Subject: Optional Mathematics

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 \(\theta\).

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° = -\(\frac{1}{\sqrt{3}}\)

- 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. - 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 \(\theta\) - 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.

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

Find out the value of sin 315° without using a calculator.

Soln

= sin(4×90°-45°)

= -sin45°

= -\(\frac{1}{\sqrt(2)}\)

Find the value of cos(-1470°)

Soln

= cos 1470° (\(\therefore\) (-\(\theta\)) = cos\(\theta\))

= cos(16 x 90° + 30°)

= cos 30°

= \(\frac{\sqrt(3)}{2}\)

Find the value of tan(-570°).

Soln

= -tan 570° (tan(-\(\theta\) = -tan \(\theta\))

= -tan(7×90° - 60°)

= {-cot60°}

= cot 60° = \(\frac{1}{\sqrt(3)}\)

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