Trigonometric Ratios of Compound Angles
Addition Formulas
sin(A + B) = sin A cos B + cos A sin B
sin(A − B) = sin A cos B − cos A sin B
cos(A + B) = cos A cos B − sin A sin B
cos(A − B) = cos A cos B + sin A sin B
Tangent Formulas
tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}
tan(A − B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}
cot(A + B) = \frac{cot A cot B - 1}{cot B + cot A}
cot(A − B) = \frac{cot A cot B + 1}{cot B - cot A}
Key Identities
sin(A+B)sin(A−B) = sin²A − sin²B = cos²B − cos²A
cos(A+B)cos(A−B) = cos²A − sin²B = cos²B − sin²A
Multiple & Sub-multiple Angles
Double Angle
sin 2A = 2 sin A cos A = \frac{2 \tan A}{1 + \tan^2 A}
cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A = \frac{1 − \tan^2 A}{1 + \tan^2 A}
tan 2A = \frac{2 \tan A}{1 − \tan^2 A}
Triple Angle
sin 3A = 3 sin A − 4 sin³ A
cos 3A = 4 cos³ A − 3 cos A
tan 3A = \frac{3 tan A - tan³ A}{1 - 3 tan² A}
Half Angle
sin \frac{A}{2} = \sqrt{\frac{1 - cos A}{2}}, \quad cos \frac{A}{2} = \sqrt{\frac{1 + cos A}{2}}
Sum ↔ Product Transformations
Product → Sum
2 sin A cos B = sin(A+B) + sin(A−B)
2 cos A sin B = sin(A+B) − sin(A−B)
2 cos A cos B = cos(A+B) + cos(A−B)
2 sin A sin B = cos(A−B) − cos(A+B)
Sum → Product
sin C + sin D = 2 sin\left(\frac{C+D}{2}\right) cos\left(\frac{C-D}{2}\right)
sin C − sin D = 2 cos\left(\frac{C+D}{2}\right) sin\left(\frac{C-D}{2}\right)
cos C + cos D = 2 cos\left(\frac{C+D}{2}\right) cos\left(\frac{C-D}{2}\right)
cos C − cos D = -2 sin\left(\frac{C+D}{2}\right) sin\left(\frac{C-D}{2}\right)
Inverse Trigonometric Functions
Principal Value Ranges
| Function | Domain | Range |
|---|---|---|
| sin⁻¹ x | [−1, 1] | [−π/2, π/2] |
| cos⁻¹ x | [−1, 1] | [0, π] |
| tan⁻¹ x | ℝ | (−π/2, π/2) |
Key Properties
sin⁻¹(−x) = −sin⁻¹ x
cos⁻¹(−x) = π − cos⁻¹ x
tan⁻¹(−x) = −tan⁻¹ x
Identities
sin⁻¹ x + cos⁻¹ x = π/2
tan⁻¹ x + cot⁻¹ x = π/2
sin⁻¹ x = tan⁻¹ \frac{x}{\sqrt{1-x^2}} = cos⁻¹ \sqrt{1-x^2}
General Solutions of Trigonometric Equations
| Equation | General Solution |
|---|---|
| sin θ = sin α | θ = nπ + (−1)ⁿ α |
| cos θ = cos α | θ = 2nπ ± α |
| tan θ = tan α | θ = nπ + α |
| sin θ = 1 | θ = π/2 + 2nπ |
| sin θ = −1 | θ = −π/2 + 2nπ = 3π/2 + 2nπ |
| cos θ = 1 | θ = 2nπ |
| cos θ = −1 | θ = (2n+1)π |
Solution of Triangle
Sine Rule
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R
Cosine Rule
cos A = \frac{b^2 + c^2 - a^2}{2bc}
cos B = \frac{c^2 + a^2 - b^2}{2ca}
cos C = \frac{a^2 + b^2 - c^2}{2ab}
Projection Formula
a = b cos C + c cos B
b = c cos A + a cos C
Half-Angle Formulas
\sin \frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{bc}}, \quad \cos \frac{A}{2} = \sqrt{\frac{s(s-a)}{bc}}, \quad \tan \frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}}
Height and Distance
Angle of Elevation
Angle between line of sight and horizontal when object is above observer.
Angle of Depression
Angle between line of sight and horizontal when object is below observer.