Chapter 3 — Trigonometric Functions

Fig 3.1 — High-Resolution Educational Poster: The Trigonometry Unit Circle, ASTC Quadrants & Sine/Cosine Graphs

An angle is the figure formed by two rays (sides) sharing a common endpoint (vertex). In trigonometry we measure angles in two main units: degrees (°) and radians (rad).

Degree measure: A full circle = 360°. Commonly used submultiples: 90° (right), 180° (straight), 270° (reflex), etc. Angles can be acute (0°<θ<90°), right (90°), obtuse (90°<θ<180°), straight (180°), reflex (180°<θ<360°).

Radian measure: One radian is the angle subtended at the center of a circle by an arc whose length equals the radius. For a circle of radius r, the measure θ (in radians) of an arc of length s is θ = s / r. A full circle = 2π rad, so π rad = 180°.

Why radians matter: Radians are the natural unit for analysis because many trigonometric limits and calculus formulas assume angles in radians (for example, lim_{θ→0} sin θ / θ = 1 only for θ in radians).

Standard position, sign and coterminal angles: An angle in standard position has its vertex at the origin and initial side along the positive x-axis; positive angles rotate counterclockwise, negative clockwise. Angles that differ by integer multiples of 2π (or 360°) are coterminal.

Unit circle and trig values: The unit circle (radius 1) links an angle θ to the point (cos θ, sin θ). Common angle measures in radians and coordinates: 0 (1,0), π/6 (√3/2,1/2), π/4 (√2/2,√2/2), π/3 (1/2,√3/2), π/2 (0,1), etc. The sign of sine and cosine depends on the quadrant.

Arc length and sector area: For radius r and angle θ (in radians): arc length s = rθ, sector area A = (1/2) r^2 θ. These are direct consequences of θ = s/r.

Angular and linear velocity: If an object moves around a circle of radius r with angular speed ω (rad/s), its linear speed v = rω. Frequency and period relate to ω by ω = 2πf = 2π / T.

Small-angle approximations (for θ in radians): sin θ ≈ θ, tan θ ≈ θ, cos θ ≈ 1 − θ^2/2 for |θ| small. These are useful in physics and engineering.

Minute and second: 1 degree = 60 minutes (′), 1 minute = 60 seconds (″). To convert deg-min-sec to decimal degrees: deg + min/60 + sec/3600.

This topic forms the foundation for trigonometric functions, graphs and calculus. Using radians simplifies formulas and connects geometry with analysis.

Definition (right-triangle context): For an acute angle θ in a right-angled triangle, the trigonometric ratios are defined as ratios of two sides of the triangle. Label the triangle so that for angle θ the sides are:

• Hypotenuse — the side opposite the right angle (longest side).
• Opposite — the side opposite angle θ.
• Adjacent — the side next to angle θ (but not the hypotenuse).

The primary six ratios are:

• sin θ = (Opposite) / (Hypotenuse)
• cos θ = (Adjacent) / (Hypotenuse)
• tan θ = (Opposite) / (Adjacent)
• cosec θ = 1 / sin θ = (Hypotenuse) / (Opposite)
• sec θ = 1 / cos θ = (Hypotenuse) / (Adjacent)
• cot θ = 1 / tan θ = (Adjacent) / (Opposite)

Mnemonic: SOH-CAH-TOA helps to remember sin = Opp/Hyp, cos = Adj/Hyp, tan = Opp/Adj.

Important properties (for 0° < θ < 90°):

• 0 < sin θ < 1 and 0 < cos θ < 1.
• tan θ ≥ 0 and increases to +∞ as θ → 90° (from left).
• Pythagorean relation: sin²θ + cos²θ = 1 (follows from Pythagoras on the triangle).
• Reciprocal relations: sec = 1/cos, cosec = 1/sin, cot = 1/tan.
• Quotient identities: tan θ = sin θ / cos θ, cot θ = cos θ / sin θ.
• Complementary-angle relations: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ.

How to use: Given two sides, compute the appropriate ratio. Given an angle and one side, use trigonometric ratios to find other sides: e.g., Opp = Hyp × sin θ.

Scope note: Right-triangle (side-based) definitions apply directly to acute angles (0° < θ < 90°). Later (unit-circle approach) these ratios extend to all real angles.

Basic idea: Place a circle of radius 1 (unit circle) centered at the origin of an xy-plane. For any angle θ measured from the positive x-axis (counterclockwise positive), the terminal side of θ meets the unit circle at a point P. The coordinates of P are (cosθ, sinθ). Thus cosine and sine are defined for every real angle by the x- and y-coordinates of P, so trig functions extend naturally to all angles (including negative and >360°).

Formal unit-circle definitions: For point P(x, y) on the unit circle corresponding to angle θ,

• sinθ = y
• cosθ = x
• tanθ = y / x, provided x ≠ 0
• secθ = 1 / cosθ, cosecθ = 1 / sinθ, cotθ = 1 / tanθ (where defined)

Radian and degree measure: Angles can be in degrees or radians. One full rotation is 360° = 2π radians.

Sign convention (quadrants): The signs of sin and cos depend on the quadrant of θ:

• Quadrant I (0 to π/2): sin+, cos+
• Quadrant II (π/2 to π): sin+, cos−
• Quadrant III (π to 3π/2): sin−, cos−
• Quadrant IV (3π/2 to 2π): sin−, cos+

Reference angle: For any angle θ, the reference angle α is the acute angle between the terminal side of θ and the x-axis. To find values of trig functions for θ, evaluate the trig function of α (a first-quadrant angle) and apply the sign appropriate to the quadrant of θ.

Pythagorean identity: From x^2 + y^2 = 1 on the unit circle, we get

• sin^2θ + cos^2θ = 1

Even-odd and periodic properties:

• sin(−θ) = −sinθ (odd), cos(−θ) = cosθ (even)
• tan(−θ) = −tanθ (odd)
• sin(θ + 2π) = sinθ, cos(θ + 2π) = cosθ, tan(θ + π) = tanθ (periodicity)

Using the unit circle to evaluate standard angles: For common angles 0, π/6, π/4, π/3, π/2 (and their multiples) the coordinates are known and give exact values (for example cosπ/3 = 1/2, sinπ/4 = √2/2).

Angles beyond one revolution and negative angles: For angles > 2π subtract multiples of 2π until in [0, 2π). For negative angles add multiples of 2π. Use reference angle and quadrant signs to get the final value.

Geometric interpretation: For a point moving on a unit circle, cosθ and sinθ are projections of the point on the x- and y-axes respectively. Tangent equals the slope of the radius line: tanθ = (y/x).

Practical tip: Always reduce the angle modulo 2π, find reference angle, determine quadrant sign, and then use the known value for the reference angle.

Overview
The sign of trigonometric functions (sin, cos, tan and their reciprocals) depends on the quadrant in which the angle lies. Use the unit circle: a point on the unit circle at angle θ (measured counterclockwise from positive x-axis) has coordinates (cos θ, sin θ). Thus the sign of cos θ is the sign of the x-coordinate, and the sign of sin θ is the sign of the y-coordinate. The sign of tan θ = sin θ / cos θ follows from the signs of sin and cos.

Quadrants and signs
There are four quadrants (I to IV):

• Quadrant I (0 < θ < π/2): x > 0, y > 0 → cos > 0, sin > 0, tan > 0
• Quadrant II (π/2 < θ < π): x < 0, y > 0 → cos < 0, sin > 0, tan < 0
• Quadrant III (π < θ < 3π/2): x < 0, y < 0 → cos < 0, sin < 0, tan > 0
• Quadrant IV (3π/2 < θ < 2π): x > 0, y < 0 → cos > 0, sin < 0, tan < 0

Mnemonic: "All Students Take Calculus" (ASTC) or "All Sin Tan Cos" depending on ordering. It means: Quadrant I: All positive; Quadrant II: Sine positive; Quadrant III: Tangent positive; Quadrant IV: Cosine positive.

Reference-angle method
If θ lies in a quadrant and α is its reference angle (the acute angle made with the x-axis), then the absolute values of trig functions equal those at α, but signs depend on the quadrant. Example: for θ in QII, sin θ = +sin α, cos θ = −cos α, tan θ = −tan α.

Reciprocals
The signs of sec, csc and cot follow the functions they reciprocate: sign(sec θ) = sign(cos θ), sign(csc θ) = sign(sin θ), sign(cot θ) = sign(tan θ).

Useful transformation/sign identities
(helpful for quickly determining signs when shifting angles):

• sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ
• sin(π − θ) = sin θ, cos(π − θ) = −cos θ, tan(π − θ) = −tan θ
• sin(π + θ) = −sin θ, cos(π + θ) = −cos θ, tan(π + θ) = tan θ
• sin(π/2 − θ) = cos θ, cos(π/2 − θ) = sin θ (co-function relations)

How to use
1) Locate the angle on the unit circle to decide the quadrant. 2) Use quadrant sign rules (or ASTC). 3) If needed, compute magnitude using the reference angle and then apply the sign.

Standard values of trigonometric functions are the exact values of sine, cosine, tangent and their reciprocals at commonly used angles on the unit circle. The most important standard angles (in degrees and radians) are 0° (0), 30° (π/6), 45° (π/4), 60° (π/3) and 90° (π/2). These values are obtained from two special right triangles: the isosceles right triangle (45°-45°-90°) and the 30°-60°-90° triangle (formed by splitting an equilateral triangle).

Key facts:

• Unit circle interpretation: for an angle θ, cosθ and sinθ are the x- and y-coordinates of the point on the unit circle at angle θ measured from the positive x-axis.
• Signs depend on the quadrant (ASTC rule: All, Sine, Tangent, Cosine positive in I, II, III, IV respectively).
• Reciprocal relations: sec = 1/cos, csc = 1/sin, cot = 1/tan. Tangent = sin/cos.

Standard-value table (degrees, radians and exact values):

Mnemonic for sine values 0, 30, 45, 60, 90: sin = (√0)/2, (√1)/2, (√2)/2, (√3)/2, (√4)/2. Cosine values are the same sequence in reverse. Tangent values follow from tan = sin/cos.

Extensions: values at 180°, 270°, 360° (and negative angles) follow from symmetry and quadrant signs: e.g., sin(180°)=0, cos(180°)=-1; sin(-30°)=-1/2, etc.

What are related angles? Related angles are angles that differ by simple amounts such as π/2, π, 2π or sign reversal. In trigonometry these lead to simple identities that let you express trig functions of one angle in terms of another (e.g. sin(π - x), cos(-x), tan(π/2 + x), etc.). These identities follow from the unit circle and symmetry.

Common types of related angles

• Negative angle: -x (reflection about x-axis on unit circle)
• Complementary: π/2 − x and π/2 + x (swap sine and cosine up to sign)
• Supplementary: π − x and π + x (change sign(s) depending on quadrant)
• Periodic shifts: x + 2π (full rotation, same values)

Why these identities hold (intuitive): On the unit circle the coordinates (cos x, sin x) represent cos and sin. Simple geometric symmetries—reflection about axes or rotation by π/2, π, 2π—give the relations. For example, the point for π - x is the reflection of the point for x across the y-axis, so sin(π - x) = sin x while cos(π - x) = −cos x.

Simple transformations of trig graphs

• Vertical scaling (amplitude): y = A sin x has amplitude |A|.
• Horizontal scaling (frequency): y = sin(kx) has period 2π/|k|.
• Phase (horizontal) shift: y = sin(x − c) shifts the graph right by c (y = sin(x + c) shifts left by c).
• Vertical shift: y = sin x + D moves the graph up by D.
• Reflection: y = −sin x reflects about the x-axis; y = sin(−x) is reflection about the y-axis (=even/odd behavior).

How related-angle identities connect with transformations: Using identities like sin(π/2 − x) = cos x lets you view cosine as a phase-shifted sine (cos x = sin(π/2 − x) = sin(x + π/2) up to sign), so trig functions are translates/reflections of one another.

Use in problem solving: Convert a trig expression of a related angle to a basic function (sin x, cos x, tan x) using identities; then apply algebra or graph transformations to analyze amplitude, period, phase and symmetry.

What they are: Fundamental trigonometric identities are basic equalities true for all angles (where both sides are defined). They form the algebraic backbone for simplifying trigonometric expressions and solving equations. The central set are the Pythagorean identities, together with reciprocal and quotient relations.

Derivation (unit circle / right triangle view):

• On the unit circle (radius = 1), a point on the circle at angle θ has coordinates (cos θ, sin θ). By the circle equation x² + y² = 1 we get
• sin²θ + cos²θ = 1 (the primary Pythagorean identity).
• Divide the primary identity by cos²θ (where cos θ ≠ 0) to obtain 1 + tan²θ = sec²θ. Divide by sin²θ (where sin θ ≠ 0) to get 1 + cot²θ = csc²θ.

Other basic relations:

• Reciprocal identities: sin θ = 1/csc θ, cos θ = 1/sec θ, tan θ = 1/cot θ (where defined).
• Quotient identities: tan θ = sin θ / cos θ, cot θ = cos θ / sin θ (where denominator ≠ 0).
• Even–odd identities: cos(−θ) = cos θ (even), sin(−θ) = −sin θ and tan(−θ) = −tan θ (odd).
• Co-function identities (relating complementary angles): sin(90°−θ) = cos θ, cos(90°−θ) = sin θ, tan(90°−θ) = cot θ, etc.

Why they matter: These identities let you convert between trig functions, reduce powers, verify or transform expressions, solve trig equations, and compute missing ratios from one known ratio and the quadrant.

Tips for using them: Always note the quadrant of the angle to determine signs; use the primary Pythagorean identity first, then divide to get other forms; when simplifying, try to express everything in terms of sin and cos (or tan and sec) to apply identities easily.

What are "Other Trigonometric Identities"? In addition to the basic relations (reciprocal, quotient and Pythagorean identities, and sum/difference formulae), this set includes identities used to convert sums into products (sum-to-product), products into sums (product-to-sum), double-/half-/triple-angle relations, the R-method (amplitude–phase form), and a few useful transformation formulas. These identities simplify algebraic manipulation, solving trig equations and modeling wave superposition.

• Why they matter: They help simplify expressions, evaluate products or sums of trig functions, solve equations, and analyze wave interference and oscillations.
• How they are obtained: Most are derived from the sum and difference formulas, e.g. sin(A±B) and cos(A±B).

Key derivations (sketches):

• Sum-to-product: add the two sum/difference formulas for sine or cosine. Example: sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2).
• Product-to-sum: from sum-to-product by algebraic rearrangement, e.g. sin A sin B = 1/2[cos(A−B) − cos(A+B)].
• R-method (amplitude–phase): any expression a sin x + b cos x = R sin(x+α), where R = √(a²+b²) and α = arctan(b/a) (choose quadrant for α). This writes a linear combination as a single sinusoid.

Common uses: simplifying integrals, proving identities, solving trig equations, analyzing beats in sound and alternating currents, and converting signals in engineering.

Overview. For trigonometric functions, the domain is the set of x-values for which the function is defined, the range is the set of possible function values, and periodicity describes repeating behaviour: f is periodic if there exists T > 0 such that f(x + T) = f(x) for all x in the domain. The smallest such positive T (if it exists) is the fundamental period.

Domain — how to determine. Check where the formula becomes undefined:

• Denominators: exclude x that make denominator 0 (e.g. tan x = sin x / cos x undefined when cos x = 0 ⇒ x ≠ π/2 + kπ).
• Square roots: require radicand ≥ 0.
• Inverse trig: argument limits (arcsin, arccos require input in [−1,1]; arctan accepts all real numbers).
• Logarithms or other operations: apply their domain conditions.

Range — typical basic results. For basic trig functions:

• sin x and cos x: range = [−1, 1].
• tan x and cot x: range = (−∞, ∞) where defined.
• sec x and csc x: range = (−∞, −1] ∪ [1, ∞) (they take values with absolute value ≥ 1).
• For a transformed function y = A·sin(Bx + C) + D, range = [D − |A|, D + |A|]. Same for cosine.

Periodicity — standard periods and transformed functions.

• sin x and cos x have fundamental period 2π: sin(x + 2π) = sin x.
• tan x and cot x have fundamental period π: tan(x + π) = tan x.
• If y = f(Bx) where f has period T0, the period of y is T = T0/|B| (so for y = sin(Bx + C), period = 2π/|B|; for y = tan(Bx + C), period = π/|B|).
• Phase shift: for y = f(Bx + C), shift = −C/B (to the right if positive).

Even/odd properties (useful for symmetry and domain/range reasoning). cos is even: cos(−x) = cos x; sin and tan are odd: sin(−x) = −sin x, tan(−x) = −tan x.

Inverse trig ranges (principal values).

• arcsin x: domain [−1,1], range [−π/2, π/2].
• arccos x: domain [−1,1], range [0, π].
• arctan x: domain (−∞, ∞), range (−π/2, π/2).

Practical tips. To find domain: list algebraic restrictions, solve them (e.g. cos x = 0 ⇒ x = π/2 + kπ). To find range of transformed sine/cosine use amplitude and vertical shift. To find period, identify multiplier B of x and use T = base_period/|B|.

Overview
Graphs of trigonometric functions show how sine, cosine, tangent (and their reciprocals) vary with the angle. The basic shapes repeat periodically and are transformed by amplitude, frequency (period), phase shift and vertical shift.

Basic functions

• y = sin x: continuous, smooth wave. Period = 2π, amplitude = 1, range = [−1, 1]. Key points: sin0 = 0, sin(π/2) = 1, sinπ = 0, sin(3π/2) = −1, sin2π = 0. Odd function: sin(−x) = −sin x.
• y = cos x: similar to sine but shifted. Cosine is even: cos(−x) = cos x. Period = 2π, amplitude = 1, range = [−1, 1]. Key points: cos0 = 1, cos(π/2) = 0, cosπ = −1, cos3π/2 = 0, cos2π = 1.
• y = tan x: ratio sinx/cosx. Period = π, no finite amplitude, range = (−∞, ∞). Vertical asymptotes where cos x = 0 (x = π/2 + nπ). Odd function: tan(−x) = −tan x. Passes through (0,0) and increases between asymptotes.

General transformed form
y = A sin(Bx + C) + D or y = A cos(Bx + C) + D. Effects:

• A = vertical stretch/compression and reflection. Amplitude = |A|.
• B = horizontal scaling. Period = 2π/|B| for sin and cos; period = π/|B| for tan.
• C = horizontal (phase) shift: shift = −C/B to the right if C>0 (depending on sign convention).
• D = vertical shift (midline y = D).

How to sketch

1. Identify A, B, C, D and compute amplitude, period, midline and phase shift.
2. For one period, mark equally spaced key x-values: for sin/cos divide period into four (quarter-period) intervals to find maxima/minima/zeros.
3. Plot midline y = D, then plot key points relative to the midline and connect smoothly for sin/cos.
4. For tan, find asymptotes at x = (π/2 + nπ − C)/B, and plot the curve between asymptotes passing through the midline point(s).

Properties to note
Zeros, maxima and minima occur at predictable x-values; sin and cos are bounded while tan is unbounded and has vertical asymptotes. Sine and cosine are phase-shifts of each other: cos x = sin(x + π/2).

What the topic is: Solving basic trigonometric equations means finding all angles x (or general values) that satisfy an equation involving sin x, cos x, tan x (or their multiples). Two common goals are (a) finding solutions in a given interval such as [0, 2π), and (b) writing the general solution (all solutions) using the periodicity of trig functions.

Key ideas / method:

• Reduce the equation to one basic trig function (sin, cos or tan). If needed use identities or algebraic manipulation (e.g., factorization, Pythagorean identities).
• Find the principal angle α in [0, π/2] (often α = arcsin|k|, α = arccos|k|, α = arctan|k|) so that the reference angle is known.
• Use the signs of the trig function in the four quadrants to place α in the correct quadrant(s) and write specific solutions in [0, 2π).
• Convert specific solutions to the general solution by adding the period (2π for sin and cos; π for tan): this gives all solutions.
• Check any restrictions (e.g., domain, division by zero) and any extraneous solutions if functions were transformed.

Common general solution forms (use integer n ∈ Z):

• sin x = k (|k| ≤ 1): x = α + 2nπ or x = π − α + 2nπ, where α = arcsin(k) in [−π/2, π/2] (equivalently x = nπ + (−1)^n α).
• cos x = k (|k| ≤ 1): x = ±α + 2nπ, where α = arccos(k) in [0, π].
• tan x = k: x = α + nπ, where α = arctan(k) in (−π/2, π/2).

Notes: Always verify |k| ≤ 1 for sin/cos. For equations with multiples (like sin(2x) or cos(x/3)) first solve for the inner angle (set y = 2x etc.) then divide the solutions by the multiplier to get x.

Strategy summary (step-by-step):

1. Simplify the equation to one trig function of a single angle: f(θ) = k.
2. Find the reference angle α = principal inverse value.
3. List solutions in [0, 2π) by using quadrant rules (ASTC: All, Sin, Tan, Cos positive in respective quadrants).
4. Write general solution by adding the function period (2π or π) to each basic solution.
5. If initial equation had a multiplied angle, divide the general solutions by that multiplier and include integer n.

What this topic covers
This topic shows how trigonometric functions model periodic phenomena and how to solve geometry and algebra problems using trigonometric reasoning. It combines understanding of identities, equations, graphs and geometric drawing to set up and solve real problems.

Applications

• Heights and distances: use right-triangle trig (sine, cosine, tangent) with angles of elevation/depression to find unknown heights or distances.
• Periodic modelling: represent oscillations (sound, tides, daylight hours, seasonal temperature) with functions y = A sin(Bx + C) + D.
• Navigation and bearings: use trig to find separation between objects given angles/bearings.
• Engineering and waves: sinusoidal functions model AC voltages, mechanical vibrations, and wave motion.
• Architecture and construction: calculating slopes, inclines, and component lengths using trig relations.

Problem-solving strategies

• Read and draw: sketch the situation (right triangles, bearings, unit circle). Label angles and sides clearly.
• Choose the right ratio: identify which trig ratio (sin, cos, tan) relates given quantities.
• Use identities to simplify: apply Pythagorean, sum/difference, double-angle, or product–sum identities to transform expressions or equations.
• Transform linear combinations: convert a sin x + b cos x into R sin(x + α) or R cos(x − α) to find amplitude, phase and solve equations easily.
• Consider domain/range and principal values: check permitted values for inverse trig and restrict solutions to the required interval; then give general solutions using periodicity (add 2π or π as appropriate).
• Use graphs: sketch or reason with graphs (amplitude, period, phase shift, vertical shift) to locate maxima/minima, zeros and asymptotes (for tan).
• Solve algebraic trig equations: reduce higher-degree trig equations using identities (e.g., convert sin^2 to 1 − cos^2), factor, and then apply basic equation solutions.
• Check units and approximations: ensure angles are in the correct unit (degrees/radians) and round final numerical answers appropriately.

How to approach a typical problem

1. Draw diagram and assign variables.
2. Write relations using trig ratios or identities.
3. Simplify and isolate the trig function (sin x, cos x, tan x).
4. Find principal solutions using inverse trig; then write general solution using periodicity.
5. Pick solutions that fit the required interval or physical constraints (positive lengths, feasible angles).