Developing the sine and cosine of the sum of two angles

This is one of the compound angle formulae in the course.

You can use paper and pencil to follow. [br]Click on the green arrow to explore the development of the sine of the sum of two angles.[br]The initial screen is step 0, with each subsequent step being 1, 2, etc. [br]Follow each step in detail so you can answer the questions that follow.

1. About the inscribed right angle triangle. Refer to step 3.

Explain why the adjacent and opposite sides of this triangle are [math]cos\beta[/math] and [math]sin\beta[/math]? [br]

2. About the right angle triangle below the initial triangle. Refer to step 4.

Explain why the adjacent and opposite sides of the triangle at the bottom (the one with angle [math]\alpha[/math] )[br]are [math]cos\alpha\cdot cos\beta[/math] and [math]sin\alpha\cdot cos\beta[/math][br][i][br]Hint: Use the definition of the cosine and sine ratios of angle [math]\alpha[/math] in this triangle and keep in mind that you are finding sides.[/i]

3. The sides shown on step 5

Explain why these sides are labeled [math]sin\alpha\cdot sin\beta[/math] and [math]sin\beta\cdot cos\alpha[/math] ?

4. The sides shown on step 6

Explain the labels on the last two sides, meaning [math]sin\left(\alpha+\beta\right)[/math] and [math]cos\left(\alpha+\beta\right)[/math]

5. Making sense of the formulae. Refer to the steps 7 and onward

Explain how the compound angle formulae for the sine and cosine of the sum of two angles are finally determined based on the diagram and on previous steps.