In these trigonometry formulas, we will focus on all the identities used for solving the complex problems in trigonometry. Firstly, we will understand the functions used in trigonometries such as \sin, \cos,\tan,\csc, \sec and \cot and the basics of it. These functions are inter-related in the following way

  • Reciprocal identity
\csc\theta = \frac{1}{\sin \theta}\\ \sec \theta = \frac{1}{\cos \theta}\\ \cot \theta = \frac{1}{\tan \theta}\\ \sin\theta = \frac{1}{\csc \theta}\\ \cos \theta = \frac{1}{\sec \theta}\\ \tan \theta = \frac{1}{\cot \theta}\\

Above we have shown the basic formulas for this function using a right angled triangle in trigonometry table. Also, few more trigonometry formulas

  • Co-functional identity

I)In radians
\sin (\Pi/2 – \theta) = \cos \theta \:\&\: \cos (\Pi/2 – \theta) = \sin \theta \\ \sin (\Pi/2 + \theta) = \cos \theta \:\&\: \cos (\Pi/2 + \theta) = – \sin \theta \\ \sin (3\Pi/2 – \theta) = – \cos \theta \:\&\: \cos (3\Pi/2 – \theta) = – \sin \theta \\ \sin (3\Pi/2 + \theta) = – \cos \theta \:\&\: \cos (3\Pi/2 + \theta) = \sin \theta \\ \sin (\Pi – \theta) = \sin \theta \:\&\: \cos (\Pi – \theta) = – \cos \theta \\ \sin (\Pi + \theta) = – \sin \theta \:\&\: \cos (\Pi + \theta) = – \cos \theta \\ \sin (2\Pi – \theta) = – \sin \theta \:\&\: \cos (2\Pi – \theta) = \cos \theta \\ \sin (2\Pi + \theta) = \sin \theta \:\&\: \cos (2\Pi + \theta) = \cos \theta \\

II)Degree
\sin(90°−\theta) = \cos \theta\\ \cos(90°−\theta) = \sin \theta\\ \tan(90°−\theta) = \cot \theta\\ \cot(90°−\theta) = \tan \theta\\ \sec(90°−\theta) =\ csc \theta\\ \csc(90°−\theta) = \sec \theta\\

  • Addition and subtraction formulas 
\sin(a+b) = \sin(a)\cos(b)+\cos(a)\sin(b)\\ \cos(a+b) = \cos(a)\cos(b)–sin(a)\sin(b)\\ \tan(a+b) = \frac{(\tan a + \tan b)}{ (1−\tan a \times \tan b)}\\ \sin(a-b) = \sin(a)\cos(b)–\cos(a)\sin(b)\\ \cos(a-b) = \cos(a)\cos(b) + \sin(a)\sin(b)\\ \tan(a-b) = \frac{(\tan a–\tan b)} {(1+\tan a \times \tan b)}\\
  • Double and triple angle formula
\sin(2\theta) = 2\sin(\theta) \cos(\theta) = [\frac{2\tan \theta}{(1+\tan2 \theta)}]\\ \cos(2\theta) = \cos2(\theta)–\sin2(\theta) = \frac{(1-\tan2 \theta)}{(1+\tan2 \theta)}\\ \cos(2\theta) = 2\cos2(x\theta)−1 = 1–2\sin2(\theta)\\ \tan(2\theta) = \frac{2\tan(\theta)}{ 1−\tan2(\theta)}\\ \sec (2\theta) =\frac {\sec2 \theta}{(2-\sec2 \theta)}\\ \csc (2\theta) = \frac{(\sec \theta. \csc \theta)}{2}\\ \sin 3\theta = 3\sin \theta – 4\sin 3\theta \\ \cos 3\theta = 4\cos 3\theta-3\cos \theta\\ \tan 3\theta = \frac{[3 \tan \theta-\tan 3\theta]}{[1-3\tan 2\theta]}\\
  • Half angle formulas
\sin \frac{\theta}{2}=\sqrt{±1−\cos\frac{\theta}{2}}\\ \cos \frac{\theta}{2}=\sqrt{±1+\cos\frac{\theta}{2}}\\ \tan\frac{\theta}{2}=\sqrt{\frac{1−\cos(\theta)}{1+\cos(\theta)}}\\ \tan\frac{\theta}{2}=\sqrt{\frac{1−\cos(\theta)}{1+\cos(\theta)}}=\sqrt{\frac{1−cos(\theta)}{sin(\theta)}}
  • Product and sum formulas
\sin \alpha .\cos \beta =\frac{\sin(\alpha +\beta)+\sin(\alpha -\beta)}{2}\\ \cos \alpha .\cos \beta =\frac{\cos(\alpha +\beta)+\cos(\alpha -\beta)}{2}\\ \sin \alpha .\sin \beta =\frac{\cos(\alpha -\beta)-\cos(\alpha +\beta)}{2}\\ \sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}\\ \sin\alpha−\sin\beta=2\cos\frac{\alpha+\beta}\sin\frac{\alpha-\beta}{2}\\ \cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}\cos\frac{\alpha-\beta}{2}\\ \cos\alpha−\cos\beta=−2\sin\frac{\alpha+\beta}\sin\frac{\alpha-\beta}{2}
  • Inverse trigonometry formula
\sin^{-1} (–\theta) = – \sin^{-1} \theta \\ \cos^{-1} (–\theta) = \Pi – \cos^{-1} \theta\\ \tan^{-1} (–\theta) = – \tan^{-1} \theta\\ \csc^{-1} (–\theta) = – \csc^{-1} \theta\\ \sec^{-1} (–\theta) = \Pi – \sec^{-1} \theta\\ \cot^{-1} (–\theta) = \Pi – \cot^{-1} \theta

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