#### Trigonometry

After completion of this chapter, students will be able to:

- know about various trigonometric ratios, trigonometric identities and equations.
- solve problems related to trigonometric ratios, trigonometric identities and equations.
- solve problems of height and distance.

#### Notes

#### Trigonometric Ratios of Compound Angles

Let A and B two angles. Then their sum A + B or the difference A - B is called a compound angle. The sum or difference of any two or more than two angles is called a compound angle.

#### Trigonometric Ratios of Multiple Angles

If A is an angle, then 2A, 3A, 4A, 5A, etc. are called multiple angles of A.

#### Trigonometric Ratios of Sub Multiple Angles

If A is an angle, then \(\frac{A}{2}\), \(\frac{A}{3}\), \(\frac{A}{4}\) etc. are called sub - multiple angles of A.

#### Transformation of Trigonometric Formulae

Transformation formulae |
Key to remember |

2sinA cosB = sin(A + B) + sin(A - B) | 2 sin. cos = sin + sin |

2 cosA sinB = sin(A + B) - sin(A - B) | 2 cos. sin = sin - sin |

2 cosA cosB = cos(A + B) + cos(A - B) | 2 cos. cos = cos + cos |

2 sinnA sinB = cos (A - B) - cos(A + B) |
2 sin. sin = cos - cos |

sinC + sinD = 2sin(\(\frac{C + D}{2}\)) cos (\(\frac{C - D}{2}\)) | sin + sin = 2sin. cos |

sinC - sinD = 2 cos (\(\frac{C + D}{2}\)) sin (\(\frac{C - D}{2}\)) | sin - sin = 2cos. sin |

cosC + cosD = 2 cos (\(\frac{C + D}{2}\)) cos (\(\frac{C - D}{2}\)) | cos + cos = 2cos. cos |

cosC - cosD = -2 sin (\(\frac{C + D}{2}\)) sin (\(\frac{C - D}{2}\)) | cos - cos = 2sin. sin |

#### Conditional Trigonometric Identities

**Conditional Trigonometric Identities.**

Identities which are true under some given conditions are termed as conditional identities and

in this section, we will deal some trigonometric identities which are bound to the condition of the sum of the angles of a triangle i.e. A + B + C =π

#### Solution of Trigonometric Equations

**A method for finding angles.**

(i) First of all, we determine the quadrant where the angle falls. For this, we use the all sin, tan, cos rule.

If sinθ is positive, the angleθ falls in the 1^{st} and 2^{nd} quadrants and if sinθ is negative, the angleθ falls in the 3^{rd} and 4^{th} quadrants. If cosθ is positive,θ lies in the 1^{st} and 4^{th} quadrants and if cosθ is negative,θ lies in the 2^{nd} and 3^{rd} quadrants. if tanθ is positive,θ lies in the first and third quadrants and if tanθ is negative,θ lies in second and fourth quadrants.

(ii) To find the angle in the first quadrant, we find the acute angle which satisfies the equation.

For example, if 2cosθ = 1

then cosθ = \(\frac{1}{2}\)

or, cosθ = cos60^{0}

So, θ = 60^{0}

(iii) To find the angle in the second quadrant, we subtract acute angleθ from 180^{0}.

(iv) To find the angle in the third quadrant, we add acute angleθ to 180^{0}

(v) To find the angle in the 4^{th} quadrant, we subtract acute angleθ to 360^{0}

(vi) To find the value ofθ from the equations like sinθ = 0, cosθ = 0, tanθ= 0, sinθ = 1, cosθ = 1, sinθ = -1, cosθ = -1. We should note the following results:

If sinθ = 0, thenθ = 0^{0}, 180^{0} or 360^{0} |
If sinθ = 1, thenθ = 90^{0} |

If tanθ = 0, thenθ = 0^{0,}180^{0} or 360^{0} |
If sinθ = -1, thenθ = 270^{0} |

If cosθ = 0, thenθ = 90^{0} or 270^{0} |
If cosθ = 1, thenθ = 0^{0} or 360^{0} |

If cosθ = -1, then θ = 180^{0}.

#### Heights and Distances

This note explains about the angle of depression and angle of elevation.