Signs of Trigonometric Ratios

Subject: Optional Mathematics

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Overview

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 \(\theta\).

Signs of Trigonometric Ratios

 Trigonometric Ratios of any angle 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 \(\theta\)

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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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Questions and Answers

Soln

 = sin(4×90°-45°)

 = -sin45°

 = -\(\frac{1}{\sqrt(2)}\)

 

 

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° = \(\frac{1}{\sqrt(3)}\)

Quiz

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