Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Trigonometric Functions with Negative Angles

(i)  \sin(-\theta)= -\sin\theta

(ii)  \cos(-\theta)= \cos\theta

(ii)  \tan(-\theta)= -\tan\theta

(iv)  \cot(-\theta)= -\cot\theta

(v)  \sec(-\theta)= \sec\theta

(vi)  \csc(-\theta)= -\csc\theta

Hence,    \sin, \tan, \cot \text { and } \csc are odd fucntions and   \cos, \sec,   are even functions.

Some useful formulas according to quadrants:

(a)   1\text {st} quadrant (I)

(i) For \left( \dfrac{\pi}{2}-\theta \right) \text { or } \left( 90^o- \theta \right)

(i)  \sin \left( \dfrac{\pi}{2}-\theta \right) = \cos\theta

(ii)  \cos \left( \dfrac{\pi}{2}-\theta \right) = \sin\theta

(iii) \tan \left( \dfrac{\pi}{2}-\theta \right) = \cot\theta

 (iv) \cot \left( \dfrac{\pi}{2}-\theta \right) = \tan\theta

(v) \sec \left(\dfrac{\pi}{2}-\theta \right) = \csc\theta

(vi) \csc \left(\dfrac{\pi}{2}-\theta \right) = \sec\theta

All are Positive.

(ii)For  (2\pi +\theta) \text { or } (360^o +\theta)

(i) \sin (2\pi+\theta)=\sin \theta

(ii)  \cos (2\pi+\theta)=\cos \theta

(iii)  \tan (2\pi+\theta)=\tan \theta

(iv) \cot (2\pi+\theta)=\cot \theta

(v)  \sec (2\pi+\theta)=\sec \theta

(vi) \csc (2\pi+\theta)=\csc \theta

All are positive.

(b)  2 \text {nd} quadrant (II)

(i) For \left( \dfrac{\pi}{2}+\theta \right) \text { or } (90^o+\theta)

(i)  \sin \left( \dfrac{\pi}{2}+\theta \right)=\cos\theta

(ii)  \cos \left( \dfrac{\pi}{2}+\theta \right)=-\sin\theta

(iii) \tan \left( \dfrac{\pi}{2}+\theta \right)=-\cot\theta

(iv) \cot \left( \dfrac{\pi}{2}+\theta \right)=-\tan\theta

(v) \sec \left( \dfrac{\pi}{2}+\theta \right)=-\csc\theta

(vi) \csc \left ( \dfrac{\pi}{2}+\theta \right)=\sec\theta

Only  \sin and  \csc are positive.

(II) For  (\pi-\theta) \text{ or } (180^o-\theta)

(i) \sin(\pi-\theta)=\sin \theta

(ii)  \cos(\pi-\theta)=-\cos \theta

(iii) \tan(\pi-\theta)=-\cot \theta

(iv)   \cot(\pi-\theta)=-\tan \theta

          (v) \sec(\pi-\theta)=-\sec \theta

(vi)  \csc(\pi-\theta)=\csc \theta

Only  \sin and  \csc are positive.

(c)  3\text {rd} quadrant (III)

(I) For   \left ( \dfrac{3\pi}{2}-\theta\right) \text{ or }  (270^o-\theta)

(i) \sin \left (\dfrac{3\pi}{2}-\theta \right)= -\cos\theta

(ii) \cos \left (\dfrac{3\pi}{2}-\theta \right)= -\sin\theta

(iii) \tan \left (\dfrac{3\pi}{2}-\theta \right)= \cot\theta

(iv) \cot \left (\dfrac{3\pi}{2}-\theta \right)= \tan\theta

(v) \sec \left (\dfrac{3\pi}{2}-\theta \right)= -\csc\theta

(vi) \csc \left (\dfrac{3\pi}{2}-\theta \right)= -\ sec\theta

Only  \tan and  \cot are positive.

(II) For  (\pi+\theta)\text { or } (180^o =\theta)

(i) \sin (\pi+\theta)=-\sin \theta

(ii) \cos (\pi+\theta)=-\cos \theta

(iii) \tan (\pi+\theta)=\tan \theta

(iv) \cot (\pi+\theta)=\cot \theta

(v) \sec (\pi+\theta)=-\sec \theta

(vi) \cos sec (\pi+\theta)=-\csc \theta

Only  \tan and  \cot are positive.

(d)  4\text {th} quadrant (IV)

(I) For  \left(\dfrac{3\pi}{2}+\theta \right)\text{ or } (270^o +\theta)

(i) \sin \left( \dfrac{3\pi}{2}+\theta \right)=-\cos \theta

(ii) \cos  \left( \dfrac{3\pi}{2}+\theta \right)=\sin \theta

(iii) \tan \left( \dfrac{3\pi}{2}+\theta \right)=-\cot \theta

(iv) \cot  \left( \dfrac{3\pi}{2}+\theta \right)=-\tan \theta

(v) \sec \left( \dfrac{3\pi}{2}+\theta \right)=\csc \theta

(vi) \csc \left( \dfrac{3\pi}{2}+\theta\right)=-\ sec \theta

Only \cos and \sec are positive.

(II )For  (2\pi-\theta)\text { or }  (360^o-\theta)

(i) \sin (2\pi-\theta)=-\sin \theta

(ii) \cos (2\pi-\theta)=\cos \theta

(iii) \tan (2\pi-\theta)=-\tan\theta

(iv) \cot (2\pi-\theta)=-\cot\theta

(v) \sec (2\pi-\theta)=\sec\theta

(vi) \csc(2\pi-\theta)=-\csc\theta

Only \cos and \sec are positive.

NOTE:

(A)

(i)  \sin(n\pi+\theta)=(-1)^{-1^{n}}\sin\theta

(ii) \cos(n\pi+\theta)=(-1)^{n}\cos\theta

(iii) \tan(n\pi+\theta)=\tan\theta

(iv) \cot(n\pi+\theta)=\cot\theta

(v) \sec(n\pi+\theta)=(-1)^{n}\sec\theta

(vi)  \csc(n\pi+\theta)=(-1)\dfrac{n-1}{2}\csc\theta

All are  n \in Z

(B)

(i) \sin \left( \dfrac{n \pi}{2}+\theta \right)=(-1)^\frac{n-1}{2}\cos \theta

(ii) \cos \left( \dfrac{n \pi}{2}+\theta \right)=(-1)^\frac{n+1}{2}\sin \theta

(iii) \tan \left( \dfrac{n \pi}{2}+\theta \right)=-\cot \theta

(iv)  \cot \left( \dfrac{n \pi}{2}+\theta \right)=-\tan \theta

(v)  \sec \left( \dfrac{n \pi}{2}+\theta \right)= (-1)^\frac{n+1}{2}\csc \theta

(vi)  \csc \left( \dfrac{n \pi}{2}+\theta \right)= (-1)^\frac{n-1}{2}\sec \theta

where  'n' is an odd integer.

Every trigonometric ratio converted into other trigonometric ratios:

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