Trigonometry Part-1 || Chapter – 8 || Class 10th (For English Medium)

Trigonometry Part-2 || Chapter – 8 || Class 10th (For English Medium)

Trigonometry Part-1 || Chapter – 8 || Class 10th (For Hindi Medium)

Trigonometry Part-2 || Chapter – 8 || Class 10th (For Hindi Medium)

__Trigonometric Ratios__

Opposite & Adjacent Sides in a Right Angled Triangle

In the ΔABC right-angled at B, BC is the side opposite to ∠A, AC is the hypotenuse and AB is the side adjacent to ∠A.

**Trigonometric Ratios**

For the right ΔABC, right-angled at ∠B, the trigonometric ratios of the ∠A are as follows:

- sin A=opposite side/hypotenuse=BC/AC
- cos A=adjacent side/hypotenuse=AB/AC
- tan A=opposite side/adjacent side=BC/AB
- cosec A=hypotenuse/opposite side=AC/BC
- sec A=hypotenuse/adjacent side=AC/AB
- cot A=adjacent side/opposite side=AB/BC

To know more about Trigonometric Ratios, visit here.

**Visualization of Trigonometric Ratios Using a Unit Circle**

Draw a circle of the unit radius with the origin as the centre. Consider a line segment OP joining a point P on the circle to the centre which makes an angle θ with the x-axis. Draw a perpendicular from P to the x-axis to cut it at Q.

- sinθ=PQ/OP=PQ/1=PQ
- cosθ=OQ/OP=OQ/1=OQ
- tanθ=PQ/OQ=sinθ/cosθ
- cosecθ=OP/PQ=1/PQ
- secθ=OP/OQ=1/OQ
- cotθ=OQ/PQ=cosθ/sinθ

**Visualisation of Trigonometric Ratios Using a Unit Circle**

Relation between Trigonometric Ratios

- cosec θ =1/sin θ
- sec θ = 1/cos θ
- tan θ = sin θ/cos θ
- cot θ = cos θ/sin θ=1/tan θ

**Trigonometric Ratios of Specific Angles**

Range of Trigonometric Ratios from 0 to 90 degrees

For 0∘≤θ≤90∘,

- 0≤sinθ≤1
- 0≤cosθ≤1
- 0≤tanθ<∞
- 1≤secθ<∞
- 0≤cotθ<∞
- 1≤cosecθ<∞

tanθ and secθ are not defined at 90∘.

cotθ and cosecθ are not defined at 0∘.

**Variation of trigonometric ratios from 0 to 90 degrees**

As θ increases from 0∘ to 90∘

- sin θ increases from 0 to 1
- cos θ decreases from 1 to 0
- tan θ increases from 0 to ∞
- cosec θ decreases from ∞ to 1
- sec θ increases from 1 to ∞
- cot θ decreases from ∞ to 0

Standard values of Trigonometric ratios

∠A | 0^{o} |
30^{o} |
45^{o} |
60^{o} |
90^{o} |

sin A | 0 | 1/2 | 1/√2 | √3/2 | 1 |

cos A | 1 | √3/2 | 1/√2 | 1/2 | 0 |

tan A | 0 | 1/√3 | 1 | √3 | not defined |

cosec A | not defined | 2 | √2 | 2/√3 | 1 |

sec A | 1 | 2/√3 | √2 | 2 | not defined |

cot A | not defined | √3 | 1 | 1/√3 | 0 |

To know more about Trigonometric Ratios of Standard Angles, visit here.

**Trigonometric Ratios of Complementary Angles**

**Complementary Trigonometric ratios**

If θ is an acute angle, its complementary angle is 90∘−θ. The following relations hold true for trigonometric ratios of complementary angles.

- sin (90∘− θ) = cos θ
- cos (90∘− θ) = sin θ
- tan (90∘− θ) = cot θ
- cot (90∘− θ) = tan θ
- cosec (90∘− θ) = sec θ
- sec (90∘− θ) = cosec θ

To know more about Trigonometric Ratios of Complementary Angles, visit here.

**Trigonometric Identities**

- sin2θ+cos2θ=1
- 1+cot2θ=coesc2θ
- 1+tan2θ=sec2θ