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.
● Trigonometric Functions
11 and 12 Grade Math
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