Find the Other Trig Values in Quadrant I cos(x)=24/25

Problem

cos(x)=24/25,0<x<π/2

Solution

1. Identify the given values for the adjacent side a and the hypotenuse h from the definition cos(x)=a/h

a=24

h=25

2. Apply the Pythagorean theorem a2+o2=h2 to find the opposite side o

24+o2=25

576+o2=625

o2=49

o=7

3. Determine the sign of the trigonometric functions based on the quadrant. Since x is in Quadrant I, all trigonometric values are positive.

Quadrant I⇒all positive

4. Calculate the sine value using sin(x)=o/h

sin(x)=7/25

5. Calculate the tangent value using tan(x)=o/a

tan(x)=7/24

6. Calculate the reciprocal identities for cosecant, secant, and cotangent.

csc(x)=1/sin(x)=25/7

sec(x)=1/cos(x)=25/24

cot(x)=1/tan(x)=24/7

Final Answer

sin(x)=7/25,tan(x)=7/24,csc(x)=25/7,sec(x)=25/24,cot(x)=24/7

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