Ch 4 · E6.3–E6.4 Exact Trigonometric Values and Trigonometric Functions · IGCSE Mathematics 0580
Topic 4 of 4 in Trigonometry — IGCSE Mathematics 0580 — 38 lessons.
Introduction
Sine, cosine and tangent are defined for any angle, not just those in right-angled triangles. Their graphs are repeating waves, and they show why an equation like sin x = 0.5 has two solutions between 0^° and 360^°. On the non-calculator paper you also need a few exact values. (Syllabus sections E6.3 and E6.4, IGCSE 0580 Extended only.)
Key ideas — Exact values
They come from half an equilateral triangle (sides 1, 2, sqrt3) and a right isosceles triangle (sides 1, 1, sqrt2).
🧑🏫 Teacher's explanation
Think of trigonometry as the study of ratios in right-angled triangles. Instead of using a calculator to get a decimal like 0.5, we want the 'exact' fraction or surd that represents the ratio perfectly. For example, sin 30^° is exactly 1/2 because in a 30-60-90 triangle, the side opposite the 30^° angle is always half the length of the hypotenuse.\n\nImagine you are looking at a Ferris wheel. As it turns, your height above the ground follows a wave pattern. The sin x function tracks this height, starting at 0, going up to 1, back to 0, down to -1, and back to 0. This periodic 'wave' behavior is the essence of trigonometric graphs. Knowing exact values allows you to plot these waves precisely without relying on decimal approximations that might lead to rounding errors in complex engineering or physics problems.
📖 Definitions
🔍 Worked example — Easy
Evaluate sin 60^° + cos 60^°.
🔍 Worked example — Standard
Solve cos x = -1/2 for 0^° ≤ x ≤ 360^°.
🔍 Worked example — Exam-style
Sketch the graph of y = tan x for 0^° ≤ x ≤ 360^°, clearly marking the vertical asymptotes.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — The graphs from 0° to 360°
🧑🏫 Teacher's explanation
Think of trigonometric functions as a description of repetitive motion, like a Ferris wheel spinning around a center point. The sine function tracks your vertical height: you start at the middle (0), rise to the top (1), drop back to the middle (0), descend to the bottom (-1), and return to the middle (0).
📖 Definitions
🔍 Worked example — Easy
Find the exact value of sin 60^° + cos 60^°.
🔍 Worked example — Standard
Solve cos x = -0.5 for 0^° ≤ x ≤ 360^°.
🔍 Worked example — Exam-style
Solve 2sin x + 1 = 0 for 0^° ≤ x ≤ 360^°.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Solving equations
Find the calculator value, then use the symmetry of the graph for the second solution:
🧑🏫 Teacher's explanation
Solving trigonometric equations is like finding the intersection between a horizontal line and a wavy curve. When you use your calculator to solve sin x = k, the calculator only gives you one single value (the principal value), but because trigonometric graphs are periodic and symmetrical, there is almost always a second angle within the 0^° ≤ x ≤ 360^° range that satisfies the equation.
📖 Definitions
🔍 Worked example — Easy
Solve sin x = 0.5 for 0^° ≤ x ≤ 360^°.
🔍 Worked example — Standard
Solve 2cos x + 1 = 0 for 0^° ≤ x ≤ 360^°.
🔍 Worked example — Exam-style
Solve tan x = -1.5 for 0^° ≤ x ≤ 360^°.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
Solve sin x = sqrt3/2: x = 60^° and 180^° - 60^° = 120^°.
🔍 Worked example — Easy
Find the exact values of x for 0^° ≤ x ≤ 360^° where sin x = 1/2.
🔍 Worked example — Standard
Solve 2cos x + 1 = 0 for 0^° ≤ x ≤ 360^°.
🔍 Worked example — Exam-style
Solve tan x = -√(3) for 0^° ≤ x ≤ 360^°.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) sqrt2/2 + tfrac12 = sqrt2 + 1/2. (2) 66.4^° and 293.6^°. (3) 53.1^° and 233.1^°. (4) sin x = -0.75: calculator -48.6^°; solutions 228.6^° and 311.4^°.