Study sheet: Trigonometric Functions

📋 Course Outline

1. Angles and Their Measurement
2. Degree and Radian Conversion
3. Arc Length and Circular Motion
4. Unit-Circle Trigonometric Functions
5. Signs Domains and Ranges
6. Periodicity and Function Behaviour
7. Sum and Difference Identities
8. Multiple-Angle and Product Identities

📖 1. Angles and Their Measurement

🔑 Key Concepts & Definitions

• Angle : the measure of rotation of a ray about its initial point, with the original ray called the initial side, the rotated ray called the terminal side, and the rotation point called the vertex

★ Must-know

📌 An anticlockwise rotation produces a positive angle, whereas a clockwise rotation produces a negative angle.

Further detail

• One degree is one three-hundred-and-sixtieth of a revolution, with 1° = 60′ and 1′ = 60″.

💡 Memory Hook

Anticlockwise rotation is positive, whereas clockwise rotation is negative.

📖 2. Degree and Radian Conversion

🔑 Key Concepts & Definitions

• Radian : the angle subtended at the centre of a unit circle by an arc of length one unit

★ Must-know

📐 Formula — The relation between degree and radian measures is $180^\circ=\pi\text{ radians}$.

📐 Formula — Radian measure equals $\frac{\pi}{180}\times\text{degree measure}$, and degree measure equals $\frac{180}{\pi}\times\text{radian measure}$.

Further detail

• The common angle conversions are 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2, and 360° = 2π.

💡 Memory Hook

Degree → radian: multiply by π/180; radian → degree: multiply by 180/π.

📖 3. Arc Length and Circular Motion

★ Must-know

📐 Formula — If an arc of length l in a circle of radius r subtends a central angle θ radians, then $l=r\theta$ and $\theta=\frac{l}{r}$.

Further detail

• The distance travelled by the tip of a rotating hand is found by expressing its rotation in radians and applying the arc-length relation.

• If two equal-length arcs subtend angles 65° and 110° at the centres of two circles, the radii are in the ratio 22:13.

💡 Memory Hook

Central angle and radius determine arc length through l = rθ.

📖 4. Unit-Circle Trigonometric Functions

🔑 Key Concepts & Definitions

• Sine and Cosine : For a point P(a,b) on the unit circle reached by an angle x, cos x = a and sin x = b.
• Quadrantal Angles : Quadrantal angles are integral multiples of π/2, including 0, π/2, π, 3π/2, and 2π.

📝 Essential Points

📐 Formula — For every real x, the fundamental identity is $\sin^2x+\cos^2x=1$.

• The values at 0, π/2, π, 3π/2, and 2π are respectively sin x = 0, 1, 0, −1, 0 and cos x = 1, 0, −1, 0, 1.

💡 Memory Hook

A point (cos x, sin x) moves around the unit circle.

📖 5. Signs Domains and Ranges

🔑 Key Concepts & Definitions

• Reciprocal Functions : The reciprocal trigonometric functions are cosec x = 1/sin x, sec x = 1/cos x, cot x = cos x/sin x, and tan x = sin x/cos x wherever the denominators are nonzero.

★ Must-know

• Sine and cosine are positive in quadrant I; sine is positive in quadrant II; tangent is positive in quadrant III; and cosine is positive in quadrant IV.

📐 Formula — The reciprocal identities are $1+\tan^2x=\sec^2x$ and $1+\cot^2x=\cosec^2x$.

Further detail

📐 Formula — For all real x, sine and cosine satisfy $-1\leq\sin x\leq1$ and $-1\leq\cos x\leq1$.

💡 Memory Hook

ASTC: All, Sine, Tangent, Cosine are positive in quadrants I, II, III, IV respectively.

📖 6. Periodicity and Function Behaviour

★ Must-know

📌 Sine and cosine are defined for every real number, while cosecant and cotangent exclude x = nπ and secant and tangent exclude x = (2n+1)π/2, where n is an integer.

• The ranges of sine and cosine are [−1,1], the ranges of tangent and cotangent are all real numbers, and the ranges of secant and cosecant are y ≤ −1 or y ≥ 1.

📌 Sine, cosine, secant, and cosecant repeat after 2π, whereas tangent and cotangent repeat after π.

Further detail

• The even-odd relations are sin(−x) = −sin x and cos(−x) = cos x.

💡 Memory Hook

Sine and cosine repeat every 2π, whereas tangent and cotangent repeat every π.

📖 7. Sum and Difference Identities

★ Must-know

📐 Formula — The addition and subtraction identities are $\sin(x+y)=\sin x\cos y+\cos x\sin y$ and $\sin(x-y)=\sin x\cos y-\cos x\sin y$.

📐 Formula — The cosine identities are $\cos(x+y)=\cos x\cos y-\sin x\sin y$ and $\cos(x-y)=\cos x\cos y+\sin x\sin y$.

📐 Formula — When the expressions are defined, $\tan(x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y}$ and $\tan(x-y)=\frac{\tan x-\tan y}{1+\tan x\tan y}$.

Further detail

📐 Formula — The complementary-angle identities are $\cos(\pi/2-x)=\sin x$ and $\sin(\pi/2-x)=\cos x$.

💡 Memory Hook

Cosine keeps the minus sign for a sum, whereas sine keeps the plus sign.

📖 8. Multiple-Angle and Product Identities

★ Must-know

📐 Formula — The double-angle identities are $\sin2x=2\sin x\cos x$ and $\cos2x=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x$.

📐 Formula — The tangent double-angle identity is $\tan2x=\frac{2\tan x}{1-\tan^2x}$ when defined.

📐 Formula — The triple-angle identities are $\sin3x=3\sin x-4\sin^3x$ and $\cos3x=4\cos^3x-3\cos x$.

Further detail

📐 Formula — The product-to-sum identities include $2\cos x\cos y=\cos(x+y)+\cos(x-y)$ and $2\sin x\cos y=\sin(x+y)+\sin(x-y)$.

📐 Formula — The sum-to-product identities include $\cos x+\cos y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}$ and $\sin x+\sin y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}$.

💡 Memory Hook

Double angle → triple angle → sum-to-product transformations.

📊 Synthesis Tables

Domains and ranges of trigonometric functions

Function	Domain	Range
sin x	All real numbers	[−1,1]
cos x	All real numbers	[−1,1]
tan x	x ≠ (2n+1)π/2	All real numbers
cot x	x ≠ nπ	All real numbers
sec x	x ≠ (2n+1)π/2	y ≤ −1 or y ≥ 1
cosec x	x ≠ nπ	y ≤ −1 or y ≥ 1