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in Trigonometry by (51.9k points)

Find the values of the following trigonometric ratios:

(i) sin (-11π/6)

(ii) cosec (-20π/3)

(iii) tan (-13π/4)

(iv) cos 19π/4

(v) sin 41π/4

(vi) cos 39π/4

(vii) sin 151π/6

1 Answer

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(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

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