Find the Other Trig Values in Quadrant IV cos(x)=8/17

Problem

cos(x)=8/17,x∈Quadrant IV

Solution

1. Identify the given values for the adjacent side a=8 and the hypotenuse h=17 based on the definition cos(x)=a/h

2. Apply the Pythagorean theorem a2+b2=h2 to find the magnitude of the opposite side b

8+b2=17

64+b2=289

b2=225

|b|=15

3. Determine the sign of the opposite side based on the location in Quadrant IV, where yvalues (sine) are negative and xvalues (cosine) are positive.

b=−15

4. Calculate the remaining trigonometric ratios using the values a=8 b=−15 and h=17

sin(x)=b/h=−15/17

tan(x)=b/a=−15/8

csc(x)=h/b=−17/15

sec(x)=h/a=17/8

cot(x)=a/b=−8/15

Final Answer

sin(x)=−15/17,tan(x)=−15/8,csc(x)=−17/15,sec(x)=17/8,cot(x)=−8/15

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