$ ext{Prove the relationship between the sine and cosine of the sum of two angles using the unit circle.}$

To prove the relationship between the sine and cosine of the sum of two angles, we use the unit circle and the definitions of sine and cosine:

Given two angles, $\alpha$ and $\beta$, we can represent their sums on the unit circle. Consider the points $(\cos(\alpha), \sin(\alpha))$ and $(\cos(\beta), \sin(\beta))$.

Using the unit circle and the angle addition formulas, we have:

$ \cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) – \sin(\alpha) \sin(\beta) $

$ \sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta) $

These relationships can be derived by examining the projections of the points on the unit circle and considering the definitions of sine and cosine in terms of coordinates.

To prove the relationship between the sine and cosine of the sum of two angles, we use the unit circle properties:

Let’s start with two angles, $alpha$ and $eta$. The unit circle helps us to visualize their sums through the coordinates:

$ (cos(alpha), sin(alpha)) $ and $ (cos(eta), sin(eta)) $

We then apply the angle addition formulas:

$ cos(alpha + eta) = cos(alpha) cos(eta) – sin(alpha) sin(eta) $

$ sin(alpha + eta) = sin(alpha) cos(eta) + cos(alpha) sin(eta) $

To understand these formulas, consider the rotational transformations and how the coordinates transform under the addition of angles.

To prove $ cos(alpha + eta) $ and $ sin(alpha + eta) $, use the unit circle and the angle addition formulas:

$ cos(alpha + eta) = cos(alpha) cos(eta) – sin(alpha) sin(eta) $

$ sin(alpha + eta) = sin(alpha) cos(eta) + cos(alpha) sin(eta) $

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