Trigonometric ratios behave in different quadrants. In some cases, they behave the same as well and that is why it becomes necessary for us to understand their behavior. To understand the quadrant effect on trigonometric ratios, you need to understand a few trigonometric identities. These trigonometric identities are split into different types:

  1. Acute Angles Trigonometric Identities
  2. Supplementary Angles Trigonometric Identities
  3. Angles Greater Than { 180 }^{\circ} or \pi Trigonometric Identities
  4. Angles Greater Than { 360 }^{\circ} or 2 \pi Trigonometric Identities
  5. Angles That Differ by { 90 }^{ \circ } or \frac { \pi }{ 2 } Trigonometric Identities
  6. Angles That Add Up to { 270 }^{ \circ } or \frac { 3 }{ 2 } \pi Trigonometric Identities
  7. Angles That Differ by { 270 }^{ \circ } or \frac { 3 }{ 2 } \pi Trigonometric Identities

Supplementary Angles

Two angles are supplementary if the sum is { 180 }^{ \circ } or \pi radians. So, if two supplementary Angles are added, a straight angle is obtained.

\sin { (\pi - \alpha) } = \sin { \alpha }

\cos { (\pi - \alpha) } = - \cos { \alpha }

\tan { (\pi - \alpha) } = - \tan { \alpha }

\sin { { 150 }^{\circ} } = \sin { ({ 180 }^{\circ} - { 30 }^{\circ}) } = \sin { {30}^{\circ} } = \frac { 1 }{ 2 }

\cos { { 150 }^{\circ} } = \cos { ({ 180 }^{\circ} - { 30 }^{\circ}) } = -\cos { {30}^{\circ} } = -\frac { \sqrt { 3 } }{ 2 }

\tan { { 150 }^{\circ} } = \tan { ({ 180 }^{\circ} - { 30 }^{\circ}) } = -\tan { {30}^{\circ} } = -\frac { \sqrt { 3 } }{ 3 }

 

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Angles That Differ by 180° or π Rad

 

\sin { (\pi + \alpha) } = - \sin { \alpha }

\cos { (\pi + \alpha) } = - \cos { \alpha }

\tan { (\pi + \alpha) } = \tan { \alpha }

\sin { { 210 }^{\circ} } = \sin { ({ 180 }^{\circ} + { 30 }^{\circ}) } = - \sin { {30}^{\circ} } = -\frac { 1 }{ 2 }

\cos { { 210 }^{\circ} } = \cos { ({ 180 }^{\circ} + { 30 }^{\circ}) } = -\cos { {30}^{\circ} } = -\frac { \sqrt { 3 } }{ 2 }

\tan { { 210 }^{\circ} } = \tan { ({ 180 }^{\circ} + { 30 }^{\circ}) } = \tan { {30}^{\circ} } = \frac { \sqrt { 3 } }{ 3 }

 

Angles Greater Than 360º

 

\sin { (\alpha + 2\pi k) } = \sin { \alpha }

\cos { (\alpha + 2\pi k) } = \cos { \alpha }

\tan { (\alpha + 2\pi k) } = \tan { \alpha }

 

 

 

 

 

 

 

 

\sin { { 750 }^{\circ} } = \sin { ({ 360 }^{\circ} \times 2 + { 30 }^{\circ}) } = \sin { { 30 }^{\circ} } = \frac { 1 }{ 2 }

\cos { { 750 }^{\circ} } = \cos { ({ 360 }^{\circ} \times 2 + { 30 }^{\circ}) } = \cos { { 30 }^{\circ} } = \frac { \sqrt { 3 } }{ 2 }

\tan { { 750 }^{\circ} } = \tan { ({ 360 }^{\circ} \times 2 + { 30 }^{\circ}) } = \tan { { 30 }^{\circ} } = \frac { \sqrt { 3 } }{ 3 }

Angles That Differ by 90° or π/2 Rad

 

\sin { (\alpha + \frac { \pi }{ 2 }) } = \cos { \alpha }

\cos { (\alpha + \frac { \pi }{ 2 }) } = -\sin { \alpha }

\tan { (\alpha + \frac { \pi }{ 2 }) } = -\cot { \alpha }

 

\sin { { 120 }^{\circ} } = \sin { ({ 90 }^{\circ} + { 30 }^{\circ}) } = \cos { { 30 }^{\circ} } = \frac { \sqrt { 3 } }{ 2 }

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

\tan { { 120 }^{\circ} } = \tan { ({ 90 }^{\circ} + { 30 }^{\circ}) } = - \cot { { 30 }^{\circ} } = - \sqrt { 3 }

 

Angles That Add Up to 270º or 3/2 π Rad

 

\sin { (\frac { 3 \pi }{ 2 } - \alpha) } = -\cos { \alpha }

\cos { (\frac { 3 \pi }{ 2 } - \alpha) } = -\sin { \alpha }

\tan { (\frac { 3 \pi }{ 2 } - \alpha) } = \cot { \alpha }

 

 

\sin { { 240 }^{ \circ } } = \sin { ({ 270 }^{ \circ } - { 30 }^{ \circ }) } = - \cos { { 30 }^{ \circ } } = - \frac { \sqrt { 3 } }{ 2 }

\cos { { 240 }^{ \circ } } = \cos { ({ 270 }^{ \circ } - { 30 }^{ \circ }) } = - \cos { { 30 }^{ \circ } } = - \frac { 1 }{ 2 }

\tan { { 240 }^{ \circ } } = \tan { ({ 270 }^{ \circ } - { 30 }^{ \circ }) } = \cos { { 30 }^{ \circ } } = \sqrt { 3 }

 

Angles That Differ by 270º or 3/2 π Rad

 

\sin { (\frac { 3 \pi }{ 2 } + \alpha) } = -\cos { \alpha }

\cos { (\frac { 3 \pi }{ 2 } + \alpha) } = -\sin { \alpha }

\tan { (\frac { 3 \pi }{ 2 } + \alpha) } = \cot { \alpha }

 

 

\sin { { 300 }^{ \circ } } = \sin { ({ 270 }^{ \circ } + { 30 }^{ \circ }) } = - \cos { { 30 }^{ \circ } } = - \frac { \sqrt { 3 } }{ 2 }

\cos { { 300 }^{ \circ } } = \cos { ({ 270 }^{ \circ } + { 30 }^{ \circ }) } = \sin { { 30 }^{ \circ } } = \frac { 1 }{ 2 }

\tan { { 300 }^{ \circ } } = \tan { ({ 270 }^{ \circ } + { 30 }^{ \circ }) } = - \cot { { 30 }^{ \circ } } = - \sqrt { 3 }

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.