**Trigonometry:Class 10 Mathematics Detailed Valuable Notes**

**1. Introduction to Trigonometry**

**Meaning:**Trigonometry is a branch of mathematics that deals with the study of relationships between the angles and sides of triangles.- It helps us understand the properties of triangles and is widely used in various fields, including science, engineering, and navigation.

**2. Trigonometric Ratios**

- Trigonometric ratios are essential concepts in trigonometry.
- They define the relationship between the angles and sides of a right triangle. There are
**six primary trigonometric ratios**, often denoted as: Trigonometric Ratio Abbreviation Formula Sine sin θ $sin\theta =\frac{\text{Perpendicular}}{\text{Hypotenuse}}$

Cosine cos θ $\mathrm{cos}\theta =\frac{\text{Base}}{\text{Hypotenuse}}$

Tangent tan θ $\mathrm{tan\theta}=\frac{\text{Perpendicular}}{\text{Base}}$

Cosecant csc θ $\mathrm{csc}\mathrm{\theta =}=\frac{1}{\mathrm{sin}\theta}$

Secant sec θ $\mathrm{sec}\theta =\frac{1}{\mathrm{cos}\theta}$

Cotangent cot θ $\mathrm{cot}\theta =\frac{1}{\mathrm{tan}\theta}$

**Sine (sin θ):** The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. The formula is given by:

$\mathrm{sin}\theta \; =\frac{\text{Opposite}}{\text{Hypotenuse}}$**Cosine (cos θ):** The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. The formula is given by:

$\mathrm{cos}\theta =\frac{\text{Adjacent}}{\text{Hypotenus}}$

$Tangent\; (tan\; \theta ):The\; tangent\; of\; an\; angle\; is\; the\; ratio\; of\; the\; length\; of\; the\; side\; opposite\; the\; angle\; to\; the\; length\; of\; the\; adjacent\; side.\; The\; formula\; is\; given\; by:$

$\mathrm{tan}\theta =\frac{\text{Opposite}}{\text{Adjacent}}$

**Cosecant (csc θ):** The cosecant of an angle is the reciprocal of the sine. It is given by:

$\mathrm{csc}\theta =\frac{1}{\mathrm{sin}\theta}$

**Secant (sec θ):** The secant of an angle is the reciprocal of the cosine. It is given by:

$\mathrm{sec\theta}=\frac{1}{\mathrm{cos}\theta}$

**Cotangent (cot θ):** The cotangent of an angle is the reciprocal of the tangent. It is given by:

$\mathrm{cot}\theta =\frac{1}{\mathrm{tan}\theta}$

T-Ratio | ${\mathrm{}}^{}$ | $\mathrm{}{}^{}$ | $\mathrm{}{}^{}$ | $\mathrm{}{}^{}$ | $\mathrm{}{}^{}$ |
---|---|---|---|---|---|

$$ | 0 | $1\mathrm{/}2$ | $1\mathrm{/}\sqrt{2}$ | $\sqrt{3}\mathrm{/}2$ | 1 |

$$ | 1 | $\sqrt{3}\mathrm{/}2$ | $1\mathrm{/}\sqrt{2}$ | $1\mathrm{/}2$ | 0 |

$$ | 0 | $\sqrt{\mathrm{1/}}$3 | 1 | $\sqrt{3}$ | Undefined |

This table provides the values of sine (

$\theta $), cosine (θ), and tangent (θ) for the specified angles in degrees.

**3. Trigonometric Ratios in Common Life**

Trigonometric ratios are used in various real-life scenarios, including:

**Construction:**Architects and engineers use trigonometry to calculate angles and distances when designing buildings and structures.**Astronomy:**Astronomers use trigonometry to calculate the positions and movements of celestial objects.**Navigation:**Sailors and pilots use trigonometry to determine their positions and course corrections.**Physics:**Trigonometry is crucial in physics to analyze motion, waves, and oscillations.

**4.Trigonometric Ratios of Allied Angles**

Trigonometric ratios of allied angles are relationships between the trigonometric functions of complementary angles.

**5. Trigonometric Identities**

Trigonometric identities are equations that are true for all values of the variable. Three fundamental trigonometric identities are:

$\frac{\text{}}{}$ $\frac{\mathrm{}}{}$