quadrant. In Quadrant I, all trigonometric functions (sin(x) cos(x) tan(x) csc(x) sec(x) cot(x) are positive.

2. Use the definition of the tangent function in a right triangle, where tan(x)=opposite/adjacent Let the opposite side a=3 and the adjacent side b=4

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

3+4=c2

9+16=c2

25=c2

c=5

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

sin(x)=3/5

cos(x)=4/5

5. Find the reciprocal identities for the remaining functions: csc(x)=1/sin(x) sec(x)=1/cos(x) and cot(x)=1/tan(x)

csc(x)=5/3

sec(x)=5/4

cot(x)=4/3

Final Answer

sin(x)=3/5,cos(x)=4/5,csc(x)=5/3,sec(x)=5/4,cot(x)=4/3

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