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IGCSE Maths · Trigonometry · Extended only

Exact Trig Values and Trigonometric Graphs Extended only

This entire note is Extended-only. Some angles give clean, exact trig values (fractions and surds) instead of messy decimals — worth memorising so you can skip the calculator entirely. Then we look at what sin, cos, and tan actually look like as graphs, and how to use that shape to find every solution to a trig equation, not just one.

1. Exact Values Table

x0°30°45°60°90°
sin x01/2√2/2√3/21
cos x1√3/2√2/21/20
tan x0√3/31√3undefined

2. Where These Values Come From — Two Special Triangles

The 45-45-90 triangle 1 1 √2 45° An isosceles right-angled triangle with both legs = 1. By Pythagoras, the hypotenuse = √(1²+1²) = √2.
sin 45° = 1/√2 = √2/2    cos 45° = 1/√2 = √2/2    tan 45° = 1/1 = 1
The 30-60-90 triangle √3 1 2 60° 30° Take an equilateral triangle with sides 2, and cut it in half. This gives a right-angled triangle with sides 1, √3, and 2 (hypotenuse).
sin 30° = 1/2,   cos 30° = √3/2,   tan 30° = 1/√3 = √3/3
sin 60° = √3/2,   cos 60° = 1/2,   tan 60° = √3
Worked example — using exact values Find the exact value of 4 sin 30° + 2 cos 60°.
4(1/2) + 2(1/2) = 2 + 1 = 3 (no calculator, no rounding — exact)
Worked example — exact side lengths A right-angled triangle has angle 30° and hypotenuse 8 cm. Find the exact opposite and adjacent sides.
Opposite = 8 × sin 30° = 8 × ½ = 4 cm (exact)
Adjacent = 8 × cos 30° = 8 × √3/2 = 4√3 cm (exact, left in surd form)
Try it — Exact Values
  1. Find the exact value of sin 60° − cos 30°. (Hint: what do you notice?)
  2. Find the exact value of 6 tan 45° + 2 sin 90°.
  3. A right triangle has angle 60° and hypotenuse 10 cm. Find the exact opposite side.
  4. A right triangle has angle 45° and adjacent side 6 cm. Find the exact hypotenuse.
Show answers
  1. √3/2 − √3/2 = 0 — they’re exactly equal, since sin 60° = cos 30° (complementary angles).
  2. 6(1) + 2(1) = 8
  3. 10 × √3/2 = 5√3 cm
  4. 6 ÷ cos 45° = 6 ÷ (√2/2) = 6√2 cm

3. The Graphs of y = sin x and y = cos x

Both graphs, 0° to 360° 1 −1 90 180 270 360 y=sin x y=cos x Both curves oscillate smoothly between −1 and 1, and repeat every 360° (their period). sin x starts at 0 and rises first; cos x starts at 1 (its maximum) and falls first — cos is really just sin shifted 90° to the left.
x=0°x=90°x=180°x=270°x=360°
sin x01 (max)0−1 (min)0
cos x1 (max)0−1 (min)01 (max)

4. The Graph of y = tan x

The tangent graph, 0° to 360° 90 270 360 Unlike sin and cos, tan x is undefined at 90° and 270° — the graph shoots off toward infinity there, shown as vertical asymptotes (dashed). Between the asymptotes, the curve rises steeply from left to right. The pattern repeats every 180° — tan has a shorter period than sin or cos.
Try it — Reading the Graphs
  1. At what value of x does y = sin x reach its maximum, for 0° ⩽ x ⩽ 360°?
  2. At what values of x is y = cos x equal to 0?
  3. At what values of x is y = tan x undefined?
  4. What is the period of y = tan x?
Show answers
  1. x = 90°
  2. x = 90° and x = 270°
  3. x = 90° and x = 270°
  4. 180°

5. Solving Trigonometric Equations

Your calculator only ever gives you one solution — but trig equations in the range 0° to 360° usually have two. Use the graph’s symmetry to find the second one.

Equation typeSecond solution (given calculator answer A)
sin x = value (positive)180° − A
cos x = value (negative)360° − A
tan x = valueA + 180°
Worked example — sine equation Solve sin x = 0.5 for 0° ⩽ x ⩽ 360°.
Calculator: sin⁻¹(0.5) = 30°
Second solution: 180 − 30 = 150°
Answer: x = 30° or x = 150°
Worked example — cosine equation Solve cos x = −0.5 for 0° ⩽ x ⩽ 360°.
Calculator: cos⁻¹(−0.5) = 120°
Second solution: 360 − 120 = 240°
Answer: x = 120° or x = 240°
Worked example — using an exact value Solve sin x = √3/2 for 0° ⩽ x ⩽ 360°.
From the exact values table: sin 60° = √3/2, so the calculator answer is 60°
Second solution: 180 − 60 = 120°
Answer: x = 60° or x = 120°
Try it — Solving Trig Equations
  1. Solve tan x = 1 for 0° ⩽ x ⩽ 360°.
  2. Solve sin x = 0.259 for 0° ⩽ x ⩽ 360° (give answers to 1 d.p.).
  3. Solve cos x = 0.5 for 0° ⩽ x ⩽ 360° (use the exact values table).
  4. Solve 2 sin x = 1.6 for 0° ⩽ x ⩽ 360° (1 d.p.).
Show answers
  1. tan⁻¹(1)=45°. Second: 45+180=225°. x=45° or 225°
  2. sin⁻¹(0.259)=15.0°. Second: 180−15=165°. x=15.0° or 165.0°
  3. cos 60°=0.5, so x=60°. Second: 360−60=300°. x=60° or 300°
  4. sin x=0.8. sin⁻¹(0.8)=53.1°. Second: 180−53.1=126.9°. x=53.1° or 126.9°
Common mistakes to watch for
  • Stopping after finding only one solution — always check whether a second one exists in the range.
  • Using the wrong “second solution” rule — sine uses 180−A, cosine uses 360−A, tangent uses A+180°; mixing these up gives a wrong second answer.
  • Forgetting tan x has period 180°, not 360° like sin and cos — this affects how many solutions appear in a given range.
  • Writing an exact value as a rounded decimal when the question asks for an exact answer — always leave surds as surds (e.g. √3/2, not 0.866).
  • Assuming a negative sine or cosine value means “no solution” — negative values are completely normal in quadrants where the function is negative.

6. Quick-Fire Challenge Round

True or False?
  1. sin 30° = cos 60°.
  2. tan 90° has a defined value.
  3. The graphs of sin x and cos x both have period 360°.
  4. A sine equation with a positive value always has exactly two solutions between 0° and 360°.
Show answers
  1. True — both equal 1/2.
  2. False — tan 90° is undefined (the tangent graph has a vertical asymptote there).
  3. True.
  4. False — most of the time yes, but if the value is exactly 1 (the maximum), there’s only one solution (x=90°), since the curve just touches that height once.
Spot the Error
  1. “Solve sin x = 0.6: calculator gives 36.9°, so the second solution is 360 − 36.9 = 323.1°.”
  2. “cos 45° = 1/2.”
  3. “Solve tan x = 2: calculator gives 63.4°, so the second solution is 360 − 63.4 = 296.6°.”
Show answers
  1. The second solution rule for sine is 180−A, not 360−A: 180−36.9 = 143.1°, not 323.1°.
  2. cos 45° = √2/2 (≈0.707), not 1/2 — that’s the value for cos 60° instead.
  3. The second solution rule for tangent is A+180°, not 360−A: 63.4+180 = 243.4°, not 296.6°.

7. Mixed Practice — Bringing It All Together

  1. Find the exact value of 2cos 30° − tan 45°.
  2. A right triangle has angle 45° and opposite side 5 cm. Find the exact hypotenuse.
  3. Solve sin x = 0.342 for 0° ⩽ x ⩽ 360° (1 d.p.).
  4. Solve cos x = −0.766 for 0° ⩽ x ⩽ 360° (1 d.p.).
  5. Solve tan x = 0.577 for 0° ⩽ x ⩽ 360° (using the exact values table, or 1 d.p. if not exact).
Show answers
  1. 2(√3/2) − 1 = √3 − 1
  2. 5 ÷ sin45° = 5 ÷ (√2/2) = 5√2 cm
  3. sin⁻¹(0.342)=20.0°. Second=160.0°. x=20.0° or 160.0°
  4. cos⁻¹(−0.766)=140.0°. Second=360−140=220.0°. x=140.0° or 220.0°
  5. 0.577≈1/√3=tan30°, so x=30°. Second=30+180=210°. x=30° or 210°

Practice at Home

A large set again — take it steadily.

Part A — Exact values
  1. Find the exact value of sin 45° + cos 45°.
  2. Find the exact value of 3 tan 60° − sin 90°.
  3. A right triangle has angle 60° and hypotenuse 12 cm. Find the exact opposite and adjacent sides.
Part B — Graphs
  1. At what value(s) of x does y = cos x reach its minimum, for 0° ⩽ x ⩽ 360°?
  2. At what value(s) of x is y = sin x equal to 0, for 0° ⩽ x ⩽ 360°?
  3. Explain why y = tan x has a shorter period than y = sin x.
Part C — Solving trig equations
  1. Solve sin x = 0.707 for 0° ⩽ x ⩽ 360° (use the exact values table).
  2. Solve cos x = 0.342 for 0° ⩽ x ⩽ 360° (1 d.p.).
  3. Solve tan x = 1.732 for 0° ⩽ x ⩽ 360° (use the exact values table).
  4. Solve 3 sin x = 1.5 for 0° ⩽ x ⩽ 360° (1 d.p.).
Part D — Applied / challenge
  1. Show that (sin 30°)² + (cos 30°)² = 1, using exact values — and explain why this makes sense given Pythagoras’ theorem and the 30-60-90 triangle.
  2. Solve 2cos x − 1 = 0 for 0° ⩽ x ⩽ 360°, giving exact values where possible.
  3. A student says “since sin 150° is positive, x=150° must be in the first quadrant.” Explain why this reasoning is incorrect, and state which quadrant 150° is actually in.

Key Vocabulary Recap

Exact value — a trig value expressed as a fraction or surd, not a rounded decimal
Period — the interval after which a graph’s pattern repeats
Asymptote — a line a graph approaches but never reaches (tan x has these at 90°, 270°)
Amplitude — half the distance between a graph’s maximum and minimum (1 for sin/cos)

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) sections E6.3 and E6.4 (Extended subject content) — Exact trigonometric values, and Trigonometric functions. There is no Core equivalent — sections C6.3 and C6.4 are explicitly marked “Extended content only” in the syllabus.

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