Deriving angle from sin or cos

Question

How can I derive the value in degrees of an angle starting from either the cos or sin value?

$$ \cos(t) = c_{1} \quad \text{or} \quad \sin(t) = c_{2} $$

Answer 1

Please refer to the trigonometric inverse functions, especially arcsine and arccosine.

Given that $\cos t = c_1$ and $\sin t = c_2$,

Fundamentally, for a certain range, $\displaystyle t = \arccos(c_1) = \arcsin(c_2)$

Since $\sin(x)$ and $\cos(x)$ are many-to-one functions, there will correspond multiple values of $x$ that yield a certain $\sin(x)$ or $\cos(x)$ and hence when we say $$f^{-1}\Big(f(x)\Big) = x $$ we mean that $x$ is the smallest positive value which produces a distinct $f(x)$

The inverse trig functions are, on most calculators, written as $\sin^{-1}$ and $\cos^{-1}$ and the functional inverses should not be confused with $\csc x$ or $\sec x$ which are the algebraic inverses.

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