## Note on Signs of Trigonometric Ratios

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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° = -$$\frac{1}{\sqrt{3}}$$

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.
.

### Very Short Questions

Soln

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

= cos(16 x 90° + 30°)

= cos 30°

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

Soln

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

= -tan(7×90° - 60°)

= {-cot60°}

= cot 60° = $$\frac{1}{\sqrt(3)}$$

0%
• ### What is the value of sin 30°?

(frac{1}{2})
(frac{1}{sqrt{3}})

1
• ### What is the value of tan 0°?

(frac{1}{sqrt{2}})
1
(frac{1}{sqrt{3}})
• ### What is the value of cosec 45°?

(frac{1}{sqrt{3}})
(frac{1}{sqrt{2}})
(sqrt{2})
1
• ### What is the value of cos 90°?

(sqrt{2})

-1
(frac{1}{sqrt{3}})
• ### What is the value of sec 30°?

(frac{2}{sqrt{3}})
(frac{1}{sqrt{3}})
(sqrt{2})
2
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