Respuesta :
The trigonometric functions provided in the options which has negative values are csc(5pi/4) and sec(120°).
Which trigonometric functions are positive in which quadrant?
- In first quadrant (0 < θ < π/2), all six trigonometric functions are positive.
- In second quadrant(π/2 < θ < π), only sin and cosec are positive.
- In the third quadrant (π < θ < 3π/2), only tangent and cotangent are positive.
- In fourth (3π/2 < θ < 2π = 0), only cos and sec are positive.
(this all positive negative refers to the fact that if you use given angle as input to these functions, then what sign will these functions will evaluate based on in which quadrant does the given angle lies.)
- The value of cot(pi)
The value of pi in degree is 180. Thus, the value of cot(pi) lies between the second and the third quadrant. The value of cotangent is not defined on (pi).
- The value of csc(5pi/4)
Convert it into the radian
[tex]\rm cosec (\dfrac{5\pi}{4})=cosec \dfrac{5\times180}{4}\\\rm cosec (\dfrac{5\pi}{4})=cosec (225^o)[/tex]
Thus, it lies in the third quadrant, where only tangent and cotangent are positive. Thus, csc(5pi/4) has negative value.
- The value of sec(-65°)
[tex]\sec(-65^o)=\sec(65^o)[/tex]
The value lies in the first quadrant, all six trigonometric functions are positive. Thus, the value of sec(-65°) is positive.
- The value of csc(340°)
[tex]\rm cosec(340^o)=\rm cosec(360^o-20^o)\\\rm cosec(340^o)=\rm -cosec(20^o)[/tex]
The value lies in the first quadrant, all six trigonometric functions are positive. Thus, the value of csc(20°) is positive and the value of -csc(20°) or csc(340°) is negative.
- The value of sec(120°)
The value lies in the second quadrant, where only sin and cosec are positive. Thus, the value of sec(120°) is negative.
Thus, the trigonometric functions provided in the options which has negative values are csc(5pi/4) and sec(120°).
Learn more about the trigonometry function here;
https://brainly.com/question/14421002
