At a glance. One lesson of about 60 minutes, with no calculator. It assumes surds from Year 10. Students derive exact values from half a square and half an equilateral triangle, then use them to find sides and areas with surds.
Calculator: no calculator. No calculator is needed or wanted: the point is that the values are exact, and a calculator's decimal hides the surd that makes them exact.
What students will learn
Know
- sin 45° = cos 45° = √2/2 and tan 45° = 1.
- sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3.
- sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.
Understand
- The values are ratios of sides of two special triangles, so they do not depend on size.
- The 30° angle is opposite the shortest side, which is half the hypotenuse.
Do
- Derive exact values from a sketch.
- Find sides of special triangles exactly.
- Combine exact values by multiplication and addition.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: exact values from special triangles, surds in answers (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students describe what stays the same as the triangle is enlarged and relate sin 30° to cos 60°.
- Criterion C. Communicating: a labelled triangle with sides in terms of a or s, and answers written in exact form.
- Criterion D. Applying mathematics in real-life contexts: the ladder, the pole and its shadow.
The one thing to do with the figure
Ask for the ratio before the number. In the first figure, ask: what does the ratio of leg to hypotenuse equal when a = 1? Then slide a and ask what happens.
In the second figure, ask students to name which side is opposite 30° and which is opposite 60°, before reading sin 30° and sin 60° from it.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: half a square. Collect predictions. |
| 10 min | First figure; settle the hook; rationalize 1/√2. |
| 10 min | Second figure; key idea; the warning card. |
| 7 min | The three worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B or C, return to which triangle is which. |
| 13 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 5√2; 2√3; sine. |
|---|---|
| Worked examples | 15 cm and 15√3 cm; 11√3 cm; 10√2 m. |
| Faded example | 13 cm and 13√3 cm. |
| Something to think about | tan 45° = 1; tan 60° = √3; sin 60° is less than tan 45°. |
| 1. The 30, 60, 90 triangle | A. B is the middle side; C uses the 45° triangle; D takes the sides in the ratio of the angles. |
| 2. A side from sin 30° | 7 cm. |
| 3. A side from sin 45° with a surd | 4. The height is 4√2 m. |
| 4. A side from tan 60° | 9. BC = 9√3 cm. |
| 5. A product of exact values | 1.5. 3/2. |
| 6. A shadow length from tan 30° | 12. The shadow is 12√3 m. |
| 7. Height of an equilateral triangle | 7. The height is 7√3 cm. |
| 8. Area of a 30, 60, 90 triangle | 18. The area is 18√3 cm². |
| 9. Hypotenuse of a right isosceles triangle | D. A doubles as for 30°; B multiplies by sin 45° instead of dividing; C uses √3. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Gave the middle side, used the 45° triangle, or took the sides in the ratio of the angles. | Draw the half equilateral triangle and label 1, √3 and 2 before answering. |
| 12.1 or 28 in question 2 | Used the 60° value, or divided by sine. | Ask: is the opposite side shorter or longer than the hypotenuse? |
| 8 or 5.7 in question 3 | Forgot the half, or gave a decimal. | Rewrite sin 45° as √2/2 before multiplying. |
| 3 or 18 in question 4 | Divided by tan 60°, or found the hypotenuse. | Mark BC, AB and the angle at A on a sketch first. |
| 3 or 0.87 in question 5 | Used √3 for cos 30°, or 1/2. | Write both values in a line before multiplying. |
| 4 or 24 in question 6 | Multiplied by tan, or used sine. | Ask which side is opposite 30°, the pole or the shadow? |
| 14 or 12.1 in question 7 | Forgot the half, or gave a decimal. | Cut the triangle in half and label 7, 7√3 and 14. |
| 36 or 31.2 in question 8 | Forgot the half of base times height, or gave a decimal. | Write the area formula before substituting. |
| A, B or C in question 9 | Doubled as for the 30° triangle, multiplied by sin 45°, or used √3. | Name the triangle (45° means 1 : 1 : √2), then ask: is the hypotenuse longer or shorter than a leg? |
Questions worth asking
- Why is sin 30° = cos 60°? The 30° angle's opposite side is the 60° angle's adjacent side.
- Why does the size of the triangle not matter for an exact value? Similar triangles have equal ratios.
- How can you tell from a ratio whether it is for 45° or for 30° and 60°? Listen for “45° gives √2; 30° and 60° give √3”.
- Why can the answer to a trig problem with exact values be left with a surd? A surd is exact; a decimal is an approximation.
Differentiation
A way in for anyone stuck
- Give two labelled triangle templates with the sides 1, 1, √2 and 1, √3, 2 and the angles marked.
- Provide a one-page table to complete, built from the triangles rather than copied.
- Restrict to the 30° and 60° triangle first, then add 45°.
Stretch
- Show that sin² 30° + cos² 30° = 1 with exact values.
- Use the 30°, 60°, 90° triangle to find the exact height of a regular hexagon of side 8 cm.
Students learning in English as an additional language
“Exact” means with no rounding, and “surd” is a root that cannot be simplified to a whole number. “Equilateral” has three equal sides. “Isosceles” has two equal sides. Draw the two triangles on the board and keep them there.
| Word | Plain meaning | Example |
|---|---|---|
| exact value | an answer written with roots or fractions and no rounding | a root left unevaluated |
| surd | a root that does not simplify to a whole number or a fraction | the root of a prime number |
| rationalize | rewrite so that the denominator is a whole number | turning a fraction over a root into one over a whole number |
| equilateral | a triangle with three equal sides | the face of a regular tetrahedron |
| isosceles | a triangle with two equal sides | a triangle cut off a square by a diagonal |
| altitude | the perpendicular distance from a vertex to the opposite side | the height of a triangle |
Sentence starters for explaining aloud or in writing:
- The sides of this triangle are in the ratio … because …
- sin of this angle is the opposite side divided by …
- The answer is exact because I have left …
Two things not to say
“Memorize the table.” Rebuild it from the triangle, and the values will stay.
“Use the calculator to check.” The exact value is the answer; a decimal is only a check.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Exact valueWrite the exact value of tan 30° with a rational denominator.
√3/3. - Earlier work (Year 10)Write 10/√5 with a whole-number denominator, then simplify.
2√5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA kite string is 40 m long at 60° to the ground. How high is the kite, exactly?
20√3 m. - Working backIn a right-angled triangle the side opposite 30° is 5√3 cm. Find the hypotenuse.
10√3 cm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that the diagonal of a square of side s makes 45° with each side, and find sin 45° from it.
The diagonal is s√2, so sin 45° = s ÷ s√2 = √2/2. - OpenFind the exact value of sin 60° cos 30° − cos 60° sin 30°.
3/4 − 1/4 = 1/2.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to the slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? A paper square folded along a diagonal and a paper equilateral triangle folded in half give the two triangles. No calculator is used.