Mathematics Advanced Year 11 - Module 2 - Lesson 5

Exact Values and Special Triangles

Use this worksheet after reading the lesson to practise the key ideas and prove you can meet the success criteria.

Name
Date
Class

1. Key Ideas

Calculators are banned in many exam questions for a reason: mathematicians want you to know the exact values. The good news is that you only need two special triangles to find the sine, cosine, and tangent of $30^\circ$, $45^\circ$, and $60^\circ$ exactly. Master these two triangles, and you have unlocked half of trigonometry.

  • The side ratios of the two special triangles
  • Why these two triangles generate all the common exact values

2. Success Criteria

By the end, you should be able to:

  • The side ratios of the two special triangles
  • The exact values of $\sin, \cos, \tan$ at $30^\circ, 45^\circ, 60^\circ$
  • How to use reference angles with exact values

3. Key Terms

Trigonometric RatioThe ratio of sides in a right-angled triangle (sin, cos, tan).
RadianA unit of angle measure where one radian subtends an arc equal to the radius.
Sine RuleA formula relating sides and angles in any triangle: a/sinA = b/sinB = c/sinC.
Cosine RuleA formula for finding sides or angles: c² = a² + b² - 2ab cosC.
PeriodThe length of one complete cycle of a periodic function.
AmplitudeThe maximum displacement from the centre line of a periodic function.

4. Activity: Build the Lesson Map

Use the lesson to complete the table. Keep answers brief but specific.

PromptYour answer
Main concept
Important example
Common mistake to avoid
How this links to the next lesson

5. Short Answer Questions

1. Explain this lesson goal in your own words: "The side ratios of the two special triangles". Use one specific example from the lesson.

Band 32 marks

2. Apply this idea to a new example: "The exact values of $\sin, \cos, \tan$ at $30^\circ, 45^\circ, 60^\circ$". Show your reasoning clearly.

Band 43 marks

3. Analyse why this idea matters for understanding Exact Values and Special Triangles: "How to use reference angles with exact values".

Band 54 marks

6. Extend: Apply the Idea

Band 5/65 marks

A student gives a memorised answer about Exact Values and Special Triangles but does not use evidence or reasoning.

Improve the answer by writing a stronger response that uses accurate terminology, a relevant example and a clear explanation.

7. Multiple Choice

1. What is the best first step when answering a question about Exact Values and Special Triangles?

A. Identify the key concept being tested

B. Write every fact from memory

C. Ignore the command word

D. Skip examples and evidence

2. Which answer would show stronger understanding of Exact Values and Special Triangles?

A. An answer with accurate terms and reasoning

B. A copied definition only

C. A single-word response

D. An answer with no example

3. What should you do if a question asks you to explain?

A. Link the idea to a reason or cause

B. List unrelated facts

C. Only draw a diagram

D. Write the shortest possible answer

8. Success Criteria Proof

Finish with evidence that you can do each success criterion.

Success criterion 1

Prove that you can: The side ratios of the two special triangles

Band 32 marks
Success criterion 2

Prove that you can: The exact values of $\sin, \cos, \tan$ at $30^\circ, 45^\circ, 60^\circ$

Band 43 marks
Success criterion 3

Prove that you can: How to use reference angles with exact values

Band 54 marks

One thing I still need help with:

HSCScience - Printable lesson worksheet - Mathematics Advanced Year 11 - Module 2 - Lesson 5

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Mathematics Advanced Year 11 - Module 2 - Lesson 5

Exact Values and Special Triangles

Use this worksheet after reading the lesson to practise the key ideas and prove you can meet the success criteria.

Name
Date
Class

1. Key Ideas

Calculators are banned in many exam questions for a reason: mathematicians want you to know the exact values. The good news is that you only need two special triangles to find the sine, cosine, and tangent of $30^\circ$, $45^\circ$, and $60^\circ$ exactly. Master these two triangles, and you have unlocked half of trigonometry.

  • The side ratios of the two special triangles
  • Why these two triangles generate all the common exact values

2. Success Criteria

By the end, you should be able to:

  • The side ratios of the two special triangles
  • The exact values of $\sin, \cos, \tan$ at $30^\circ, 45^\circ, 60^\circ$
  • How to use reference angles with exact values

3. Key Terms

Trigonometric RatioThe ratio of sides in a right-angled triangle (sin, cos, tan).
RadianA unit of angle measure where one radian subtends an arc equal to the radius.
Sine RuleA formula relating sides and angles in any triangle: a/sinA = b/sinB = c/sinC.
Cosine RuleA formula for finding sides or angles: c² = a² + b² - 2ab cosC.
PeriodThe length of one complete cycle of a periodic function.
AmplitudeThe maximum displacement from the centre line of a periodic function.

4. Activity: Build the Lesson Map

Use the lesson to complete the table. Keep answers brief but specific.

PromptYour answer
Main concept
Important example
Common mistake to avoid
How this links to the next lesson

5. Short Answer Questions

1. Explain this lesson goal in your own words: "The side ratios of the two special triangles". Use one specific example from the lesson.

Band 32 marks

2. Apply this idea to a new example: "The exact values of $\sin, \cos, \tan$ at $30^\circ, 45^\circ, 60^\circ$". Show your reasoning clearly.

Band 43 marks

3. Analyse why this idea matters for understanding Exact Values and Special Triangles: "How to use reference angles with exact values".

Band 54 marks

6. Extend: Apply the Idea

Band 5/65 marks

A student gives a memorised answer about Exact Values and Special Triangles but does not use evidence or reasoning.

Improve the answer by writing a stronger response that uses accurate terminology, a relevant example and a clear explanation.

7. Multiple Choice

1. What is the best first step when answering a question about Exact Values and Special Triangles?

A. Identify the key concept being tested

B. Write every fact from memory

C. Ignore the command word

D. Skip examples and evidence

2. Which answer would show stronger understanding of Exact Values and Special Triangles?

A. An answer with accurate terms and reasoning

B. A copied definition only

C. A single-word response

D. An answer with no example

3. What should you do if a question asks you to explain?

A. Link the idea to a reason or cause

B. List unrelated facts

C. Only draw a diagram

D. Write the shortest possible answer

8. Success Criteria Proof

Finish with evidence that you can do each success criterion.

Success criterion 1

Prove that you can: The side ratios of the two special triangles

Band 32 marks
Success criterion 2

Prove that you can: The exact values of $\sin, \cos, \tan$ at $30^\circ, 45^\circ, 60^\circ$

Band 43 marks
Success criterion 3

Prove that you can: How to use reference angles with exact values

Band 54 marks

One thing I still need help with:

HSCScience - Printable lesson worksheet - Mathematics Advanced Year 11 - Module 2 - Lesson 5

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Mathematics Advanced Year 11 - Module 2 - Lesson 5

Exact Values and Special Triangles

Use this worksheet after reading the lesson to practise the key ideas and prove you can meet the success criteria.

Name
Date
Class

1. Key Ideas

Calculators are banned in many exam questions for a reason: mathematicians want you to know the exact values. The good news is that you only need two special triangles to find the sine, cosine, and tangent of $30^\circ$, $45^\circ$, and $60^\circ$ exactly. Master these two triangles, and you have unlocked half of trigonometry.

  • The side ratios of the two special triangles
  • Why these two triangles generate all the common exact values

2. Success Criteria

By the end, you should be able to:

  • The side ratios of the two special triangles
  • The exact values of $\sin, \cos, \tan$ at $30^\circ, 45^\circ, 60^\circ$
  • How to use reference angles with exact values

3. Key Terms

Trigonometric RatioThe ratio of sides in a right-angled triangle (sin, cos, tan).
RadianA unit of angle measure where one radian subtends an arc equal to the radius.
Sine RuleA formula relating sides and angles in any triangle: a/sinA = b/sinB = c/sinC.
Cosine RuleA formula for finding sides or angles: c² = a² + b² - 2ab cosC.
PeriodThe length of one complete cycle of a periodic function.
AmplitudeThe maximum displacement from the centre line of a periodic function.

4. Activity: Build the Lesson Map

Use the lesson to complete the table. Keep answers brief but specific.

PromptYour answer
Main concept
Important example
Common mistake to avoid
How this links to the next lesson

5. Short Answer Questions

1. Explain this lesson goal in your own words: "The side ratios of the two special triangles". Use one specific example from the lesson.

Band 32 marks

2. Apply this idea to a new example: "The exact values of $\sin, \cos, \tan$ at $30^\circ, 45^\circ, 60^\circ$". Show your reasoning clearly.

Band 43 marks

3. Analyse why this idea matters for understanding Exact Values and Special Triangles: "How to use reference angles with exact values".

Band 54 marks

6. Extend: Apply the Idea

Band 5/65 marks

A student gives a memorised answer about Exact Values and Special Triangles but does not use evidence or reasoning.

Improve the answer by writing a stronger response that uses accurate terminology, a relevant example and a clear explanation.

7. Multiple Choice

1. What is the best first step when answering a question about Exact Values and Special Triangles?

A. Identify the key concept being tested

B. Write every fact from memory

C. Ignore the command word

D. Skip examples and evidence

2. Which answer would show stronger understanding of Exact Values and Special Triangles?

A. An answer with accurate terms and reasoning

B. A copied definition only

C. A single-word response

D. An answer with no example

3. What should you do if a question asks you to explain?

A. Link the idea to a reason or cause

B. List unrelated facts

C. Only draw a diagram

D. Write the shortest possible answer

8. Success Criteria Proof

Finish with evidence that you can do each success criterion.

Success criterion 1

Prove that you can: The side ratios of the two special triangles

Band 32 marks
Success criterion 2

Prove that you can: The exact values of $\sin, \cos, \tan$ at $30^\circ, 45^\circ, 60^\circ$

Band 43 marks
Success criterion 3

Prove that you can: How to use reference angles with exact values

Band 54 marks

One thing I still need help with:

HSCScience - Printable lesson worksheet - Mathematics Advanced Year 11 - Module 2 - Lesson 5