Exact trigonometric values from special triangles · Higher
| English | 中文 | Pinyin |
|---|---|---|
| exact value/eɡˈzækt ˈvæljuː/ | 精确值 | jīng què zhí |
A calculator shows sin30° as 0.5, but a special triangle explains why the value is exactly one half and how the other values connect.
- A calculator shows sin30° as 0.5, but a special triangle explains why the value is exactly one half and how the other values connect.
- This lesson studies exact value 精确值: A value retained as a fraction or surd rather than a rounded decimal.
Choose the mathematical structure
- Bisect an equilateral triangle of side 2 to obtain a 30–60–90 triangle with sides 1,√3,2. A right isosceles triangle has sides 1,1,√2. Apply opposite/hypotenuse, adjacent/hypotenuse and opposite/adjacent to derive sine, cosine and tangent. Use degree angles and retain surds exactly; tan90° is undefined.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines exact value?
A value retained as a fraction or surd rather than a rounded decimal.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For angles 0°,30°,45°,60°,90°, sine values are 0,1/2,√2/2,√3/2,1; cosine values are 1,√3/2,√2/2,1/2,0. For 0°,30°,45°,60°, tangent values are 0,1/√3,1,√3. The side 1 opposite 30° in the bisected equilateral triangle gives sin30°=1/2; the adjacent √3 gives cos30°=√3/2 and tan30°=1/√3. In the isosceles right triangle, sin45°=cos45°=1/√2=√2/2. A right triangle with hypotenuse 10 and angle 30° has opposite side 5 and adjacent side 5√3. Since sin90°=1 and cos90°=0, tangent at 90° would divide by zero.
Exact trigonometric values from special triangles
Bisect an equilateral triangle of side 2 to obtain a 30–60–90 triangle with sides 1,√3,2
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find sin30°.
Opposite 1 divided by hypotenuse 2.
Test a tempting shortcut
- Sine and cosine interchange when the chosen acute angle changes to its complement. Do not turn √3 into a rounded decimal when an exact answer is required. Tangent at 90° is not zero.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
All trigonometric functions have finite values at 90°. This claim is false. Explain which definition or assumption it violates.
Find tan45°.
Equal opposite and adjacent sides give ratio 1.
All trigonometric functions have finite values at 90°.
Sine and cosine interchange when the chosen acute angle changes to its complement. Do not turn √3 into a rounded decimal when an exact answer is required. Tangent at 90° is not zero.
Interpret a new situation
- AQA G21 is additional Foundation and includes the listed exact values. Derive them from labelled special triangles, then use them in G20 right-triangle calculations.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the opposite length for hypotenuse 10 and angle 30°.
10×sin30°=5.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A value retained as a fraction or surd rather than a rounded decimal. Choose the relationship, show the method, check its assumptions and interpret the result.