Basic Trigonometric formulas

right angled triangle with theta as an angle

Definition

The basic relation between any two or more trigonometric functions is called basic trigonometric identity or basic trigonometric formula.

List of Basic Identities

In trigonometry, the six trigonometric ratios form four types of basic trigonometric formulas. Here is the list of four types of basic trigonometric identities, which are derived by assuming theta ($\theta$) as the angle of the right angled triangle.

1

Reciprocal identities

Trigonometric ratios form six identities in reciprocal form and learn proofs of these reciprocal formulas.

$(1)\,\,\,\,$ $\sin \theta \,=\, \dfrac{1}{\csc \theta}$

$(2)\,\,\,\,$ $\cos \theta \,=\, \dfrac{1}{\sec \theta}$

$(3)\,\,\,\,$ $\tan \theta \,=\, \dfrac{1}{\cot \theta}$

$(4)\,\,\,\,$ $\cot \theta \,=\, \dfrac{1}{\tan \theta}$

$(5)\,\,\,\,$ $\sec \theta \,=\, \dfrac{1}{\cos \theta}$

$(6)\,\,\,\,$ $\csc \theta \,=\, \dfrac{1}{\sin \theta}$

2

Product identities

Trigonometric functions form three formulas in product form and learn the proofs of product identities.

$(1)\,\,\,\,$ $\sin \theta \times \csc \theta = 1 $

$(2)\,\,\,\,$ $\cos \theta \times \sec \theta = 1 $

$(3)\,\,\,\,$ $\tan \theta \times \cot \theta = 1 $

3

Quotient identities

The six trigonometric functions involve in two relations in quotient form and learn the proofs of quotient identities.

$(1)\,\,\,\,$ $\dfrac{\sin \theta}{\cos \theta} = \tan \theta$

$(2)\,\,\,\,$ $\dfrac{\cos \theta}{\sin \theta} = \cot \theta$

4

Pythagorean identities

The six trigonometric ratios form three formulas, which are derived by the Pythagoras theorem and learn proofs of Pythagorean formulas.

$(1)\,\,\,\,$ $\sin^2 \theta \,+\, \cos^2 \theta = 1$

$(2)\,\,\,\,$ $\sec^2 \theta \,-\, \tan^2 \theta = 1$

$(3)\,\,\,\,$ $\csc^2 \theta \,-\, \cot^2 \theta = 1$

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