ReoCLIK - Some useful formulas - Trigonometry

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• Mathematics > Trigonometry > Some useful formulas

Some useful formulas

Cosine function

Some properties of the cosine function (cos) :

c o s (- x) = c o s (x)

c o s (π + x) = - c o s (x)

c o s (π - x) = - c o s (x)

c o s (\frac{π}{2} + x) = - s i n (x)

c o s (\frac{π}{2} - x) = s i n (x)

c o s (x + y) = c o s (x) * c o s (y) - s i n (x) * s i n (y)

c o s (x - y) = c o s (x) * c o s (y) + s i n (x) * s i n (y)

\begin{aligned} c o s (2 x) & = c o s^{2} (x) - s i n^{2} (x) \\ = 1 - 2 * s i n^{2} (x) \\ = 2 * c o s^{2} (x) - 1 \end{aligned}

\begin{aligned} c o s (3 x) & = c o s (x) * (1 - 4 * s i n^{2} (x)) \\ = c o s (x) * (4 * c o s^{2} (x) - 3) \end{aligned}

c o s^{2} (\frac{x}{2}) = \frac{1 + c o s (x)}{2}

c o s (x) + c o s (y) = 2 * c o s (\frac{x + y}{2}) * c o s (\frac{x - y}{2})

c o s (x) - c o s (y) = - 2 * s i n (\frac{x + y}{2}) * s i n (\frac{x - y}{2})

c o s (x) * c o s (y) = \frac{1}{2} (c o s (x + y) + c o s (x - y))

c o s (x) * s i n (y) = \frac{1}{2} (s i n (x + y) - s i n (x - y))

Sine function

Some properties of the sine function (sin) :

s i n (- x) = - s i n (x)

s i n (π + x) = - s i n (x)

s i n (π - x) = s i n (x)

s i n (\frac{π}{2} + x) = c o s (x)

s i n (\frac{π}{2} - x) = c o s (x)

s i n (x + y) = s i n (x) * c o s (y) + c o s (x) * s i n (y)

s i n (x - y) = s i n (x) * c o s (y) - c o s (x) * s i n (y)

s i n (2 x) = 2 * s i n (x) * c o s (x)

\begin{aligned} s i n (3 x) & = s i n (x) (4 * c o s^{2} (x) - 1) \\ = s i n (x) * (3 - 4 * s i n^{2} (x)) \end{aligned}

s i n^{2} (\frac{x}{2}) = \frac{1 - c o s (x)}{2}

s i n (x) + s i n (y) = 2 * s i n (\frac{x + y}{2}) * c o s (\frac{x - y}{2})

s i n (x) - s i n (y) = 2 * c o s (\frac{x + y}{2}) * s i n (\frac{x - y}{2})

s i n (x) * s i n (y) = \frac{1}{2} (- c o s (x + y) + c o s (x - y))

s i n (x) * c o s (y) = \frac{1}{2} (s i n (x + y) + s i n (x - y))

Tangent function

Some properties of the tangent function (tan) :

t a n (- x) = - t a n (x)

t a n (π + x) = t a n (x)

t a n (π - x) = - t a n (x)

t a n (\frac{π}{2} + x) = - c o t a n (x)

t a n (\frac{π}{2} - x) = c o t a n (x)

t a n (x + y) = \frac{t a n (x) + t a n (y)}{1 - t a n (x) * t a n (y)}

t a n (x - y) = \frac{t a n (x) - t a n (y)}{1 + t a n (x) * t a n (y)}

t a n (2 x) = \frac{2 * t a n (x)}{1 - t a n^{2} (x)}

t a n (3 x) = \frac{t a n (x) (3 - t a n^{2} (x))}{1 - 3 * t a n^{2} (x)}

t a n^{2} (\frac{x}{2}) = \frac{1 - c o s (x)}{1 + c o s (x)}

\begin{aligned} t a n (\frac{x}{2}) & = \frac{1 - c o s (x)}{s i n (x)} \\ = \frac{s i n (x)}{1 + c o s (x)} \end{aligned}

t a n (x) + t a n (y) = \frac{s i n (x + y)}{c o s (x) * c o s (y)}

t a n (x) - t a n (y) = \frac{s i n (x - y)}{c o s (x) * c o s (y)}

If you have any comments or questions about the trigonometry formulas, you can discuss them in the forum: Discussion forums.

CHALLENGE : If you want to practice demonstrating any of these formulas, you can propose a proof in the forum: Discussion forums.

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