Discover double angles (AASL 3.6)

Keywords

[br][table][br][tr][/tr][br][tr][br][td]Double Angle Formulas[/td][br][td]二重角公式（にじゅうかくこうしき）[/td][br][td]双角公式（shuāng jiǎo gōng shì）[/td][br][td]이중 각도 공식 (i-jung gakdo gongsik)[/td][br][/tr][br][tr][br][td]Sine[/td][br][td]サイン（さいん）[/td][br][td]正弦（zhèng xián）[/td][br][td]사인 (sain)[/td][br][/tr][br][tr][br][td]Cosine[/td][br][td]コサイン（こさいん）[/td][br][td]余弦（yú xián）[/td][br][td]코사인 (kosain)[/td][br][/tr][br][tr][br][td]Tangent[/td][br][td]タンジェント（たんじぇんと）[/td][br][td]正切（zhèng qiè）[/td][br][td]탄젠트 (tanjenteu)[/td][br][/tr][br][tr][br][td]Trigonometric Functions[/td][br][td]三角関数（さんかくかんすう）[/td][br][td]三角函数（sān jiǎo hán shù）[/td][br][td]삼각 함수 (samgak chuso)[/td][br][/tr][br][tr][br][td]Trigonometric Identities[/td][br][td]三角関数の恒等式（さんかくかんすうのこうとうしき）[/td][br][td]三角恒等式（sān jiǎo hán děng shì）[/td][br][td]삼각함수 정체성 (samgakhamsoo jeongchejung)[/td][br][/tr][br][tr][br][td]Pythagorean Identity[/td][br][td]ピタゴラスの定理（ぴたごらすのていり）[/td][br][td]勾股定理（gōu gǔ dìng lǐ）[/td][br][td]피타고라스 정리 (pitagoras jeongri)[/td][br][/tr][br][/table]

Inquiry questions

[br][table][br][tr][br][td][b]Factual Inquiry Questions[/b][br]What are the double angle formulas for sine, cosine, and tangent?[br][br]How can the double angle formulas be derived from the sum of angles formulas (HL)?[br][/td][br][br][td][b]Conceptual Inquiry Questions[/b][br]Why are double angle formulas important in simplifying expressions involving trigonometric functions?[br][br]How can double angle formulas be used to solve trigonometric equations that involve squared terms?[br][/td][br][br][td][b]Debatable Inquiry Questions[/b][br]Is the use of double angle formulas more efficient than other trigonometric identities in solving complex trigonometric problems?[br][/td][br][/tr][br][/table][br]

Guided Exploration: Discovering Trigonometric Identities[br][br]Objective: Match equivalent trigonometric expressions and uncover the double angle formulas and the Pythagorean identity.[br][br]Introduction: Trigonometry is full of patterns and identities that can simplify how we work with angles and sides of triangles. Today, we're going to explore some of these patterns.[br][br]Step 1: Initial Observations[br]- Look at the expressions from the applet. What do you notice about the results when you compare different expressions?[br]Step 2: Make Predictions[br]- Before matching, predict which expressions might be equivalent. What is your reasoning?[br]Step 3: Matching Game[br]- Begin matching expressions that you believe are equivalent. Discuss with your partner why you think they match.[br]

Step 4: Identify the Formulas[br]- Using the matches you've made, try to identify the expressions that seem to involve doubling an angle.[br][br]Question 2: Which expressions could represent the double angle formulas? Hint: Look for expressions where the angle is doubled (like[math]sin(2x)[/math]) and try to match them with their equivalent forms.[br]

Step 5: Explore the Pythagorean Identity[br]- Which expression from the applet adds up to [math]1[/math].They are linked to a famous trigonometric identity.[br][br]Question 3: Which expression pairs add up to [math]1[/math]? How do they demonstrate the Pythagorean identity [math]sin^2(x)+cos^2(x)=1[/math]? By considering a right triangle with hypotenuse [math]1[/math] can you prove this identity?

Step 6: Confirm Your Findings[br]- Use a calculator to confirm the equivalences you've found. Do the values agree with your predictions?[br][br]Step 7: Reflection[br]- Reflect on how the expressions relate to the double angle formulas and the Pythagorean identity.[br][br]Question 4: Can you write down the double angle formulas for sine and cosine using the expressions you've matched?

Conclusion: Share your findings with the class. Discuss how these identities can be useful tools in trigonometry.[br]

Part 2 - Checking for understanding

Watch the below video for the some worked examples of how these formulas are used, and how they appear in your formulae booklet.

If [math]sin(x)=\frac{3}{5}[/math], what is [math]cos^2(x)[/math]?

[math]\frac{9}{25}[/math] [math]\frac{4}{5}[/math] [math]\frac{24}{25}[/math] [math]\frac{16}{25}[/math]

If [math]cos(x)=\frac{1}{2}[/math], what is [math]cos(2x)[/math]?

[math]0[/math] [math]-\frac{1}{2}[/math] [math]\frac{1}{4}[/math] [math]\frac{1}{2}[/math]

Given [math]sin(x)=\frac{1}{\sqrt{2}}[/math], what is [math]\sin(2x)[/math]?[br]

[math]0[/math] [math]\frac{\sqrt{2}}{2}[/math] [math]1[/math] [math]\frac{1}{2}[/math]

Part 3 - Exam style questions

[b][u]Trig double angles and Pythagoran identity to solve trigonometric equations[br][br][/u][/b]Q22, Q23, Q24, Q25, Q28, Q29, Q30, Q31, Q32, Q35

[MAA 3.6] TRIGONOMETRIC EQUATIONS

[MAA 3.6] TRIGONOMETRIC EQUATIONS_solutions

Optional extension

Proving double angle identities. This will not examined but it's useful to know how these formulaes can be proven.

Why does [math]BE=2cos\theta[/math]?

Why is [math]\angle EAC[/math] [math]2\theta[/math]?

Why does [math]EF=\sin2\theta[/math]?

Why does [math]AF=\cos2\theta[/math]?

In question 3 you explained why [math]EF=\sin2\theta[/math]. Why is it also true that [math]EF=2\sin\theta\cos\theta[/math] ?

Explain why is [math]\cos2\theta+1=2\cos\theta\cos\theta=2\cos^2\theta[/math] ?