Find the Other Trig Values in Quadrant I tan(x)=3/5

Problem

tan(x)=3/5,0<x<π/2

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. Define the sides of a right triangle using the definition tan(x)=opposite/adjacent Let the opposite side be a=3 and the adjacent side be b=5

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

3+5=c2

9+25=c2

34=c2

c=√(,34)

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

sin(x)=3/√(,34)

cos(x)=5/√(,34)

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

sin(x)=(3√(,34))/34

cos(x)=(5√(,34))/34

6. Find the reciprocal functions csc(x) sec(x) and cot(x) by inverting the sine, cosine, and tangent values.

csc(x)=√(,34)/3

sec(x)=√(,34)/5

cot(x)=5/3

Final Answer

sin(x)=(3√(,34))/34,cos(x)=(5√(,34))/34,csc(x)=√(,34)/3,sec(x)=√(,34)/5,cot(x)=5/3

Want more problems? Check here!