(i) Given as
sin (-11π/6)
sin (-11π/6) = sin (-330°)
= – sin (90 × 3 + 60)°
Here, 330° lies in the IV quadrant in which the sine function is negative.
sin (-330°) = – sin (90 × 3 + 60)°
= – (-cos 60°)
= – (-1/2)
Thus, 1/2
(ii) Given as
cosec (-20π/3)
cosec (-20π/3) = cosec (-1200)°
= – cosec (1200)°
= – cosec (90 × 13 + 30)°
Here, 1200° lies in the II quadrant in which cosec function is positive.
cosec (-1200)° = – cosec (90 × 13 + 30)°
= – sec 30°
= -2/√3
(iii) Given as
tan (-13π/4)
tan (-13π/4) = tan (-585)°
= – tan (90 × 6 + 45)°
Here, 585° lies in the III quadrant in which the tangent function is positive.
tan (-585)° = – tan (90 × 6 + 45)°
= – tan 45°
= -1
(iv) Given as
cos 19π/4
cos 19π/4 = cos 855°
= cos (90 × 9 + 45)°
Here, 855° lies in the II quadrant in which the cosine function is negative.
cos 855° = cos (90 × 9 + 45)°
= – sin 45°
= – 1/√2
(v) Given as
sin 41π/4
sin 41π/4 = sin 1845°
= sin (90 × 20 + 45)°
Here, 1845° lies in the I quadrant in which the sine function is positive.
sin 1845° = sin (90 × 20 + 45)°
= sin 45°
= 1/√2
(vi) Given as
cos 39π/4
cos 39π/4 = cos 1755°
= cos (90 × 19 + 45)°
Here, 1755° lies in the IV quadrant in which the cosine function is positive.
cos 1755° = cos (90 × 19 + 45)°
= sin 45°
= 1/√2
(vii) Given as
sin 151π/6
sin 151π/6 = sin 4530°
= sin (90 × 50 + 30)°
Here, 4530° lies in the III quadrant in which the sine function is negative.
sin 4530° = sin (90 × 50 + 30)°
= – sin 30°
Thus, -1/2