1 Angle Measurement
Degrees
A full circle = 360°. A right angle = 90°. Degrees can be subdivided into minutes (') and seconds ("):
- 1° = 60 minutes (60')
- 1' = 60 seconds (60")
- Example: 2° 5' 30" means 2 degrees, 5 minutes, 30 seconds
Radians
A radian is defined so that an angle of 1 rad at the centre of a unit circle (r = 1) produces an arc of exactly length 1. Mathematicians prefer radians because they simplify many formulas.
🔑The golden bridge: π radians = 180°. Use this to convert in both directions.
Degrees → Radians
radians = degrees × (π ÷ 180)
Radians → Degrees
degrees = radians × (180 ÷ π)
Example
300° × (π ÷ 180) = 5π/3 rad
Common Angle Conversions
0°30°45°60°90°180°
0
π/6
π/4
π/3
π/2
π
| Degrees | Radians | Degrees | Radians |
|---|---|---|---|
| 120° | 2π/3 | 240° | 4π/3 |
| 135° | 3π/4 | 270° | 3π/2 |
| 150° | 5π/6 | 360° | 2π |
2 The Unit Circle
A circle with radius = 1 centred at the origin. For any angle θ, the point on the circle is (cos θ, sin θ).
💡Memory trick: the x-coordinate IS the cosine and the y-coordinate IS the sine. Always.
The CAST Rule — Signs by Quadrant
All Students Take Calculus — tells you which functions are positive in each quadrant.
Q II · 90° – 180°
sin +, cos −, tan −
→ Sine is positive
Q I · 0° – 90°
sin +, cos +, tan +
→ All positive
Q III · 180° – 270°
sin −, cos −, tan +
→ Tangent is positive
Q IV · 270° – 360°
sin −, cos +, tan −
→ Cosine is positive
Key Unit Circle Values
| Angle (°) | Angle (rad) | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 = √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 120° | 2π/3 | √3/2 | −1/2 | −√3 |
| 135° | 3π/4 | √2/2 | −√2/2 | −1 |
| 150° | 5π/6 | 1/2 | −√3/2 | −1/√3 |
| 180° | π | 0 | −1 | 0 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 360° | 2π | 0 | 1 | 0 |
Reference Angles
The reference angle is the acute angle (< 90°) between the terminal arm and the x-axis. Use the reference angle to find trig values in any quadrant.
| Quadrant | Reference angle formula | Example |
|---|---|---|
| Q I | ref = θ | 150° → not Q1 |
| Q II | ref = 180° − θ | 150° → ref = 30° |
| Q III | ref = θ − 180° | 210° → ref = 30° |
| Q IV | ref = 360° − θ | 330° → ref = 30° |
Arc Length
Arc length formula
s = r × θ (θ must be in radians)
⚠️This formula ONLY works when θ is in radians. Convert from degrees first if needed.
3 The Six Trigonometric Functions
Right Triangle Definitions (SOH-CAH-TOA)
💡SOH-CAH-TOA: Sin = Opposite/Hypotenuse · Cos = Adjacent/Hypotenuse · Tan = Opposite/Adjacent
| Function | Right triangle | Unit circle | Reciprocal of |
|---|---|---|---|
| sin(θ) | opposite / hypotenuse | y-coordinate | csc |
| cos(θ) | adjacent / hypotenuse | x-coordinate | sec |
| tan(θ) | opposite / adjacent | sin θ / cos θ | cot |
| csc(θ) | hypotenuse / opposite | 1/y | sin |
| sec(θ) | hypotenuse / adjacent | 1/x | cos |
| cot(θ) | adjacent / opposite | cos θ / sin θ | tan |
Domains and Ranges
| Function | Domain | Range |
|---|---|---|
| sin(θ) | All real numbers | −1 ≤ y ≤ 1 |
| cos(θ) | All real numbers | −1 ≤ y ≤ 1 |
| tan(θ) | θ ≠ π/2 + nπ (n integer) | All real numbers |
| csc(θ) | θ ≠ nπ (n integer) | y ≤ −1 or y ≥ 1 |
| sec(θ) | θ ≠ π/2 + nπ (n integer) | y ≤ −1 or y ≥ 1 |
| cot(θ) | θ ≠ nπ (n integer) | All real numbers |
4 The Sinusoidal Function
📐f(x) = a · sin( b(x − h) ) + k
What Each Parameter Does
| Parameter | Name | Effect | Formula |
|---|---|---|---|
| a | Amplitude | Vertical stretch. If a < 0, reflects over x-axis. | Amplitude = |a| |
| b | Frequency factor | Horizontal stretch/compression. | Period = 2π / |b| |
| h | Phase shift | Horizontal shift. Positive h = shift RIGHT. | Shift = h |
| k | Vertical shift | Moves entire wave up or down. | Midline at y = k |
Key Characteristics
Amplitude
|a| — half the distance from min to max
Period
2π / |b| — length of one complete wave
Frequency
|b| / (2π) — number of full cycles in 2π
Maximum value
k + |a|
Minimum value
k − |a|
Worked Example
✏️
y = 3 sin(0.5(x − π)) + 2
a = 3 → Amplitude = 3
b = 0.5 → Period = 2π / 0.5 = 4π
h = π → Phase shift = π to the right
k = 2 → Vertical shift = 2 up (midline at y = 2)
Maximum = 2 + 3 = 5 · Minimum = 2 − 3 = −1
a = 3 → Amplitude = 3
b = 0.5 → Period = 2π / 0.5 = 4π
h = π → Phase shift = π to the right
k = 2 → Vertical shift = 2 up (midline at y = 2)
Maximum = 2 + 3 = 5 · Minimum = 2 − 3 = −1
Periodicity & Symmetry
- sin(θ + 2πn) = sin(θ) for any integer n
- cos(θ + 2πn) = cos(θ) for any integer n
- tan(θ + πn) = tan(θ) for any integer n
- sin(−θ) = −sin(θ) [odd function — symmetric about origin]
- cos(−θ) = cos(θ) [even function — symmetric about y-axis]
5 Trigonometric Identities
Reciprocal & Quotient Identities
| Identity | Identity |
|---|---|
| csc θ = 1 / sin θ | sin θ = 1 / csc θ |
| sec θ = 1 / cos θ | cos θ = 1 / sec θ |
| cot θ = 1 / tan θ | tan θ = sin θ / cos θ |
| cot θ = cos θ / sin θ | tan θ = 1 / cot θ |
Pythagorean Identities
📐All three come from a² + b² = c² applied to the unit circle where c = 1.
Primary (memorise this)
sin²(θ) + cos²(θ) = 1
Divide by cos²(θ)
tan²(θ) + 1 = sec²(θ)
Divide by sin²(θ)
1 + cot²(θ) = csc²(θ)
Useful rearrangements:
- sin²θ = 1 − cos²θ = (1 − cos θ)(1 + cos θ)
- cos²θ = 1 − sin²θ = (1 − sin θ)(1 + sin θ)
- tan²θ = sec²θ − 1
- cot²θ = csc²θ − 1
Double Angle Formulas
sin(2θ)
2 sin(θ) cos(θ)
cos(2θ) — three forms
cos²(θ) − sin²(θ) = 2cos²(θ) − 1 = 1 − 2sin²(θ)
tan(2θ)
2 tan(θ) / (1 − tan²(θ))
Sum and Difference Formulas
sin(α ± β)
sin α cos β ± cos α sin β
cos(α ± β)
cos α cos β ∓ sin α sin β
tan(α ± β)
(tan α ± tan β) / (1 ∓ tan α tan β)
Proving Identities — Strategy
🚫Never move terms across the = sign. That assumes what you're trying to prove. Work on ONE side only.
- Always work on the more complicated side
- Convert everything to sin and cos when stuck
- Look for Pythagorean identity substitutions
- Try factoring — especially difference of squares: (1 − cos θ)(1 + cos θ) = sin²θ
- Multiply top and bottom by a conjugate if needed
6 Inverse Trigonometric Functions
Inverse trig functions give you an angle when you know a ratio. They undo the trig function. Also written arcsin, arccos, arctan.
| Function | Also written | Domain (input) | Range (output) |
|---|---|---|---|
| sin⁻¹(x) | arcsin(x) | −1 ≤ x ≤ 1 | −π/2 ≤ y ≤ π/2 (−90° to 90°) |
| cos⁻¹(x) | arccos(x) | −1 ≤ x ≤ 1 | 0 ≤ y ≤ π (0° to 180°) |
| tan⁻¹(x) | arctan(x) | all real numbers | −π/2 < y < π/2 (−90° to 90°) |
⚠️
Watch out for multiple solutions!
sin⁻¹ gives only ONE angle (in its restricted range). But the equation sin θ = 0.5 has two solutions in [0°, 360°]: 30° and 150°. Always check the CAST rule for the second solution.
sin⁻¹ gives only ONE angle (in its restricted range). But the equation sin θ = 0.5 has two solutions in [0°, 360°]: 30° and 150°. Always check the CAST rule for the second solution.
Cancellation (careful!)
sin(sin⁻¹(x)) = x always
Restricted range
sin⁻¹(sin θ) = θ only if θ is in [−π/2, π/2]
7 Law of Sines & Law of Cosines
Used for non-right triangles. Label the triangle: sides a, b, c opposite to angles A, B, C.
Law of Sines
Formula
sin(A) / a = sin(B) / b = sin(C) / c
Use when you know: 2 angles + 1 side (AAS or ASA) or 2 sides + an opposite angle (SSA).
Law of Cosines
Finding side a
a² = b² + c² − 2bc · cos(A)
Finding side b
b² = a² + c² − 2ac · cos(B)
Finding side c
c² = a² + b² − 2ab · cos(C)
Use when you know: 3 sides (SSS) or 2 sides + included angle (SAS).
💡
Which law to use?
Know 2 angles → Law of Sines is easier.
Know all 3 sides, or 2 sides with the angle between them → Law of Cosines.
Note: Law of Cosines reduces to the Pythagorean theorem when C = 90°.
Know 2 angles → Law of Sines is easier.
Know all 3 sides, or 2 sides with the angle between them → Law of Cosines.
Note: Law of Cosines reduces to the Pythagorean theorem when C = 90°.
8 Common Mistakes to Avoid
| Mistake | What to do instead |
|---|---|
| Forgetting the CAST rule | Always check the quadrant — sin is negative in Q3 and Q4. |
| Mixing radians and degrees | Check your calculator mode (DEG or RAD) before computing. |
| Period = b, not 2π/b | Period = 2π / |b|. Frequency = |b| / (2π). |
| Phase shift sign error | In a·sin(b(x − h)) + k, the shift is +h (right when h > 0). |
| Proving identities across = | Work on one side only — never add/subtract across the equals sign. |
| Only one solution from inverse trig | sin⁻¹ gives one value. Use CAST to find the second solution in [0°, 360°]. |
| Arc length formula in degrees | s = r·θ requires θ in radians — always convert first. |