TRIGONOMETRY

TRIGONOMETRY

Some Important Formulas: 

· A line is a path joining a set of points in a plane.

· A portion of a line together with its dividing point is known as a ray.

· A join or intersection of two rays having a common endpoint gives an angle. The common endpoint of an angle is called the vertex of the angle. The rays that form the angle are called the sides of the angle.

· 90° is one measure of a Right Angle.

· Any angle measuring between 0° and 90° is called an acute angle.

· Any angle whose measure exceeds 90° but falls below 180° is called an obtuse angle.

· An angle between 180° and 360° is called a reflex angle.

· When the rotating ray stops after making an angle of rotation ϑ, the place where the rotating ray stops is called the terminal side.

· If the ray has completed one full revolution about the vertex, then the angle turned is 360°

Measurement of angles:
For the measurement of angles, there are three known systems. They are

• Sexagesimal measurement (British System):

In this system 1/360 of a turn is called a degree(°), 1/60 of a degree is called a minute(') and 1/60 of a minute is called a second("). 

• Centesimal measurement (French system):

this system 1/400 of a turn is called a grade(g), 1/100 of a grade is called a minute(') and 1/100 of a minute is called a second ("). 
Right angle = 100g, 1g = 100'; 1' = 100".

• Note:-

The minute and seconds in the Sexagesime system are different from the minutes and seconds respectively in the Centesima system.

Circular measurement (Radian measure):
The angle subtended by an area of a circle whose length is equal to the radius of the circle at the center of the circle is called a radian. In this system, the unit of measurement is radian(c). 
1 right angle = π/2 radians(πC/2)
                                                         
Trigonometric Ratios:
Let ABC be a right angle triangle. Then with reference to the angle A, we have the following. 
Sin A = oppositeside/hypotenuse = BC/AC 
Cos A = adjacentside/hypotenuse = AB/AC 
Tan A = oppositeside/adjacentside = BC/AB

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Sign of the trigonometric ratios in quadrants:-

1. All trigonometric ratios are positive in the I quadrant.

2. Sine and cosec are positive in II quadrant.

3. Tan and cot are positive in III quadrant.

4. Cos and Sec are positive in the IV quadrant.

5. Sexagesimal (English) system

6. Centesimal (French) system

7. The radian or circular measure.

Comeertion of Trigonometric Ratio to quadrants:-

• Angles of the form ( nπ/2 + or - θ)

1. If n is even Sin Remains as sin : 
Cos -- cos 
Tan -- Tan 
Cosec -- cosec 
Sec -- sec 
Cot -- cot

 

2. If n is odd Sin changes as cos : 
Cos -- as Sin 
Tan -- as Cot 
Cot -- as Tan 
Cosec -- as Sec 
Sec -- as Cosec 

The values of trigonometric ratios of some standard angles:-

Some standard formulae:

· Sin θ = 1/cosec θ => sinθ.cosecθ = 1

· Cosθ = 1/secθ => cosθ.secθ = 1

· tanθ = 1/ cot => tanθ.cotθ = 1

· sin²θ+cos²θ = 1 => 1-cos²θ = sin² and 1 - sin²θ = cos²θ

· sec²θ - tan²θ = 1 => 1+tan²θ = sec²θ and sec²θ - 1 = tan²θ

· cosec²θ - cot²θ = 1 => 1 + cot²θ = cosec²θ and cosec²θ - 1 = tan²θ

· sin2θ = 2sin θ cos θ

· cos2θ = 2sinθcosθ

· tan2θ = 2tanθ/1-tan2θ

· Sin (A+B) = SinA CosB + CosA SinB

· Sin (A-B) = SinA CosB - CosA SinB

· Cos (A+B) = CosA CosB - SinA SinB

· Cos (A-B) = CosA CosB + SinA SinB

· Tan (A+B) = (tan A + tan B)/(1-tan A tan B)

· Tan (A-B) = (tan A - tan B)/(1+tan A tan B)

· Sin 15° = Cos 75° = (√3 - 1)/2√2

· Cos 15° = Sin 75° = (√3 + 1)/2√2

· Tan 15° = Cot 75° = (√3 - 1)/(√23 + 1)

The values of trigonometric ratios of some standard angles:-

Complementary angles: Two angles α,β are said to be the complementary angles if α + β = 90°
Supplementary angles: Two angles α,β are said to be the supplementary angles if α + β = 180°
 

· If α,β are the complementary angles then

1. sinα = cosβ

2. cosα = sinβ

3. tanα = cotβ

4. cotα = tanβ

5. sin^2α + sin^2β = 1

6. cos^2α + cos^2β = 1

7. tanα,tanβ = 1

8. cotα,cotβ = 1

1. If α,β are the supplementary angles then
 

1. sinα = sinβ

2. Sin (A-B) = SinA CosB - CosA SinB

3. cosα + cosβ = 0

4. tanα + tanβ = 0

5. cotα + cotβ = 0

6. sin2α + cos2β = 0

7. cos2α + sin2β = 0

 

1. Sides of few right-angled triangles:

1. 3, 4, 5

2. 8, 15, 17

3. 9, 40, 41