​ Find the other trigonometric functions if i. cot θ = -3/5, and 180 < θ < 270 ​

Question

Find the other trigonometric functions if 

i. cot θ = -\(\frac{3}{5}\), and 180 < θ < 270 

ii. Sec A = -\(\frac{25}{7}\)and A lies in the second quadrant. 

iii cot x = \(\frac{3}{4}\), x lies in the third quadrant. 

iv. tan x = \(\frac{-5}{12}\)x lies in the fourth quadrant.

Answer 1

i. cot θ = - \(\frac{3}{5}\)

we know that, 

sin² θ = 1 – cos² θ 

= 1 –\((-\frac{3}{5})^2\) 

= 1 – \(\frac{9}{25}\)= \(\frac{16}{25}\)

∴ sin θ = ± \(\frac{4}{5}\)

Since 180° < 0 < 270°, 

θ lies in the 3rd quadrant.

∴ sin θ < 0

Since A lies in the 2nd quadrant,

ii. tan A< 0

iii. Given, cot x = \(\frac{3}{4}\)

We know that, 

cosec² x = 1 + cot² x 

= 1 + \((\frac{3}{4})^2 = 1 + \frac {9}{11} = \frac {25}{16}\)

∴ cosec x = ± \( \frac {5}{4}\)

Since x lies in the 3rd quadrant, cosec x < 0

iv. Given, tan x = -\( \frac {5}{12}\)

sec² x = 1 + tan²

^{= 1 + \((-\frac {5}{12})^2\)}

= 1 + ^{\(\frac {25}{144} = \frac {169}{144}\)}

∴ sec x = ± ^{\(\frac {13}{12} \)}

Since x lies in the 4th quadrant,

sec x > 0