Signs of Trigonometric Ratios

 Trigonometric Ratios of any angle

Any allied angle can be in the form (n × 90° ± \(\theta\)) where n is an integer. We can change the trigonometric ratios of the angle (n × 90° ± \(\theta\)) into the trigonometric ratio of an angle \(\theta\).

1. If n is even, there will be no change in the trigonometric ratios.
i.e. sin (n × 90° ± \(\theta\)) ⇒ sin \(\theta\)
cos (n × 90° ± \(\theta\)) ⇒ cos \(\theta\), etc.

2. If n is odd, then the trigonometric ratios change as follows:
sin (n × 90° ± \(\theta\)) ⇒ cos \(\theta\)
cos (n × 90° ± \(\theta\)) ⇒ sin \(\theta\)
tan (n × 90° ± \(\theta\)) ⇒ cot \(\theta\)
cosec (n × 90° ± \(\theta\)) ⇒sec \(\theta\)
sec (n × 90° ± \(\theta\)) ⇒ cosec \(\theta\)
cot (n × 90° ± \(\theta\)) ⇒ tan \(\theta\)

3. The sign of the trigonometric ratio of the angle (n × 90° ± \(\theta\)) is determined by taking into consideration that in which quadrant that angle (n × 90° ± \(\theta\)) lies.

Ratios of 120°

sin 120° = sin (2 × 90° - 60°) = sin 60° = \(\frac{\sqrt{3}}{2}\)

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

tan 120° = tan (2 × 90° - 60°) = -tan 60° = - \(\sqrt{3}\)

Ratios of 135°

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

cos 135° = cos (2 × 90° - 45°) = -cos 45 = -\(\frac{1}{\sqrt{2}}\)

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

Ratios of 150°

sin 150° = sin (2 × 90° - 30°) = sin 30° = \(\frac{1}{2}\)

cos 150° = cos (1 × 90° +60°) = -sin 60° = -\(\frac{\sqrt{3}}{2}\)

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

Things to remember

1. If n is even, there will be no change in the trigonometric ratios.
   i.e. sin(n × 90° ± \(\theta\))⇒ sin \(\theta\)
   cos(n × 90° ± \(\theta\))⇒ cos \(\theta\), etc.
2. If n is odd, then the trigonometric ratios change as follows:
   sin(n × 90° ± \(\theta\))⇒ cos \(\theta\)
   cos(n × 90° ± \(\theta\))⇒ sin \(\theta\)
   tan(n × 90° ± \(\theta\))⇒ cot \(\theta\)
   cosec(n × 90° ± \(\theta\))⇒sec \(\theta\)
   sec(n × 90° ± \(\theta\))⇒ cosec \(\theta\)
   cot(n × 90° ± \(\theta\))⇒ tan \(\theta\)
3. The sign of the trigonometric ratio of the angle(n × 90° ± \(\theta\)) is determined by taking into consideration that in which quadrant that angle(n × 90° ± \(\theta\)) lies.

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