Sum/Difference & Double Angle Formulas

The double angle formula for sine is given by: $$\text{sin}(2\theta) = 2\text{sin}(\theta)\text{cos}(\theta)$$ .

The double angle formula for cosine is given by: $$\text{cos}(2\theta) = \text{cos}^2(\theta) - \text{sin}^2(\theta)$$ .

The sum formula for sine is given by: $$\text{sin}(\theta_1 + \theta_2) = \text{sin}(\theta_1)\text{cos}(\theta_2) + \text{cos}(\theta_1)\text{sin}(\theta_2)$$ .

The difference formula for sine is given by: $$\text{sin}(\theta_1 - \theta_2) = \text{sin}(\theta_1)\text{cos}(\theta_2) - \text{cos}(\theta_1)\text{sin}(\theta_2)$$ .

The sum formula for cosine is given by: $$\text{cos}(\theta_1 + \theta_2) = \text{cos}(\theta_1)\text{cos}(\theta_2) - \text{sin}(\theta_1)\text{sin}(\theta_2)$$ .

The difference formula for cosine is given by: $$\text{cos}(\theta_1 - \theta_2) = \text{cos}(\theta_1)\text{cos}(\theta_2) + \text{sin}(\theta_1)\text{sin}(\theta_2)$$ .

The identity is: $$\text{sin}^2(\theta) + \text{cos}^2(\theta) = 1$$ .

Use the formula: $$\text{sin}(2\theta) = 2\text{sin}(\theta)\text{cos}(\theta)$$ .

Using the sum formula: $$\text{sin}(105°) = \text{sin}(60° + 45°) = \text{sin}(60°)\text{cos}(45°) + \text{cos}(60°)\text{sin}(45°) = \frac{\text{√3}}{2} \times \frac{1}{\text{√2}} + \frac{1}{2} \times \frac{1}{\text{√2}} = \frac{\text{√3} + 1}{2\text{√2}}$$ .

The formula is: $$\text{tan}(\theta_1 - \theta_2) = \frac{\text{tan}(\theta_1) - \text{tan}(\theta_2)}{1 + \text{tan}(\theta_1)\text{tan}(\theta_2)}$$ .