# Study Notes: Proving the Addition Formulas for Sine and Cosine In this guide, we will derive the **addition formulas** for sine and cosine, which express $\sin(a + b)$ and $\cos(a + b)$ in terms of the sines and cosines of the individual angles $a$ and $b$. These formulas are fundamental in trigonometry and have wide applications in mathematics, physics, and engineering. --- ## Key Concepts: Sine and Cosine Definitions Before diving into the proof, let’s recall the definitions of sine and cosine in a right triangle: - **Cosine**: $\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{x}{r}$ - **Sine**: $\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{y}{r}$ If we solve for the legs of the triangle in terms of the hypotenuse $r$ and the angle $\theta$: - $x = r \cos(\theta)$ - $y = r \sin(\theta)$ These relationships allow us to calculate the sides of a right triangle if we know the hypotenuse and the angle. --- ## The Diagram Setup To prove the addition formulas, we use a geometric approach involving two right triangles: 1. **Lower Triangle**: - Angle: $a$ - Hypotenuse: $r$ - Sides: $r \cos(a)$ (adjacent) and $r \sin(a)$ (opposite) 2. **Upper Triangle**: - Angle: $b$ - Hypotenuse: $1$ - Sides: $\cos(b)$ (adjacent) and $\sin(b)$ (opposite) We combine these triangles into a single diagram and add auxiliary lines to create additional right triangles. This setup allows us to express the sides of the triangles in terms of $a$, $b$, $\sin$, and $\cos$. --- ## Proving $\sin(a + b)$ ### Step 1: Identify the Target The goal is to find the length of the vertical side of the large triangle with: - Hypotenuse = $1$ - Angle = $a + b$ This vertical side is equal to $\sin(a + b)$. ### Step 2: Break the Vertical Side into Two Segments The vertical side is composed of two smaller segments: 1. The vertical segment from the lower triangle: $\sin(b) \cos(a)$ 2. The vertical segment from the upper triangle: $\cos(b) \sin(a)$ ### Step 3: Add the Segments The total vertical side is the sum of these two segments: $$ \sin(a + b) = \sin(b) \cos(a) + \cos(b) \sin(a) $$ ### Final Formula Rewriting the terms in a standard order: $$ \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) $$ If $a = 30^\circ$ and $b = 45^\circ$, then: - $\sin(30^\circ) = \frac{1}{2}$, $\cos(30^\circ) = \frac{\sqrt{3}}{2}$ - $\sin(45^\circ) = \frac{\sqrt{2}}{2}$, $\cos(45^\circ) = \frac{\sqrt{2}}{2}$ Using the formula: $$ \sin(30^\circ + 45^\circ) = \sin(30^\circ)\cos(45^\circ) + \cos(30^\circ)\sin(45^\circ) $$ Substitute values: $$ \sin(75^\circ) = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4} $$ --- ## Proving $\cos(a + b)$ ### Step 1: Identify the Target The goal is to find the length of the horizontal side of the large triangle with: - Hypotenuse = $1$ - Angle = $a + b$ This horizontal side is equal to $\cos(a + b)$. ### Step 2: Break the Horizontal Side into Two Segments The horizontal side is composed of two smaller segments: 1. The horizontal segment from the lower triangle: $\cos(b) \cos(a)$ 2. The horizontal segment from the upper triangle: $\sin(b) \sin(a)$ ### Step 3: Subtract the Segments The total horizontal side is the difference between these two segments: $$ \cos(a + b) = \cos(b) \cos(a) - \sin(b) \sin(a) $$ ### Final Formula Rewriting the terms in a standard order: $$ \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) $$ If $a = 30^\circ$ and $b = 45^\circ$, then: - $\sin(30^\circ) = \frac{1}{2}$, $\cos(30^\circ) = \frac{\sqrt{3}}{2}$ - $\sin(45^\circ) = \frac{\sqrt{2}}{2}$, $\cos(45^\circ) = \frac{\sqrt{2}}{2}$ Using the formula: $$ \cos(30^\circ + 45^\circ) = \cos(30^\circ)\cos(45^\circ) - \sin(30^\circ)\sin(45^\circ) $$ Substitute values: $$ \cos(75^\circ) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} $$ --- ## Key Takeaways 1. **Addition Formulas**: - $\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b)$ - $\cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b)$ 2. **Geometric Insight**: - The proofs rely on breaking down the sides of a large triangle into smaller segments using trigonometric relationships. 3. **Applications**: - These formulas are essential for solving problems involving angle sums, wave interference, and oscillatory motion. A common mistake is to confuse the signs in the formulas. Remember: - For $\sin(a + b)$, the terms are **added**. - For $\cos(a + b)$, the terms are **subtracted**. To remember the formulas, think of "Sine Same, Cosine Change": - $\sin(a + b)$ has the **same** sign as the operation (+). - $\cos(a + b)$ has the **opposite** sign (−). --- ## Practice Problems 1. Prove $\sin(a - b)$ and $\cos(a - b)$ using similar reasoning. 2. Calculate $\sin(105^\circ)$ and $\cos(105^\circ)$ using the addition formulas. 3. Verify the formulas for specific angles, such as $a = 60^\circ$ and $b = 30^\circ$.