Trigonometry — Sin, Cos, Tan, and Identities
The trig ratios for right triangles, the key identities, and the laws for solving any triangle.
The Right Triangle Ratios
For a right triangle with angle θ (not the right angle):
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Mnemonic: SOH CAH TOA
The reciprocals:
csc θ = 1/sin θ = hypotenuse / opposite
sec θ = 1/cos θ = hypotenuse / adjacent
cot θ = 1/tan θ = adjacent / opposite
And the quotient:
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
Solving Right Triangles
Given one side and one angle (or two sides), find everything else.
Right triangle, angle = 35°, hypotenuse = 10
opposite = 10 × sin(35°) ≈ 10 × 0.574 = 5.74
adjacent = 10 × cos(35°) ≈ 10 × 0.819 = 8.19
Inverse trig functions — find the angle from the ratio:
sin⁻¹(0.5) = 30°
cos⁻¹(√2/2) = 45°
tan⁻¹(1) = 45°
Also written arcsin, arccos, arctan. The output of arcsin is in [−90°, 90°]; arccos in [0°, 180°]; arctan in (−90°, 90°).
The Fundamental Identity and Derived Identities
Pythagorean identities (from cos²θ + sin²θ = 1):
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
cot²θ + 1 = csc²θ
Angle 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
tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
Double angle formulas (set B = A in addition formulas):
sin(2A) = 2 sin A cos A
cos(2A) = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1
tan(2A) = 2 tan A / (1 − tan²A)
Half angle formulas:
sin²A = (1 − cos 2A) / 2
cos²A = (1 + cos 2A) / 2
Product-to-sum and sum-to-product — useful in signal processing and integration:
sin A cos B = ½[sin(A+B) + sin(A−B)]
cos A cos B = ½[cos(A−B) + cos(A+B)]
sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2)
cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2)
Solving Non-Right Triangles
Right-triangle trig only handles right triangles. For any triangle with sides a, b, c and opposite angles A, B, C:
The Sine Rule
a/sin A = b/sin B = c/sin C
Use when you know: two angles and a side (AAS or ASA), or two sides and a non-included angle (SSA — watch for ambiguous case).
Triangle with A = 40°, B = 75°, a = 8
C = 180° − 40° − 75° = 65°
b = 8 × sin(75°)/sin(40°) ≈ 8 × 0.966/0.643 ≈ 12.0
The Cosine Rule
a² = b² + c² − 2bc cos A
b² = a² + c² − 2ac cos B
c² = a² + b² − 2ab cos C
Use when you know: two sides and the included angle (SAS), or all three sides (SSS).
Rearranged for finding an angle:
cos A = (b² + c² − a²) / (2bc)
Note: when A = 90°, cos A = 0, and the formula reduces to Pythagoras. The cosine rule is the generalisation of Pythagoras.
Area of a Triangle via Trig
When you know two sides and the included angle:
Area = ½ ab sin C
More general than base × height — you don’t need the height directly.
Graphs of Trig Functions
sin(x):
- Period: 2π
- Range: [−1, 1]
- Starts at 0, rises to 1 at π/2, back to 0 at π, down to −1 at 3π/2, back to 0 at 2π
cos(x):
- Period: 2π
- Range: [−1, 1]
- Starts at 1, same shape as sin but shifted left by π/2 (cos(x) = sin(x + π/2))
tan(x):
- Period: π
- Range: (−∞, ∞)
- Vertical asymptotes at x = π/2 + nπ (where cos = 0)
Transformations of trig functions
y = A sin(Bx + C) + D
- A — amplitude (peak height from midline)
- B — angular frequency; period = 2π/B
- C — phase shift (horizontal shift = −C/B)
- D — vertical shift (midline)
y = 3 sin(2x − π/4) + 1
Amplitude = 3, period = π, phase shift = π/8, midline = 1
Inverse Trig Functions — Ranges
Because trig functions aren’t one-to-one, their inverses have restricted domains:
| Function | Domain | Range |
|---|---|---|
| arcsin(x) | [−1, 1] | [−π/2, π/2] |
| arccos(x) | [−1, 1] | [0, π] |
| arctan(x) | (−∞, ∞) | (−π/2, π/2) |
arctan is especially useful in physics and programming — it converts (x, y) coordinates to angles. Most programming languages have atan2(y, x) which handles all four quadrants correctly.
Where Trig Actually Appears
Waves and oscillations: any periodic signal is a sum of sine waves (Fourier’s theorem). Sound, light, AC electricity, radio — all described with sin and cos.
Rotation and vectors: rotating a vector (x, y) by angle θ:
x' = x cos θ − y sin θ
y' = x sin θ + y cos θ
This is the rotation matrix. It’s pure trig.
Navigation and surveying: distances and bearings. The sine and cosine rules are used to calculate positions from angles — GPS, dead reckoning, triangulation.
Complex numbers: Euler’s formula ties it all together:
e^(iθ) = cos θ + i sin θ
Multiplication of complex numbers is rotation in the plane. Trig is the geometry of the complex numbers.