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° = -\(\frac{1}{\sqrt{3}}\)
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° = \(\frac{1}{\sqrt(3)}\)
What is the value of sin 30°?
What is the value of tan 0°?
What is the value of cosec 45°?
What is the value of cos 90°?
What is the value of sec 30°?
ASK ANY QUESTION ON Signs of Trigonometric Ratios
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Ashok
Tan45
Mar 15, 2017
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Mar 08, 2017
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