Trigonometrical Ratios of some Particular Angles

Trigonometrical ratios of some particular angles i.e., 120°, -135°, 150° and 180° are given below.

1. sin 120° = sin (1 × 90° + 30°) = cos 30° = \(\frac{√3}{2}\);

cos 120° = cos (1 × 90° + 30°) = - sin 30° = - \(\frac{1}{2}\);

tan 120° = tan (1 × 90° + 30°) = - cot 30° = - √3;

csc 120° = csc (1 × 90° + 30°) = sec 30° = \(\frac{2}{√3}\);

sec 120° = sec (1 × 90° + 30°) = - csc 30° = - 2;

tan 120° = tan (1 × 90° + 30°) = - cot 30° = - √3;

cot 120° = cot (1 × 90° + 30°) = - tan 30° = - \(\frac{1}{√3}\).


2. sin (- 135°)= - sin 135°= - sin (1 × 90°+ 45°) = - cos 45° = - \(\frac{1}{√2}\);

cos (- 135°)= cos 135°= cos (1 × 90°+ 45°) = - sin 45°= - \(\frac{1}{√2}\);

tan (- 135°) = - tan 135° = - tan ( 1 × 90° + 45°) = - (- cot 45°) = 1;

csc (- 135°)= - csc 135°= - csc (1 × 90°+ 45°)= - sec 45° = - √2;

sec (- 135°)= sec 135°= sec (1 × 90°+ 45°)= - csc 45°= - √2;

cot (- 135°) = - cot 135° = - cot ( 1 × 90° + 45°) = - (-tan 45°) = 1.


3. sin 150° = sin (2 × 90° - 30°) = sin 30° = 1/2;

cos 150° = cos (2 × 90° - 30°) = cos 30° = - \(\frac{√3}{2}\);

tan 150° tan (2 × 90° - 30°) = - tan 30° = - \(\frac{1}{√3}\);

csc 150° = csc (2 × 90° - 30°) = csc 30° = 2;

sec 150° = sec (2 × 90° - 30°) = sec 30° = - \(\frac{2}{√3}\);

cot 150° = cot (2 × 90° - 30°) = - cot 300 = - √3.


4. sin 180° = sin (2 × 90° - 0°) = sin 0° = 0;

cos 180° = cos (2 × 90° - 0°) = - cos 0° = - 1;

tan 180° = tan (2 × 90° + 0°) = tan 0° = 0;

csc 180° = csc (2 × 90° - 0°) = csc 0° = Undefined;

sec 180° = sec (2 × 90° - 0°) = - sec 0° = - 1;

cot 180° = cot (2 × 90° + 0°) = cot 0° = Undefined.

 

5. sin 270° = sin (3 × 90° + 0°) = - cos 0° = - 1;

cos 270° = cos (3 × 90° + 0°) = sin 0° = 0;

tan 270° = tan (3 × 90° + 0°) = - cot 0° = Undefined;

csc 270° = csc (3 × 90° + 0°) = - sec 0° = - 1;

sec 270° = sec (3 × 90° + 0°) = csc 0° = Undefined;

cot 270° = cot (3 × 90° + 0°) = - tan 0° = 0.

These trigonometrical ratios of some particular angles (120°, -135°, 150° and 180°) are required to solve various problems.





11 and 12 Grade Math

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