“All Students Take Calculus”
Which ratios are positive in each quadrant
Quadrant I all positive, II sine, III tangent, IV cosine. Going anticlockwise from the first quadrant.
Mathematics, Class 11 and 12 · Gujarat Board, CBSE, GUJCET and JEE
Trigonometry formulas
Trigonometry is the chapter the rest of mathematics keeps borrowing from. Calculus, vectors and three-dimensional geometry all assume you know these identities cold, so time spent here is repaid three times over.
22 formulas, 4 memory hooks, and the 5 mistakes that cost the most marks.
Standard values
| What | Formula | Symbols and notes |
|---|---|---|
| sin | 0, ½, 1/√2, √3/2, 1 | at 0°, 30°, 45°, 60°, 90° |
| cos | 1, √3/2, 1/√2, ½, 0 | the sine row read backwards |
| tan | 0, 1/√3, 1, √3, not defined | tan = sin/cos, and cos 90° = 0 |
| Radians | 180° = π radian | 1 radian is about 57.3° |
Identities
| What | Formula | Symbols and notes |
|---|---|---|
| Pythagorean | sin²θ + cos²θ = 1 | the identity everything else is built from |
| Secant form | 1 + tan²θ = sec²θ | divide the first by cos²θ |
| Cosecant form | 1 + cot²θ = cosec²θ | divide the first by sin²θ |
| Sum of angles | sin(A + B) = sinA cosB + cosA sinB | cos(A + B) = cosA cosB − sinA sinB |
| Difference of angles | sin(A − B) = sinA cosB − cosA sinB | cos(A − B) = cosA cosB + sinA sinB |
| Tangent sum | tan(A + B) = (tanA + tanB)/(1 − tanA tanB) | undefined when the denominator is zero |
Multiple and half angles
| What | Formula | Symbols and notes |
|---|---|---|
| Double angle, sine | sin2A = 2 sinA cosA | also 2tanA/(1 + tan²A) |
| Double angle, cosine | cos2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A | three forms, choose what the question gives |
| Double angle, tangent | tan2A = 2tanA/(1 − tan²A) | undefined at 45° |
| Triple angle | sin3A = 3sinA − 4sin³A, cos3A = 4cos³A − 3cosA | the 3-4 and 4-3 pattern |
| Product to sum | 2 sinA cosB = sin(A + B) + sin(A − B) | used constantly in integration |
| Sum to product | sinC + sinD = 2 sin((C + D)/2) cos((C − D)/2) | and cosC + cosD = 2 cos((C+D)/2) cos((C−D)/2) |
Triangles and general solutions
| What | Formula | Symbols and notes |
|---|---|---|
| Sine rule | a/sinA = b/sinB = c/sinC = 2R | R is the circumradius |
| Cosine rule | cosA = (b² + c² − a²)/2bc | use when two sides and the included angle are known |
| Area of a triangle | Δ = ½ab sinC = √(s(s−a)(s−b)(s−c)) | the second is Heron's formula |
| General solution, sine | θ = nπ + (−1)nα | when sinθ = sinα |
| General solution, cosine | θ = 2nπ ± α | when cosθ = cosα |
| General solution, tangent | θ = nπ + α | when tanθ = tanα |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“SOH CAH TOA”
The three basic ratios
Sine is Opposite over Hypotenuse, Cosine Adjacent over Hypotenuse, Tangent Opposite over Adjacent.
“Write the roots, then simplify”
The standard values
Write √0/2, √1/2, √2/2, √3/2, √4/2 for sine at 0, 30, 45, 60, 90 and simplify. Cosine is the same list backwards, so nothing needs memorising.
“Sine is odd, cosine is even”
Signs of negative angles
sin(−θ) = −sinθ but cos(−θ) = cosθ. Tangent, like sine, is odd.
Where marks go
The mistakes that cost the most.
- Writing sin(A + B) as sinA + sinB. None of the ratios distribute over addition.
- Forgetting the (−1)n in the general solution for sine.
- Using the cosine rule with the wrong angle. The angle must be the one opposite the side that stands alone.
- Cancelling sinθ from both sides of an equation, which silently loses the roots where sinθ = 0.
- Working in degrees inside a calculus question. Differentiation formulas assume radians.
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