MathematicsChapter 6: Trigonometry and Bearing
Standard Angles 30 45 60
Three acute angles appear in almost every exercise: 30 deg, 45 deg, and 60 deg.
Learning objectives
- Explain standard angles 30 45 60 in clear mathematical language
- Use the main rules and formulas in worked examples
- Recognise and correct common errors in this topic
What this grid covers
Three acute angles appear in almost every exercise: 30 deg, 45 deg, and 60 deg. Their sine, cosine, and tangent values come from two right triangles: a 45-45-90 triangle with equal legs, and a 30-60-90 triangle whose sides are 1, sqrt(3), and 2. Learn the table in both directions: given the angle, quote the ratio; given the ratio, name the angle.
Where / symbols
- 30 deg
- Three acute angles appear in almost every exercise: 30 deg, 45 deg, and 60 deg.
- 45 deg
- Three acute angles appear in almost every exercise: 30 deg, 45 deg, and 60 deg.
- 60 deg
- Three acute angles appear in almost every exercise: 30 deg, 45 deg, and 60 deg.
- sqrt(3)
- Their sine, cosine, and tangent values come from two right triangles: a 45-45-90 triangle with equal legs, and a 30-60-90 triangle whose sides are 1, sqrt(3), and 2.
What it means
Three acute angles appear in almost every exercise: 30 deg, 45 deg, and 60 deg.
Main ideas
30 deg: sin 30 = 12, cos 30 = sqrt(3)/2; tan 30 = 1/sqrt(3) 45 deg: sin 45 = 1/sqrt(2), cos 45 = 1/sqrt(2); tan 45 = 1 60 deg: sin 60 = sqrt(3)/2, cos 60 = 12; tan 60 = sqrt(3) 0 deg (edge): sin 0 = 0, cos 0 = 1; tan 0 = 0
Why it matters
This method connects the definition to worked questions and later mathematics.
How to answer in an exam
sin 30 = 12 and cos 30 = sqrt(3)/2; do not swap them. sin 45 = cos 45 = 1/sqrt(2) and tan 45 = 1. sin 60 = sqrt(3)/2, cos 60 = 12, tan 60 = sqrt(3). tan 30 = 1/sqrt(3). Leave sqrt(3) in the answer; do not force a decimal unless asked.
Exact values you must write without a calculator:
| Angle | sin and cos | tan |
|---|---|---|
| 1. 30 deg | sin 30 = 1/2, cos 30 = sqrt(3)/2 | tan 30 = 1/sqrt(3) |
| 2. 45 deg | sin 45 = 1/sqrt(2), cos 45 = 1/sqrt(2) | tan 45 = 1 |
| 3. 60 deg | sin 60 = sqrt(3)/2, cos 60 = 1/2 | tan 60 = sqrt(3) |
| 4. 0 deg (edge) | sin 0 = 0, cos 0 = 1 | tan 0 = 0 |
| 5. 90 deg (edge) | sin 90 = 1, cos 90 = 0 | tan 90 undefined |
| 6. Pattern | sin 30 = cos 60, sin 45 = cos 45, sin 60 = cos 30 | tan = sin/cos for each row |
In simple words
sin 30 = 12 and cos 30 = sqrt(3)/2; do not swap them. sin 45 = cos 45 = 1/sqrt(2) and tan 45 = 1. sin 60 = sqrt(3)/2, cos 60 = 12, tan 60 = sqrt(3). tan 30 = 1/sqrt(3). Leave sqrt(3) in the answer; do not force a decimal unless asked.
Key points
- 30 deg
- 45 deg
- 60 deg
- sqrt(3)
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Standard Angles 30 45 60
Sticky Notes · Quick Revision
Core Idea
- Three acute angles appear in almost every exercise: 30 deg, 45 deg, and 60 deg.
Key Terms
- 30 deg: Three acute angles appear in almost every exercise: 30 deg, 45 deg, and 60 deg.
- 45 deg: Three acute angles appear in almost every exercise: 30 deg, 45 deg, and 60 deg.
Main Pattern
- 30 deg: sin 30 = 12, cos 30 = sqrt(3)/2
- 45 deg: sin 45 = 1/sqrt(2), cos 45 = 1/sqrt(2)
Worked Reminder
- 30 deg: sin 30 = 12, cos 30 = sqrt(3)/2; tan 30 = 1/sqrt(3)
- 45 deg: sin 45 = 1/sqrt(2), cos 45 = 1/sqrt(2); tan 45 = 1
- 60 deg: sin 60 = sqrt(3)/2, cos 60 = 12; tan 60 = sqrt(3)
Common Mistake
- sin 30 = 12 and cos 30 = sqrt(3)/2; do not swap them. sin 45 = cos 45 = 1/sqrt(2) and tan 45 = 1. sin 60 = sqrt(3)/2, cos 60 = 12, tan 60 = sqrt(3). tan 30 = 1/sqrt(3). Leave sqrt(3) in the answer; do not force a decimal unless asked.
Exam Tip
- State the rule for standard angles 30 45 60, show each algebraic step, and check the final answer against the question.
60-second revision
- Three acute angles appear in almost every exercise: 30 deg, 45 deg, and 60 deg.
- 30 deg: sin 30 = 12, cos 30 = sqrt(3)/2; tan 30 = 1/sqrt(3)
- 45 deg: sin 45 = 1/sqrt(2), cos 45 = 1/sqrt(2); tan 45 = 1
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