Sin = Opposite/Hypotenuse. Cos = Adjacent/Hypotenuse. Tan = Opposite/Adjacent. Every right-triangle trig problem starts here.
S
Sin = Opposite ÷ Hypotenuse
C
Cos = Adjacent ÷ Hypotenuse
T
Tan = Opposite ÷ Adjacent
🔢 Trigonometry
"All Students Take Calculus"
ASTC — Signs by Quadrant
Which trig functions are positive in each quadrant
Quadrant I: All positive. Quadrant II: Sine positive. Quadrant III: Tangent positive. Quadrant IV: Cosine positive.
A
All — Quadrant I, all functions positive
S
Students — Quadrant II, Sine positive
T
Take — Quadrant III, Tangent positive
C
Calculus — Quadrant IV, Cosine positive
🔢 Trigonometry
sin²θ + cos²θ = 1
Pythagorean Identity
The most fundamental trig identity — memorize it cold
Every other Pythagorean identity derives from this. Divide by cos²θ to get tan²θ+1=sec²θ. Divide by sin²θ to get 1+cot²θ=csc²θ.
🔢 Trigonometry
"0, 30, 45, 60, 90 — know your squad"
Special Angle Values
The 5 angles you must know on the unit circle
sin(0)=0, sin(30)=½, sin(45)=√2/2, sin(60)=√3/2, sin(90)=1. Cos goes in reverse. Learn these 5 and you can handle any unit circle question.
🔢 Trigonometry
Tan = Sin / Cos
Tangent Identity
Tangent is just sine divided by cosine — always
tan(θ) = sin(θ)/cos(θ). This single identity lets you convert any tangent expression into sine and cosine, which are almost always easier to work with.
Unit Circle
Unit circle: radius=1. Know exact values at 0, 30, 45, 60, 90° and their radian equivalents.
Unit Circle
The foundation of trigonometry — coordinates give sin and cos
On the unit circle, the coordinates of any point are (cos θ, sin θ). At 0°: (1,0). At 90°: (0,1). At 180°: (-1,0). At 270°: (0,-1). Memorize the 30-60-90 and 45-45-90 values. Everything else follows from symmetry.
Converting between the two angle measurement systems
180° = π rad. 90° = π/2. 45° = π/4. 60° = π/3. 30° = π/6. 360° = 2π. Memory: multiply by π/180 to go to radians (π is the destination). Multiply by 180/π to go to degrees.
Law of Sines
Law of Sines: a/sinA = b/sinB = c/sinC — use when you have angle-side pairs
Law of Sines
For non-right triangles — use when you have an angle and its opposite side
Use when you know: AAS, ASA, or SSA (ambiguous case — may have 0, 1, or 2 solutions). a, b, c are sides opposite angles A, B, C respectively. Cross-multiply to solve for unknown sides or angles.
Law of Cosines
Law of Cosines: c² = a² + b² - 2ab·cosC — use when you have two sides and included angle
Law of Cosines
For non-right triangles — use when the Law of Sines won't work
Use when you know: SAS or SSS. When C = 90°, cosC = 0, reduces to Pythagorean theorem. For finding angles: cosC = (a² + b² - c²)/(2ab). Law of Cosines is the generalized Pythagorean theorem.
Start with sin²θ + cos²θ = 1. Divide every term by cos²θ: tan²θ + 1 = sec²θ. Divide every term by sin²θ: 1 + cot²θ = csc²θ. These three are derived from ONE identity. On exams, manipulate whichever form has the functions you need.
Graphing Trig Functions
Graphing trig: y = A·sin(Bx + C) + D. Amplitude=|A|. Period=2π/B. Phase shift=-C/B. Vertical shift=D.
Graphing Trig Functions
Four parameters that transform any sine or cosine graph
Amplitude |A|: distance from midline to peak. Period 2π/B: length of one complete cycle (B compresses or stretches horizontally). Phase shift -C/B: horizontal shift (positive C shifts left). Vertical shift D: moves the entire graph up or down.
A
Amplitude — height from midline
B
Period = 2π/B — horizontal stretch
C
Phase shift = -C/B — horizontal slide
D
Vertical shift — up or down
Inverse Trig Functions
Inverse trig: sin⁻¹(x) gives angle whose sine is x. Range: [-π/2, π/2] for sin⁻¹ and tan⁻¹, [0,π] for cos⁻¹.
Inverse Trig Functions
Undoing trig functions — and their restricted ranges
sin⁻¹(0.5) = 30° (π/6) — the angle whose sine equals 0.5. Restricted ranges avoid ambiguity (sine is not one-to-one over all reals). sin⁻¹ and tan⁻¹: output between -90° and 90°. cos⁻¹: output between 0° and 180°.