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 Identities

(i)  \sin^{2}\theta+\cos^{2}\theta=1

 \implies\sin^{2}\theta =1-\cos^2\theta      \implies \cos^2\theta=1-\sin^2\theta

\implies \sin\theta =\sqrt{1-\cos^2\theta}    \implies\cos\theta=\sqrt{1-\sin^2\theta}

(ii)  \sec^2\theta-\tan^2\theta=1

 \implies \sec^2\theta=1+\tan^2}\theta                \implies\tan^2\theta=\sec^2\theta-1

 \implies\sec\theta=\sqrt{1+\tan^2\theta}           \implies \tan\theta=\sqrt{\sec^{2}\theta-1}

(iii)  \csc^{2}\theta-\cot^{2}\theta=1

 \implies\csc^{2}\theta=1+\cot^{2}\theta             \implies\cot^{2}\theta=\csc^{2}\theta-1

 \implies \csc\theta=\sqrt{1+\cot^{2}\theta}            \implies\cot\theta=\sqrt{\csc^{2}\theta-1}

Trigonometric values: (We can write undefined in place of  '\infty'

  0 \dfrac{\pi}{6} \dfrac{\pi}{4}  \dfrac{\pi}{3} \dfrac{\pi}{2} \pi  \dfrac{3 \pi}{2}  2 \pi
\sin  0  \dfrac{1}{2} \dfrac{1}{\sqrt{2}}  \dfrac{\sqrt{3}}{2}  1 0 -1 0
\cos  1 \dfrac{\sqrt{3}}{2} \dfrac{1}{\sqrt{2}} \dfrac{1}{2}  0 -1 0 1
\tan  0 \dfrac{1}{\sqrt{3}}  1  \sqrt{3} \infty 0 \infty 0
\cot  \infty  \sqrt{3}  1 \dfrac{1}{\sqrt{3}}  0 \infty 0 \infty
\sec 1 \dfrac{2}{\sqrt{3}} \sqrt{2}  2 \infty -1 \infty 1
\csc  \infty  2  \sqrt{2}  \dfrac{2}{\sqrt{3}} 1 \infty -1  \infty

Signs of Trigonometric Functions:

The ASTC (All,\sin, \tan, \cos ) RULE

(i)  0 <\theta <\dfrac {\pi}{2}\implies (r,\theta)\in \text{1st quadrant, for } r > 0 \implies (r, \theta)_x, (r, \theta)_y > 0

 \implies  All of  \sin\theta, \cos\theta, \tan\theta, \cot\theta,\sec\theta,\text {and}\csc \theta are positive.

 

(ii)  \dfrac{\pi}{2} < \theta < \pi \implies (r,\theta) \in\text{ 2nd quadrant, for r > 0}

               \implies (r,\theta)_{x} < 0, (r,\theta)_{y}, > 0

 \implies \sin\theta > 0, \csc \theta > 0, ( All the rest are negative)

 

(iii)  \pi < \theta < \dfrac{3\pi}{2}\implies (r,\theta), \in \text{ 3rd quadrant} ,

for  r > \theta\implies (r,\theta)_{x}, (r,\theta)_{y} < 0

 \implies \tan\theta > 0, \cot\theta> 0 ( All the rest are negative)

 

(iv)  \dfrac{3\pi}{2} < \theta < 2\pi\implies (r,\theta) \in \text{ 4th quadrant} ,

for  r > 0\implies (r,\theta)_{x} > 0, (r,\theta)_{y} < 0

 \implies\cos\theta > 0, \sec\theta > 0, ( All the rest are negative)

Pictorial Representation:

 

Scroll to Top