Trigonometry: Ratios and Graphs

What you'll learn

This revision guide covers all testable trigonometric ratios and graphs for CXC CSEC Additional Mathematics. You will master the six trigonometric ratios, evaluate exact values without a calculator, solve trigonometric equations, and sketch and transform trigonometric graphs. These skills are essential for Paper 1 (multiple choice) and Paper 2 (extended response) questions worth approximately 15-20% of your examination marks.

Key terms and definitions

Sine (sin θ) — the ratio of the opposite side to the hypotenuse in a right-angled triangle; also defined as the y-coordinate on the unit circle.

Cosine (cos θ) — the ratio of the adjacent side to the hypotenuse in a right-angled triangle; also defined as the x-coordinate on the unit circle.

Tangent (tan θ) — the ratio of the opposite side to the adjacent side in a right-angled triangle; equal to sin θ / cos θ.

Reciprocal ratios — cosecant (cosec θ = 1/sin θ), secant (sec θ = 1/cos θ), and cotangent (cot θ = 1/tan θ).

Amplitude — the maximum displacement from the central axis of a trigonometric graph; for y = a sin x, the amplitude is |a|.

Period — the horizontal length of one complete cycle of a trigonometric wave; for y = sin bx, the period is 360°/b or 2π/b radians.

Phase shift — the horizontal translation of a trigonometric graph from its standard position.

Pythagorean identities — fundamental trigonometric relationships derived from Pythagoras' theorem, particularly sin²θ + cos²θ = 1.

Core concepts

The six trigonometric ratios

For any angle θ in standard position (measured anticlockwise from the positive x-axis), six trigonometric ratios exist:

Primary ratios:

• sin θ = opposite/hypotenuse = y/r
• cos θ = adjacent/hypotenuse = x/r
• tan θ = opposite/adjacent = y/x

Secondary (reciprocal) ratios:

• cosec θ = 1/sin θ = r/y
• sec θ = 1/cos θ = r/x
• cot θ = 1/tan θ = x/y

Remember that tan θ = sin θ / cos θ and cot θ = cos θ / sin θ.

Exact values and special angles

You must memorize exact trigonometric values for angles 0°, 30°, 45°, 60°, and 90° (or 0, π/6, π/4, π/3, and π/2 radians). These appear frequently in CXC examinations.

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3 or √3/3
45°	1/√2 or √2/2	1/√2 or √2/2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

To derive these values, use:

• The equilateral triangle method (for 30° and 60°)
• The isosceles right triangle method (for 45°)

CAST diagram and signs in quadrants

The CAST diagram indicates which trigonometric ratios are positive in each quadrant:

• Quadrant I (0° to 90°): All ratios positive
• Quadrant II (90° to 180°): Sine (and cosec) positive only
• Quadrant III (180° to 270°): Tangent (and cot) positive only
• Quadrant IV (270° to 360°): Cosine (and sec) positive only

Starting from Quadrant IV and moving anticlockwise: Cosine, All, Sine, Tangent.

For example, sin 150° is positive (Quadrant II), while cos 150° is negative. Use reference angles to evaluate: sin 150° = sin 30° = 1/2, but cos 150° = -cos 30° = -√3/2.

Trigonometric identities

Fundamental identity: sin²θ + cos²θ = 1

Derived identities:

• 1 + tan²θ = sec²θ (divide the fundamental identity by cos²θ)
• 1 + cot²θ = cosec²θ (divide the fundamental identity by sin²θ)

Compound angle formulae:

• sin(A ± B) = sin A cos B ± cos 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 formulae:

• sin 2A = 2 sin A cos A
• cos 2A = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A
• tan 2A = 2 tan A/(1 - tan²A)

Graphs of sine, cosine, and tangent

y = sin x:

• Domain: all real numbers
• Range: -1 ≤ y ≤ 1
• Period: 360° (2π radians)
• Amplitude: 1
• Passes through origin (0, 0)
• Maximum at 90°, minimum at 270°

y = cos x:

• Domain: all real numbers
• Range: -1 ≤ y ≤ 1
• Period: 360° (2π radians)
• Amplitude: 1
• Starts at maximum (0, 1)
• Zero crossings at 90° and 270°

y = tan x:

• Domain: all real numbers except odd multiples of 90° (π/2 radians)
• Range: all real numbers
• Period: 180° (π radians)
• Vertical asymptotes at x = ..., -90°, 90°, 270°, ...
• Passes through origin (0, 0)
• Increases continuously between asymptotes

Transformations of trigonometric graphs

For y = a sin b(x - c) + d:

a (vertical stretch):

• Changes amplitude to |a|
• If a < 0, graph reflects in the x-axis
• Example: y = 3 sin x has amplitude 3, oscillating between -3 and 3

b (horizontal stretch):

• Changes period to 360°/b (or 2π/b)
• If b = 2, the period halves to 180°
• If b = 1/2, the period doubles to 720°

c (horizontal translation/phase shift):

• Shifts graph c units right if positive
• Shifts graph c units left if negative
• Example: y = sin(x - 30°) shifts right by 30°

d (vertical translation):

• Shifts entire graph d units up if positive
• Shifts entire graph d units down if negative
• Example: y = sin x + 2 oscillates between 1 and 3

Consider a coconut palm swaying in Caribbean trade winds. If its displacement follows y = 0.5 sin 2(t - 5) + 1.2 metres at time t seconds, the amplitude is 0.5 m, period is 180 seconds, phase shift is 5 seconds right, and central position is 1.2 m from the base reference.

Solving trigonometric equations

General procedure:

1. Isolate the trigonometric ratio
2. Find the principal (reference) angle using inverse functions
3. Use CAST diagram to find all solutions in the specified interval
4. Apply the period to find additional solutions if required

Example approach for sin x = 0.5, 0° ≤ x ≤ 360°:

• Principal angle: x = 30° (Quadrant I)
• Sine is also positive in Quadrant II
• Second solution: x = 180° - 30° = 150°
• Solutions: x = 30°, 150°

For equations with compound angles:

• Let the compound expression equal a new variable
• Solve for that variable
• Back-substitute to find the original variable

Worked examples

Example 1: Exact values and identities

Question: Without using a calculator, find the exact value of: (a) sin 60° cos 30° + cos 60° sin 30° [3 marks] (b) Given that tan θ = 3/4 and θ is acute, find sec θ [3 marks]

Solution:

(a) Recognize the compound angle formula: sin(A + B) = sin A cos B + cos A sin B

sin 60° cos 30° + cos 60° sin 30° = sin(60° + 30°) = sin 90° = 1 ✓✓✓

Alternative: Direct substitution = (√3/2)(√3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1

(b) Use the identity 1 + tan²θ = sec²θ

1 + (3/4)² = sec²θ ✓ 1 + 9/16 = sec²θ 25/16 = sec²θ ✓ sec θ = 5/4 (positive since θ is acute) ✓

sec θ = 5/4 or 1.25

Example 2: Solving trigonometric equations

Question: Solve the equation 2 sin²x - 3 sin x + 1 = 0 for 0° ≤ x ≤ 360° [6 marks]

Solution:

This is a quadratic equation in sin x. Let y = sin x: 2y² - 3y + 1 = 0 ✓

Factorize: (2y - 1)(y - 1) = 0 ✓

Therefore: 2y - 1 = 0 or y - 1 = 0 y = 1/2 or y = 1 ✓

When sin x = 1/2: x = 30° (Quadrant I) ✓ x = 150° (Quadrant II, sine positive) ✓

When sin x = 1: x = 90° ✓

Solutions: x = 30°, 90°, 150°

Example 3: Graph transformations

Question: A fishing boat in Barbados waters bobs on waves modeled by the equation h = 2 sin(30t)° + 3, where h is height in metres above sea level and t is time in seconds.

(a) State the amplitude and period of the motion [2 marks] (b) What are the maximum and minimum heights of the boat? [2 marks] (c) Sketch the graph for 0 ≤ t ≤ 12 seconds [3 marks]

Solution:

(a) Amplitude = |a| = 2 metres ✓ Period = 360°/30 = 12 seconds ✓

(b) Maximum height = d + amplitude = 3 + 2 = 5 metres ✓ Minimum height = d - amplitude = 3 - 2 = 1 metre ✓

(c) Key features for sketch: ✓

• Starts at h = 3 when t = 0
• First maximum at t = 3 s, h = 5 m
• Returns to h = 3 at t = 6 s
• Minimum at t = 9 s, h = 1 m
• Completes cycle at t = 12 s, h = 3 m
• Smooth sinusoidal curve through these points ✓✓

Common mistakes and how to avoid them

• Confusing degrees and radians: Always check whether the question specifies degrees or radians. If no unit is given in equations, assume radians. Calculator mode must match the unit.

• Missing solutions in trigonometric equations: Remember that sine and cosine are positive in two quadrants each. Always use the CAST diagram to find all solutions within the specified range. Don't stop after finding the principal angle.

• Incorrect reciprocal ratios: sec θ is 1/cos θ, NOT 1/sin θ. Students frequently confuse reciprocal pairs. Write the definition before calculating.

• Sign errors with transformations: In y = sin(x - c), a positive c shifts RIGHT, not left (counterintuitive). Test with a known point to verify direction.

• Forgetting domain restrictions for tan x: Tangent is undefined at odd multiples of 90° (or π/2). When sketching or solving equations involving tan x, mark asymptotes clearly.

• Amplitude confusion with negative coefficients: For y = -3 sin x, the amplitude is 3 (always positive), but the graph is reflected. Maximum y-value is 0, minimum is -3 if no vertical shift exists.

Exam technique for "Trigonometry: Ratios and Graphs"

• "Find the exact value" command word: Do NOT use a calculator. Express answers with surds (√2, √3) and fractions. Marks deducted for decimal approximations like 0.707 instead of √2/2.

• "Solve" or "Find all values" questions: Identify the specified interval carefully (e.g., 0° ≤ x ≤ 360° or -π ≤ x ≤ π). Find ALL solutions in that range. Method marks available even if arithmetic is incorrect—always show the CAST diagram approach.

• "Sketch the graph" command: Mark key features: amplitude, period, intercepts, maximum/minimum points, and asymptotes (for tan). Label axes with scales. Even rough sketches earn marks if features are correctly positioned.

• Multi-step identity proofs: Work on ONE side of the equation only (usually the more complex side). Show each transformation step clearly. Use standard identities explicitly rather than jumping steps—each line earns potential marks.

Quick revision summary

Master the six trigonometric ratios and their reciprocals. Memorize exact values for 0°, 30°, 45°, 60°, 90°. Use CAST to determine signs in each quadrant and find all solutions to equations. The three fundamental identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ. For graphs, identify how coefficients transform amplitude (vertical stretch), period (horizontal stretch), phase shift (horizontal translation), and vertical translation. Practice sketching all three basic graphs and their transformations, marking key features clearly.