Trigonometric Functions, Identities & Equations

🦉 Special Angles

\begin{array}{|c|c|c|c|c|c|} \hline \theta & 0^{\circ}& 30^{\circ} & 45^{\circ} & 60^{\circ} & 90^{\circ} \\ \hline \sin\theta & \frac{\sqrt{0}}{2}=0 & \frac{\sqrt{1}}{2}=\frac{1}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{2} & \frac{\sqrt{4}}{2}=1 \\ \hline \cos\theta & \frac{\sqrt{4}}{2}=1 & \frac{\sqrt{3}}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{1}}{2}=\frac{1}{2} & \frac{\sqrt{0}}{2}=0 \\ \hline \tan\theta & 0 & \frac{1}{\sqrt{3}} & 1 & \sqrt{3} & - \\ \hline \end{array}

🦉 Reciprocal Functions

1.\,\text{cosec }\theta=\dfrac{1}{\sin\theta}

2.\,\sec\theta=\dfrac{1}{\cos\theta}

3.\,\cot\theta=\dfrac{1}{\tan\theta}

🦉 Negative Functions

1.\,\sin(-\theta)=-\sin\theta

2.\,\cos(-\theta)=\cos\theta

3.\,\tan(-\theta)=-\tan\theta

🦉 Tangent & Cotangent

1.\,\tan\theta=\dfrac{\sin\theta}{\cos\theta}

2.\,\cot\theta=\dfrac{\cos\theta}{\sin\theta}

🦉 Trigonometric Identities

1.\,\sin^2{\theta} + \cos^2{\theta} = 1

2.\,\sec^2{\theta}=1+\tan^2{\theta}

3.\,\text{cosec}^2{\theta}=1 + \cot^2{\theta}

🦉 Addition Formulae

1.\,\sin(A \pm B) = \sin A\cos B \pm \cos A \sin B

2.\,\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

3.\,\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

🦉 Double Angle Formulae

1.\,\sin 2A = 2\sin A \cos A

2.\,\cos2A = \cos^2A - \sin^2A

= 2 \cos^2A -1

= 1 - 2\sin^2A

3.\,\tan 2A = \dfrac{2\tan A}{1-\tan^2A}

🦉 R-Formulae

For $a > 0, b > 0, 0^\circ < \alpha < 90^\circ$,

1.\,a\sin \theta \pm b \cos \theta = R \sin (\theta \pm \alpha)

2.\,a \cos \theta \pm b \sin \theta = R \cos (\theta \mp \alpha)

where $R=\sqrt{a^2+b^2}, \tan\alpha=\dfrac{b}{a}$.