Find the Other Trig Values in Quadrant II cos(x)=-4/5

Problem

cos(x)=−4/5,x∈Quadrant II

Solution

1. Identify the given values for the reference triangle. In the coordinate plane, cos(x)=a/r where a is the adjacent side (x-coordinate) and r is the hypotenuse (radius).

a=−4

r=5

2. Apply the Pythagorean theorem to find the opposite side o (y-coordinate). The formula is a2+o2=r2

(−4)2+o2=5

16+o2=25

o2=9

3. Determine the sign of the opposite side. Since the angle x is in Quadrant II, the y-coordinate must be positive.

o=√(,9)

o=3

4. Calculate the sine using the ratio sin(x)=o/r

sin(x)=3/5

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

tan(x)=3/(−4)

tan(x)=−3/4

6. Calculate the reciprocal functions by inverting the primary ratios.

csc(x)=1/sin(x)=5/3

sec(x)=1/cos(x)=−5/4

cot(x)=1/tan(x)=−4/3

Final Answer

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

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