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How are sine and cosine connected through Euler’s formula?

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Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

Euler’s formula connects sine and cosine through the complex exponential function using the equation:

$e^{i θ} = \cos θ + i \sin θ$

In this formula, $e$ is the base of the natural logarithm, $i$ is the imaginary unit ( $\sqrt{- 1}$ ), and $θ$ is an angle. By combining this expression with its companion formula for a negative angle, $e^{- i θ} = \cos θ - i \sin θ$ , one can derive explicit expressions for both functions in terms of complex exponentials:12

$\cos θ = \frac{e^{i θ} + e^{- i θ}}{2}$
$\sin θ = \frac{e^{i θ} - e^{- i θ}}{2 i}$

These relationships demonstrate that sine and cosine are essentially components of a single complex exponential function. Geometrically, this can be visualized on the unit circle, where $\cos θ$ represents the horizontal coordinate and $\sin θ$ represents the vertical coordinate of a point corresponding to the angle $θ$ . This unification of trigonometry and algebra is a cornerstone of modern analytic trigonometry.231