Trigonometry

It is the study of angles and sides and relationship between them in a triangle.

• The angle is measured in degree or radian.

Conversion :

Angle in radian     = 𝜋/180° ✕ Angle in degree

Angle in degrees   = 180°/𝜋 ✕ Angle in radian

Trigonometric Ratios 

sinθ = perpendicular/hypotenuse = 1/cosecθ

cosθ = base/hypotenuse = 1/secθ

tanθ = perpendicular/base = 1/cotθ = sinθ/cosθ

Sign of Trigonometric Ratios

• In first quadrant all are +ve
• In second quadrant sin and cosec are +ve
• In third quadrant tan and cot are +ve
• In fourth quadrant cos and sec are +ve 

It can be remembered as "Add Sugar To Coffee".

Trigonometric Ratios of Some angles

Trick (For sine) :

1. Write numbers 0,1,2,3,4
2. Divide all numbers by 4. You will get 0, 1/4, 1/2, 3/4, 1.
3. Take square root of all numbers. Now, you will get 0, 1/2, 1/√2, √3/2, 1 which is the required trigonometric ratio of sine for angles 0°, 30°, 45°, 60°, 90° respectively.
4. For cos repeat the above three steps with numbers in opposite order i.e, 4,3,2,1,0.

Note: Value of sinA and cosA always lie between -1 and +1.

Trigonometric Identities

• sin²A + cos²A = 1
• sec²A - tan²A = 1
• cosec²A - cot²A = 1

Sum and Difference Formulae :

• sin(A + B) = sinAcosB + cosAsinB
• cos(A + B) = cosAcosB ∓ sinAsinB
• tan(A + B) = (tanA + tanB)/(1 ∓ tanAtanB)

Trigonometric Ratios of Multiple Angles :

• sin2𝛳 = 2sin𝛳cos𝛳 = 2tan𝛳/(1 + tan²𝛳)
• sin3𝛳 = 3sin𝛳 - 4sin³𝛳
• cos2𝛳 = cos²𝛳 - sin²𝛳 = 1 - 2sin²𝛳 = 2cos²𝛳 - 1 = (1 - tan²𝛳)/(1 + tan²𝛳)
• cos3𝛳 = 4cos³𝛳 - 3cos𝛳
• tan2𝛳 = 2tan𝛳/(1 - tan²𝛳)
• tan3𝛳 = (3tan𝛳 - tan³𝛳)/(1 - 3tan²𝛳)

Operations on Allied angles :

• Trigonometric ratios changes for angles (90 + 𝛳) and (270 + 𝛳). For example, sin changes to cos and vice versa, tan changes to cot and vice versa, sec changes to cosec and vice versa.
• Trigonometric ratios remains same for (180 + 𝛳) and (360 + 𝛳).
• Angle 𝛳 is an acute angle.
• Sign of Trigonometric ratios must be kept in mind. 

1. Quadrant I : 90 - 𝛳, 360 + 𝛳
2. Quadrant II : 90 + 𝛳, 180 - 𝛳
3. Quadrant III : 180 + 𝛳, 270 - 𝛳
4. Quadrant IV : 270 + 𝛳, -𝛳 or 360 - 𝛳