Trigonometric Ratios

Where 𝞱 is an angle between two sides of a right angled triangle and P, B, H are the respective sides of that triangle

cosec 𝞱, sec 𝞱 and cot 𝞱 are the reciprocal ratios of sin 𝞱, cos 𝞱 and tan 𝞱 respectively

Standard and Important formulae:

sin 2a + cos2b = 1

sec2a = 1 + tan2b

cosec2a = 1 + cot2b

Examples
Complementary angles:
  • sin(56) = cos(34)
  • cos(72) = sin(18)
  • sec(47) = cosec(43)
  • cosec(26) = sec(64)
  • tan(84) = cot(6)
  • cot(31) = tan(59)
Complementary Angles Image

Angle of elevation:

The angle between the horizontal and the line of sight joining an observation point to an elevated object.

Angle of depression:

The angle between the horizontal and the line of sight joining an observation point to an object below the horizontal level.

Examples

1. A man standing at a certain distance from a building, observe the angle of elevation of its top to be 60 degrees. He walks 30 meters away from the building. Now the angle of elevation is 30 degrees. Calculate the height of the building.


2. From an observation tower, the angle of depression of 2 cars on opposite sides of the tower are Ι‘ and Ξ². If the height of the tower is h meters, find the distance between the cars.

Quadrants:

For angles greater than 90, the signs of each trig function changes. This variation can be seen in trigonometric quadrants. There are 4 quadrants.

cosec, sec and cot have the same signs as sin, cos and tan respectively in each quadrant

Important Trigonometric Formule

Examples

1.Prove that:tan(a) + sec(a) βˆ’ 1/tan(a) βˆ’ sec(a) + 1 = 1 + sin(a)/cos(a)


2. Find the value of √3 cosec 20° - sec 20°


3.If cos(πœƒ+Ο•)=π‘š cos(πœƒ-Ο•),then prove that tan⁑ πœƒ = 1-π‘š/1+m cot πœƒ

Exercise Problems

1. Prove

2.Prove

3.Prove