# Sub-Multiple Angles: Complete Study Guide
## Master Half-Angle and Fractional Angle Identities with Confidence

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## 🎯 Introduction: Why Sub-Multiple Angles are Essential

Welcome to the **comprehensive guide** for mastering sub-multiple angles! This crucial topic bridges basic trigonometry with advanced identities, focusing on relationships between angles and their fractions (A/2, A/3, etc.). These formulas are fundamental for solving complex trigonometric problems and simplifying expressions.

**Real-World Applications:**
- 🔬 **Physics**: Wave interference and harmonic analysis
- 🏗️ **Engineering**: Structural oscillations and resonance calculations  
- 📡 **Signal Processing**: Frequency division and modulation
- 🎵 **Acoustics**: Musical intervals and sound wave analysis
- 🚀 **Astronomy**: Orbital mechanics and celestial calculations
- 💻 **Computer Graphics**: Rotation transformations and animations

**Key Learning Objective**: Master the art of converting between angles and their sub-multiples using half-angle formulas, third-angle relationships, and systematic algebraic manipulation.

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## 🧠 Essential Formula Arsenal: Your Sub-Multiple Toolkit

### 🎯 **Half-Angle Formulas (The Foundation)**

#### **Primary Half-Angle Identities:**
$$\sin \frac{A}{2} = \pm\sqrt{\frac{1 - \cos A}{2}}$$
$$\cos \frac{A}{2} = \pm\sqrt{\frac{1 + \cos A}{2}}$$
$$\tan \frac{A}{2} = \pm\sqrt{\frac{1 - \cos A}{1 + \cos A}} = \frac{\sin A}{1 + \cos A} = \frac{1 - \cos A}{\sin A}$$

#### **Alternative Half-Angle Forms:**
$$\tan \frac{A}{2} = \frac{\sin A}{1 + \cos A}$$
$$\cot \frac{A}{2} = \frac{1 + \cos A}{\sin A}$$
$$\sin A = \frac{2\tan \frac{A}{2}}{1 + \tan^2 \frac{A}{2}}$$
$$\cos A = \frac{1 - \tan^2 \frac{A}{2}}{1 + \tan^2 \frac{A}{2}}$$

### ⚡ **Triple-Angle to Sub-Multiple Formulas**

#### **From Triple Angles to Third-Angles:**
$$\sin A = \sin 3 \cdot \frac{A}{3} = 3\sin \frac{A}{3} - 4\sin^3 \frac{A}{3}$$
$$\cos A = \cos 3 \cdot \frac{A}{3} = 4\cos^3 \frac{A}{3} - 3\cos \frac{A}{3}$$

#### **Reverse Relationships:**
If $\sin \frac{A}{3} = x$, then $\sin A = 3x - 4x^3$
If $\cos \frac{A}{3} = y$, then $\cos A = 4y^3 - 3y$

### 🔄 **Essential Transformation Identities**

#### **Half-Angle Expressions:**
$$1 - \cos A = 2\sin^2 \frac{A}{2}$$
$$1 + \cos A = 2\cos^2 \frac{A}{2}$$
$$\sin A = 2\sin \frac{A}{2} \cos \frac{A}{2}$$

#### **Useful Reciprocal Forms:**
$$\frac{1 - \cos A + \sin A}{1 + \cos A + \sin A} = \tan \frac{A}{2}$$
$$\frac{1 + \cos A + \sin A}{1 - \cos A + \sin A} = \cot \frac{A}{2}$$
$$\cosec A - \cot A = \tan \frac{A}{2}$$

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## 🎯 The SUBMULT Strategy for Sub-Multiple Problems

### 📋 **S**-**U**-**B**-**M**-**U**-**L**-**T** Method

#### **S**elect the Appropriate Formula
- Identify whether dealing with A/2, A/3, or other fractions
- Choose half-angle or triple-to-sub formulas accordingly

#### **U**se Given Information Wisely
- Extract exact values from given conditions
- Apply algebraic substitution systematically

#### **B**reak Down Complex Expressions
- Separate numerators and denominators
- Factor common terms when possible

#### **M**anipulate Using Identities
- Apply Pythagorean relationships
- Use complementary angle formulas

#### **U**nify Terms with Common Denominators
- Combine fractions systematically
- Look for telescoping patterns

#### **L**inearize Using Standard Forms
- Convert to standard half-angle expressions
- Apply known trigonometric values

#### **T**ransform to Final Answer
- Simplify radicals and fractions
- Verify using alternative methods

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## 📚 Complete Problems from Raw Data

*All 15 sub-multiple angle problems with detailed step-by-step solutions*

**📝 Source Note:** All problems below are extracted directly from the raw data, covering half-angle identities, third-angle relationships, and specific value calculations.

### Problem 1: Complex Third-Angle Identity ⭐ (ADVANCED PROOF)
**Question:** If $\sin \frac{A}{3} = \frac{1}{2}\left(a + \frac{1}{a}\right)$, prove that:
$$\sin A = -\frac{1}{2}\left(a^3 + \frac{1}{a^3}\right)$$

**🎯 Problem Type:** Third-angle to full angle conversion
**🧠 Key Insight:** Use triple angle formula for sine

**Complete Solution:**

**Step 1: Apply triple angle formula**
$$\sin A = \sin\left(3 \cdot \frac{A}{3}\right) = 3\sin \frac{A}{3} - 4\sin^3 \frac{A}{3}$$

**Step 2: Substitute given value**
$$= 3 \cdot \frac{1}{2}\left(a + \frac{1}{a}\right) - 4 \left[\frac{1}{2}\left(a + \frac{1}{a}\right)\right]^3$$

**Step 3: Simplify the first term**
$$= \frac{3}{2}\left(a + \frac{1}{a}\right) - 4 \cdot \frac{1}{8}\left(a + \frac{1}{a}\right)^3$$
$$= \frac{3}{2}\left(a + \frac{1}{a}\right) - \frac{1}{2}\left(a + \frac{1}{a}\right)^3$$

**Step 4: Expand the cubic term**
$$\left(a + \frac{1}{a}\right)^3 = a^3 + \frac{1}{a^3} + 3a \cdot \frac{1}{a}\left(a + \frac{1}{a}\right) = a^3 + \frac{1}{a^3} + 3\left(a + \frac{1}{a}\right)$$

**Step 5: Substitute back**
$$= \frac{3}{2}\left(a + \frac{1}{a}\right) - \frac{1}{2}\left[a^3 + \frac{1}{a^3} + 3\left(a + \frac{1}{a}\right)\right]$$

**Step 6: Distribute and simplify**
$$= \frac{3}{2}\left(a + \frac{1}{a}\right) - \frac{1}{2}\left(a^3 + \frac{1}{a^3}\right) - \frac{3}{2}\left(a + \frac{1}{a}\right)$$

**Step 7: Cancel like terms**
$$= -\frac{1}{2}\left(a^3 + \frac{1}{a^3}\right)$$

**Final Answer:** $$\boxed{-\frac{1}{2}\left(a^3 + \frac{1}{a^3}\right)}$$ ✓

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### Problem 2: Cosine Third-Angle Identity 🔥
**Question:** If $\cos \frac{B}{3} = \frac{1}{2}\left(m + \frac{1}{m}\right)$, prove that:
$$\cos B = \frac{1}{2}\left(m^3 + \frac{1}{m^3}\right)$$

**🎯 Problem Type:** Third-angle cosine conversion
**🧠 Key Insight:** Use triple angle formula for cosine

**Complete Solution:**

**Step 1: Apply triple angle formula**
$$\cos B = \cos\left(3 \cdot \frac{B}{3}\right) = 4\cos^3 \frac{B}{3} - 3\cos \frac{B}{3}$$

**Step 2: Substitute given value**
$$= 4 \left[\frac{1}{2}\left(m + \frac{1}{m}\right)\right]^3 - 3 \cdot \frac{1}{2}\left(m + \frac{1}{m}\right)$$

**Step 3: Simplify**
$$= 4 \cdot \frac{1}{8}\left(m + \frac{1}{m}\right)^3 - \frac{3}{2}\left(m + \frac{1}{m}\right)$$
$$= \frac{1}{2}\left(m + \frac{1}{m}\right)^3 - \frac{3}{2}\left(m + \frac{1}{m}\right)$$

**Step 4: Expand the cubic term**
$$\left(m + \frac{1}{m}\right)^3 = m^3 + \frac{1}{m^3} + 3m \cdot \frac{1}{m}\left(m + \frac{1}{m}\right) = m^3 + \frac{1}{m^3} + 3\left(m + \frac{1}{m}\right)$$

**Step 5: Substitute and simplify**
$$= \frac{1}{2}\left[m^3 + \frac{1}{m^3} + 3\left(m + \frac{1}{m}\right)\right] - \frac{3}{2}\left(m + \frac{1}{m}\right)$$
$$= \frac{1}{2}\left(m^3 + \frac{1}{m^3}\right) + \frac{3}{2}\left(m + \frac{1}{m}\right) - \frac{3}{2}\left(m + \frac{1}{m}\right)$$

**Step 6: Cancel like terms**
$$= \frac{1}{2}\left(m^3 + \frac{1}{m^3}\right)$$

**Final Answer:** $$\boxed{\frac{1}{2}\left(m^3 + \frac{1}{m^3}\right)}$$ ✓

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### Problem 3: Half-Angle Tangent Application 💫
**Question:** If $\tan \frac{A}{2} = \frac{3}{4}$, find the value of $\sin A$.

**🎯 Problem Type:** Half-angle to full angle sine
**🧠 Key Insight:** Use double angle formula with half-angle tangent

**Complete Solution:**

**Step 1: Apply double angle formula**
$$\sin A = \sin\left(2 \cdot \frac{A}{2}\right) = \frac{2\tan \frac{A}{2}}{1 + \tan^2 \frac{A}{2}}$$

**Step 2: Substitute given value**
$$= \frac{2 \cdot \frac{3}{4}}{1 + \left(\frac{3}{4}\right)^2} = \frac{\frac{6}{4}}{1 + \frac{9}{16}}$$

**Step 3: Simplify denominator**
$$= \frac{\frac{3}{2}}{1 + \frac{9}{16}} = \frac{\frac{3}{2}}{\frac{16 + 9}{16}} = \frac{\frac{3}{2}}{\frac{25}{16}}$$

**Step 4: Complete the division**
$$= \frac{3}{2} \times \frac{16}{25} = \frac{48}{50} = \frac{24}{25}$$

**Final Answer:** $$\boxed{\frac{24}{25}}$$ ✓

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### Problem 4: Third-Angle Sine Calculation ⚡
**Question:** If $\sin \frac{A}{3} = \frac{4}{5}$, find the value of $\sin A$.

**🎯 Problem Type:** Direct third-angle sine conversion
**🧠 Key Insight:** Apply triple angle formula systematically

**Complete Solution:**

**Step 1: Use triple angle formula**
$$\sin A = 3\sin \frac{A}{3} - 4\sin^3 \frac{A}{3}$$

**Step 2: Substitute given value**
$$= 3 \cdot \frac{4}{5} - 4 \left(\frac{4}{5}\right)^3$$

**Step 3: Calculate each term**
$$= \frac{12}{5} - 4 \cdot \frac{64}{125} = \frac{12}{5} - \frac{256}{125}$$

**Step 4: Find common denominator**
$$= \frac{12 \times 25}{125} - \frac{256}{125} = \frac{300 - 256}{125} = \frac{44}{125}$$

**Final Answer:** $$\boxed{\frac{44}{125}}$$ ✓

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### Problem 5: Classic Half-Angle Identity 🌟
**Question:** Prove that:
$$\frac{1 - \cos θ + \sin θ}{1 + \cos θ + \sin θ} = \tan \frac{θ}{2}$$

**🎯 Problem Type:** Half-angle identity proof
**🧠 Key Insight:** Use half-angle substitutions

**Complete Solution:**

**Step 1: Apply half-angle substitutions**
Using $\cos θ = 1 - 2\sin^2 \frac{θ}{2}$ and $\sin θ = 2\sin \frac{θ}{2} \cos \frac{θ}{2}$

**Step 2: Substitute into numerator**
$$1 - \cos θ + \sin θ = 1 - (1 - 2\sin^2 \frac{θ}{2}) + 2\sin \frac{θ}{2} \cos \frac{θ}{2}$$
$$= 2\sin^2 \frac{θ}{2} + 2\sin \frac{θ}{2} \cos \frac{θ}{2}$$

**Step 3: Substitute into denominator**
$$1 + \cos θ + \sin θ = 1 + (1 - 2\sin^2 \frac{θ}{2}) + 2\sin \frac{θ}{2} \cos \frac{θ}{2}$$
$$= 2 - 2\sin^2 \frac{θ}{2} + 2\sin \frac{θ}{2} \cos \frac{θ}{2} = 2\cos^2 \frac{θ}{2} + 2\sin \frac{θ}{2} \cos \frac{θ}{2}$$

**Step 4: Factor both terms**
$$= \frac{2\sin \frac{θ}{2}(\sin \frac{θ}{2} + \cos \frac{θ}{2})}{2\cos \frac{θ}{2}(\cos \frac{θ}{2} + \sin \frac{θ}{2})}$$

**Step 5: Cancel common factors**
$$= \frac{2\sin \frac{θ}{2}}{2\cos \frac{θ}{2}} = \frac{\sin \frac{θ}{2}}{\cos \frac{θ}{2}}$$

**Final Answer:** $$\boxed{\tan \frac{θ}{2}}$$ ✓

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### Problem 6: Alternative Half-Angle Form 📐
**Question:** Prove that:
$$\frac{\sin \frac{θ}{2} + \sin θ}{1 + \cos \frac{θ}{2} + \cos θ} = \tan \frac{θ}{2}$$

**🎯 Problem Type:** Complex half-angle identity
**🧠 Key Insight:** Use double angle substitutions

**Complete Solution:**

**Step 1: Express sin θ and cos θ in terms of θ/2**
$$\sin θ = 2\sin \frac{θ}{2} \cos \frac{θ}{2}$$
$$\cos θ = 2\cos^2 \frac{θ}{2} - 1$$

**Step 2: Substitute into the expression**
$$= \frac{\sin \frac{θ}{2} + 2\sin \frac{θ}{2} \cos \frac{θ}{2}}{1 + \cos \frac{θ}{2} + 2\cos^2 \frac{θ}{2} - 1}$$
$$= \frac{\sin \frac{θ}{2}(1 + 2\cos \frac{θ}{2})}{\cos \frac{θ}{2} + 2\cos^2 \frac{θ}{2}} = \frac{\sin \frac{θ}{2}(1 + 2\cos \frac{θ}{2})}{\cos \frac{θ}{2}(1 + 2\cos \frac{θ}{2})}$$

**Step 3: Cancel common factors**
$$= \frac{\sin \frac{θ}{2}}{\cos \frac{θ}{2}}$$

**Final Answer:** $$\boxed{\tan \frac{θ}{2}}$$ ✓

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### Problem 7: Reciprocal Function Identity 💪
**Question:** Prove that:
$$\cosec θ - \cot θ = \tan \frac{θ}{2}$$

**🎯 Problem Type:** Reciprocal to half-angle conversion
**🧠 Key Insight:** Convert to sine and cosine, then use half-angle substitutions

**Complete Solution:**

**Step 1: Convert to sine and cosine**
$$\cosec θ - \cot θ = \frac{1}{\sin θ} - \frac{\cos θ}{\sin θ} = \frac{1 - \cos θ}{\sin θ}$$

**Step 2: Apply half-angle substitutions**
$$1 - \cos θ = 2\sin^2 \frac{θ}{2}$$
$$\sin θ = 2\sin \frac{θ}{2} \cos \frac{θ}{2}$$

**Step 3: Substitute**
$$= \frac{2\sin^2 \frac{θ}{2}}{2\sin \frac{θ}{2} \cos \frac{θ}{2}} = \frac{\sin \frac{θ}{2}}{\cos \frac{θ}{2}}$$

**Final Answer:** $$\boxed{\tan \frac{θ}{2}}$$ ✓

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### Problem 8: Double Cotangent Identity 🎯
**Question:** Prove that:
$$\cot \frac{A}{2} - \tan \frac{A}{2} = 2\cot A$$

**🎯 Problem Type:** Half-angle difference to full angle
**🧠 Key Insight:** Convert to common denominator, then apply double angle

**Complete Solution:**

**Step 1: Express in terms of sine and cosine**
$$\cot \frac{A}{2} - \tan \frac{A}{2} = \frac{\cos \frac{A}{2}}{\sin \frac{A}{2}} - \frac{\sin \frac{A}{2}}{\cos \frac{A}{2}}$$

**Step 2: Find common denominator**
$$= \frac{\cos^2 \frac{A}{2} - \sin^2 \frac{A}{2}}{\sin \frac{A}{2} \cos \frac{A}{2}}$$

**Step 3: Apply double angle formulas**
$$\cos^2 \frac{A}{2} - \sin^2 \frac{A}{2} = \cos A$$
$$\sin \frac{A}{2} \cos \frac{A}{2} = \frac{1}{2}\sin A$$

**Step 4: Substitute and simplify**
$$= \frac{\cos A}{\frac{1}{2}\sin A} = \frac{2\cos A}{\sin A}$$

**Final Answer:** $$\boxed{2\cot A}$$ ✓

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### Problem 9: Cotangent Half-Angle Identity ⚡
**Question:** Prove that:
$$\frac{1 + \cos θ + \sin θ}{1 - \cos θ + \sin θ} = \cot \frac{θ}{2}$$

**🎯 Problem Type:** Complex half-angle cotangent proof
**🧠 Key Insight:** Use half-angle substitutions systematically

**Complete Solution:**

**Step 1: Apply half-angle substitutions**
$$1 + \cos θ = 2\cos^2 \frac{θ}{2}, \quad 1 - \cos θ = 2\sin^2 \frac{θ}{2}$$
$$\sin θ = 2\sin \frac{θ}{2} \cos \frac{θ}{2}$$

**Step 2: Substitute into the expression**
$$= \frac{2\cos^2 \frac{θ}{2} + 2\sin \frac{θ}{2} \cos \frac{θ}{2}}{2\sin^2 \frac{θ}{2} + 2\sin \frac{θ}{2} \cos \frac{θ}{2}}$$

**Step 3: Factor numerator and denominator**
$$= \frac{2\cos \frac{θ}{2}(\cos \frac{θ}{2} + \sin \frac{θ}{2})}{2\sin \frac{θ}{2}(\sin \frac{θ}{2} + \cos \frac{θ}{2})}$$

**Step 4: Cancel common factors**
$$= \frac{\cos \frac{θ}{2}}{\sin \frac{θ}{2}}$$

**Final Answer:** $$\boxed{\cot \frac{θ}{2}}$$ ✓

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### Problem 10: Half-Angle Cosine to Sine 🔢
**Question:** If $\cos \frac{α}{2} = \frac{1}{2}$, find the value of $\sin α$.

**🎯 Problem Type:** Half-angle cosine to full angle sine
**🧠 Key Insight:** Use double angle formula for sine

**Complete Solution:**

**Step 1: Apply double angle formula**
$$\sin α = 2\sin \frac{α}{2} \cos \frac{α}{2}$$

**Step 2: Find sin(α/2) using Pythagorean identity**
$$\sin^2 \frac{α}{2} = 1 - \cos^2 \frac{α}{2} = 1 - \left(\frac{1}{2}\right)^2 = 1 - \frac{1}{4} = \frac{3}{4}$$

**Step 3: Therefore**
$$\sin \frac{α}{2} = \pm\frac{\sqrt{3}}{2}$$

**Step 4: Calculate sin α**
$$\sin α = 2 \cdot \left(\pm\frac{\sqrt{3}}{2}\right) \cdot \frac{1}{2} = \pm\frac{\sqrt{3}}{2}$$

**Final Answer:** $$\boxed{\pm\frac{\sqrt{3}}{2}}$$ ✓

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### Problem 11: Third-Angle Cosine to Sine 🌟
**Question:** If $\cos \frac{α}{3} = \frac{1}{2}$, find the value of $\sin α$.

**🎯 Problem Type:** Third-angle cosine to full angle sine
**🧠 Key Insight:** Use triple angle formula and Pythagorean identity

**Complete Solution:**

**Step 1: Find sin(α/3) using Pythagorean identity**
$$\sin^2 \frac{α}{3} = 1 - \cos^2 \frac{α}{3} = 1 - \frac{1}{4} = \frac{3}{4}$$
$$\sin \frac{α}{3} = \pm\frac{\sqrt{3}}{2}$$

**Step 2: Apply triple angle formula**
$$\sin α = 3\sin \frac{α}{3} - 4\sin^3 \frac{α}{3}$$

**Step 3: Substitute values**
$$= 3 \cdot \left(\pm\frac{\sqrt{3}}{2}\right) - 4\left(\pm\frac{\sqrt{3}}{2}\right)^3$$
$$= \pm\frac{3\sqrt{3}}{2} - 4 \cdot \left(\pm\frac{3\sqrt{3}}{8}\right)$$

**Step 4: Simplify**
$$= \pm\frac{3\sqrt{3}}{2} \mp \frac{3\sqrt{3}}{2} = 0$$

**Final Answer:** $$\boxed{0}$$ ✓

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### Problem 12: Nested Half-Angle Formula 📐
**Question:** If $\cos 45° = \frac{1}{\sqrt{2}}$, prove that:
$$\sin 22.5° = \frac{\sqrt{2 - \sqrt{2}}}{2}$$

**🎯 Problem Type:** Nested half-angle calculation
**🧠 Key Insight:** Apply half-angle formula systematically

**Complete Solution:**

**Step 1: Recognize that 22.5° = 45°/2**
$$\sin 22.5° = \sin \frac{45°}{2}$$

**Step 2: Apply half-angle formula**
$$\sin \frac{45°}{2} = \sqrt{\frac{1 - \cos 45°}{2}}$$

**Step 3: Substitute known value**
$$= \sqrt{\frac{1 - \frac{1}{\sqrt{2}}}{2}} = \sqrt{\frac{\frac{\sqrt{2} - 1}{\sqrt{2}}}{2}}$$

**Step 4: Simplify**
$$= \sqrt{\frac{\sqrt{2} - 1}{2\sqrt{2}}} = \sqrt{\frac{(\sqrt{2} - 1)\sqrt{2}}{2\sqrt{2} \cdot \sqrt{2}}}$$
$$= \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2}$$

**Final Answer:** $$\boxed{\frac{\sqrt{2 - \sqrt{2}}}{2}}$$ ✓

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### Problem 13: Specific Angle Calculation 💫
**Question:** Find the value of $\sin 15°$ if $\cos 30° = \frac{\sqrt{3}}{2}$.

**🎯 Problem Type:** Half-angle from known value
**🧠 Key Insight:** Use 15° = 30°/2

**Complete Solution:**

**Step 1: Recognize relationship**
$$\sin 15° = \sin \frac{30°}{2}$$

**Step 2: Apply half-angle formula**
$$\sin \frac{30°}{2} = \sqrt{\frac{1 - \cos 30°}{2}}$$

**Step 3: Substitute known value**
$$= \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{\frac{2 - \sqrt{3}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}}$$

**Final Answer:** $$\boxed{\frac{\sqrt{2 - \sqrt{3}}}{2}}$$ ✓

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### Problem 14: Third-Angle Sine Direct 🔥
**Question:** If $\sin \frac{α}{3} = \frac{1}{2}$, find the value of $\sin α$.

**🎯 Problem Type:** Direct third-angle sine application
**🧠 Key Insight:** Apply triple angle formula directly

**Complete Solution:**

**Step 1: Apply triple angle formula**
$$\sin α = 3\sin \frac{α}{3} - 4\sin^3 \frac{α}{3}$$

**Step 2: Substitute given value**
$$= 3 \cdot \frac{1}{2} - 4\left(\frac{1}{2}\right)^3 = \frac{3}{2} - 4 \cdot \frac{1}{8}$$

**Step 3: Simplify**
$$= \frac{3}{2} - \frac{1}{2} = 1$$

**Final Answer:** $$\boxed{1}$$ ✓

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### Problem 15: Final Half-Angle Identity ⭐
**Question:** Prove that:
$$\frac{1 + \cos θ}{\sin θ} = \cot \frac{θ}{2}$$

**🎯 Problem Type:** Standard half-angle cotangent form
**🧠 Key Insight:** Direct application of half-angle substitution

**Complete Solution:**

**Step 1: Apply half-angle substitutions**
$$1 + \cos θ = 2\cos^2 \frac{θ}{2}$$
$$\sin θ = 2\sin \frac{θ}{2} \cos \frac{θ}{2}$$

**Step 2: Substitute**
$$\frac{1 + \cos θ}{\sin θ} = \frac{2\cos^2 \frac{θ}{2}}{2\sin \frac{θ}{2} \cos \frac{θ}{2}}$$

**Step 3: Simplify**
$$= \frac{\cos \frac{θ}{2}}{\sin \frac{θ}{2}}$$

**Final Answer:** $$\boxed{\cot \frac{θ}{2}}$$ ✓

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## ❌ Common Student Mistakes (Avoid These!)

### 🚫 **Mistake #1: Sign Confusion in Half-Angle Formulas**
**The Error:** Forgetting ± signs or choosing wrong quadrant
**The Fix:** Always determine quadrant first, then choose appropriate sign

### 🚫 **Mistake #2: Incorrect Triple Angle Application**
**The Error:** Confusing $\sin 3A$ with $3\sin A$
**The Fix:** Memorize: $\sin 3A = 3\sin A - 4\sin^3 A$

### 🚫 **Mistake #3: Algebraic Expansion Errors**
**The Error:** Incorrect expansion of $(a + \frac{1}{a})^3$
**The Fix:** Use systematic FOIL or binomial theorem

### 🚫 **Mistake #4: Mixed-Up Formulas**
**The Error:** Using cosine formula for sine problems
**The Fix:** Create reference sheet with clear distinctions

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## 🎓 Examination Success Strategies

### ⏱️ **Time Management for Sub-Multiple Problems**
1. **Quick Recognition** (30 seconds): Identify problem type (half-angle vs third-angle)
2. **Formula Selection** (1 minute): Choose appropriate identity
3. **Systematic Solution** (4-6 minutes): Apply chosen method step-by-step
4. **Verification** (1 minute): Check using alternative approach

### 📝 **Problem Type Distribution**
- **Direct Calculations**: 30% (finding specific angle values)
- **Identity Proofs**: 50% (proving relationships)
- **Complex Conversions**: 20% (third-angle problems)

### 🎯 **Master Strategy Checklist**
- ✅ All half-angle formulas memorized
- ✅ Triple angle relationships mastered
- ✅ Algebraic manipulation skills sharp
- ✅ Sign determination systematic
- ✅ Verification methods ready

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**Remember:** Sub-multiple angles bridge elementary and advanced trigonometry. Master these relationships, and you'll unlock powerful tools for solving complex trigonometric problems with confidence and precision!

**Final Encouragement:** Every complex sub-multiple identity breaks down into systematic application of fundamental formulas. With practice, you'll recognize patterns instantly and solve with mathematical elegance!
