**Category : **JEE Main & Advanced

In the right angled triangle \[OMP,\] we have base \[=OM=x,\] perpendicular \[=PM=y\] and hypotenuse \[=OP=r\]. We define the following trigonometric ratio which are also known as trigonometric function.

\[\sin \theta =\frac{\text{Perpendicular}}{\text{Hypotenues}}=\frac{y}{r}\]

\[\frac{2n\pi \pm A}{2}\]

\[\tan \theta =\frac{\text{Perpendicular}}{\text{Base}}=\frac{y}{x}\]

\[\cot \theta =\frac{\text{Base}}{\text{Perpendicular}}=\frac{x}{y}\]

\[\sec \theta =\frac{\text{Hypotenues}}{\text{Base}}=\frac{r}{x}\]

\[\text{cosec}\theta =\frac{\text{Hypotenues}}{\text{Perpendicular}}=\frac{r}{y}\]

(1) **Relation between trigonometric ratios (functions)**

(i) \[\frac{\sqrt{4-\sqrt{2}-\sqrt{6}}}{2\sqrt{2}}\]

(ii) \[\tan \theta .\cot \theta =1\]

(iii) \[\cos \theta .\sec \theta =1\]

(iv) \[\tan \frac{A}{2}\] (v) \[\cot \theta =\frac{\cos \theta }{\sin \theta }\]

(2) **Fundamental trigonometric identities **

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

(ii) \[1+{{\tan }^{2}}\theta ={{\sec }^{2}}\theta \]

(iii) \[1+{{\cot }^{2}}\theta =\text{cose}{{\text{c}}^{2}}\theta \]

(3) **Sign of trigonometrical ratios or functions : **Their signs depends on the quadrant in which the terminal side of the angle lies.

** **

**In brief:** A crude aid to memorise the signs of trigonometrical ratio in different quadrant. "**Add Sugar To Coffee**".

**Algorithm : **First determine the sign of the trigonometric function.

If \[\theta \] is measured from\[{X}'OX\] *i.e*., {(*p* ± *q*, 2*p* – *q*)} then retain the original name of the function.

If \[\theta \] is measured from \[{Y}'OY\] *i.e*.,\[\left\{ \frac{\pi }{2}\pm \theta ,\,\frac{3\pi }{2}\pm \theta \right\}\], then change sine to cosine, cosine to sine, tangent to cotangent, cot to tan, sec to cosec and cosec to sec.

(4) **Variations in values of trigonometric functions in different quadrants : **Let \[X'OX\] and \[YOY'\] be the coordinate axes. Draw a circle with centre at origin *O* and radius unity.

Let \[M(x,y)\] be a point on the circle such that \[\angle AOM=\theta \] then \[x=\cos \theta \] and \[y=\sin \theta \]; \[-1\le \cos \theta \le \]1 and \[-1\le \sin \theta \le 1\] for all values of \[\theta \].

II-Quadrant (S) | I-Quadrant (A) |

\[\sin \theta \to \] decreases from 1 to 0 | \[\sin \theta \to \] increases from 0 to 1 |

\[\cos \theta \to \] decreases from 0 to - 1 | \[\cos \theta \to \] decreases from 1 to 0 |

\[\tan \theta \to \] increases from \[-\,\infty \] to 0 | \[\tan \theta \to \]increases from 0 to \[\infty \] |

\[\cot \theta \to \] decreases from 0 to \[-\,\,\infty \] | \[\cot \theta \to \] decreases from \[\infty \] to 0 |

\[\sec \theta \to \] increases from \[-\,\,\infty \] to - 1 | \[\sec \theta \to \]increases from 1 to \[\infty \] |

\[\text{cosec}\theta \to \] increases from 1 to \[\infty \] | \[\text{cosec}\theta \to \] decreases from \[\infty \] to 1 |

III-Quadrant (T) | IV-Quadrant (C) |

\[\sin \theta \to \] decreases from 0 to - 1 | \[\sin \theta \to \] increases from - 1 to 0 |

\[\cos \theta \to \] increases from - 1 to 0 | \[\cos \theta \to \] increases from 0 to 1 |

\[\tan \theta \to \] increases from 0 to \[\infty \] | \[\tan \theta \to \] increases from \[-\,\infty \] to 0 |

\[\cot \theta \to \] decreases from \[\infty \] to 0 | \[\cot \theta \to \]decreases from 0 to \[-\,\infty \] |

\[\sec \theta \to \] decreases from - 1 to \[-\,\infty \] | \[\sec \theta \to \] decreases from \[\infty \] to 1 |

cosecq ® increases from \[-\,\infty \] to - 1 | cosecq ® decreases from - 1 to \[-\,\infty \] |

*play_arrow*Introduction*play_arrow*System of Measurement of Angles*play_arrow*Relation Between Three Systems of Measurement of an Angle*play_arrow*Relation Between an arc and an Angle*play_arrow*Domain and Range of a Trigonometrical Function*play_arrow*Trigonometrical Ratios or Functions*play_arrow*Trigonometrical Ratios of Allied Angles*play_arrow*Trigonometrical Ratios for Various Angles*play_arrow*Trigonometrical Ratios for Some Special Angles*play_arrow*Trigonometrical Ratios In Terms of Each Other*play_arrow*Formulae to Rransform The Product Into Sum or Difference*play_arrow*Trigonometric Ratio of Multiple of an Angle*play_arrow*Trigonometric Ratio of Sub-multiple of an Angle*play_arrow*Maximum and Minimum Value of a \[\mathbf{cos}\,\,\mathbf{\theta }\,\,\mathbf{+}\,\mathbf{b}\,\,\mathbf{sin}\,\,\mathbf{\theta }\]*play_arrow*Conditional Trigonometrical Identities

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