# Trigonometry

## Trigonometry

### Identities:

sin2 θ + cos2 θ = 1
tan2 θ + 1 = sec2 θ
cot2 θ + 1 = cosec2 θ

## Trigonometric Ratio of Allied Angles:

Two angles are said to be allied when their sum or difference is either zero or multiple of 90°. -θ, 90°+θ, 180°-θ, 180°+θ, 360°+θ, 360°-θ are allied to the angle θ

### Trigonometric ratio of (-θ) in terms of θ

If θ is -ve, then only cos & sec are positive as -θ lies in 4th Quadrant.
sin(-θ) = - sin θ
cos(-θ) = cos θ
tan(-θ) = -tan θ
cot(-θ) = -cot θ
sec(-0) = sec θ
cosec(-θ) = -cosec θ

### Trigonometric ratio of (90°-θ) in terms of θ

All T-ratios are positive as 90°-θ lies in 1st Quadrant
sin(90°-θ) = cos θ
cos(90°-θ) = sin θ
tan(90°-θ) = cot θ
cot(90°-θ) = tan θ
sec(90°-0) = cosec θ
cosec(90°-θ) = sec θ

### Trigonometric ratio of (90°+θ) in terms of θ

90°+θ lies in 2nd Quadrant so only sin(90°+θ) & cosec(90°+θ) are positive
sin(90°+θ) = cos θ
cos(90°+θ) = -sin θ
tan(90°+θ) = -cot θ
cot(90°+θ) = -tan θ
sec(90°+0) = -cosec θ
cosec(90°+θ) = sec θ

### Trigonometric ratio of (180°-θ) in terms of θ

180°-θ lies in 2nd Quadrant so only sin(180°-θ) & cosec(180°-θ) are positive
sin(180°-θ) = sin θ
cos(180°-θ) = -cos θ
tan(180°-θ) = -tan θ
cot(180°-θ) = -cot θ
sec(180°-0) = -sec θ
cosec(180°-θ) = cosec θ

### Trigonometric ratio of (180°+θ) in terms of θ

180°+θ lies in 3rd Quadrant so only tan(180°+θ) & cot(180°-θ) are positive
sin(180°+θ) = -sin θ
cos(180°+θ) = -cos θ
tan(180°+θ) = tan θ
cot(180°+θ) = cot θ
sec(180°+0) = -sec θ
cosec(180°+θ) = -cosec θ

## Maximum and Minimum values of Trigonometrical Functions:

As we know,

Maximum and Minimum value of a trigonometrical function of the form sinθ+b cosθ are:

Maximum value of a trigonometrical function of the form sinθ-b cosθ are: