Trigonometry

Arc half division and tangent formulas

Isolate the squares We start with the identities $\cos2u=1-2\sin^2u$ and $\cos2u=2\cos^2u-1$. Making $u=x/2$ and isolating: $$\sin^2\frac x2=\frac{1-\cos x}{2},\qquad \cos^2\frac x2=\frac{1+\cos x}{2}.$$Consequently: $$\sin\frac x2=\pm\sqrt{\frac{1-\cos x}{2}},\qquad \cos\frac x2=\pm\sqrt{\frac{1+\cos x}{2}}.$$The signs are not a free choice: they are determined by the image quadrant of $x/2$. If $0

October 21, 2026 Read

Sine, cosine and tangent of double and triple arc

An addition formula applied to itself Replace $b=a$ in addition formulas: $$\sin2a=2\sin a\cos a,$$ $$\cos2a=\cos^2a-\sin^2a.$$The fundamental identity allows two other forms for the cosine: $$\cos2a=1-2\sin^2a=2\cos^2a-1.$$The choice depends on the data available. If we only know the sine, the first alternative expression avoids calculating the cosine and choosing its sign. If tangents exist: $$\tan2a=\frac{2\tan a}{1-\tan^2a}.$$The form with $\tan a$ requires $\cos a\ne0$ and $1-\tan^2a\ne0$; does not cover all cases where $\tan2a$ exists.

October 21, 2026 Read

Arc addition and subtraction formulas

Rotate a vector Consider the horizontal and vertical unit vectors. After a rotation of $a$, they start to point, respectively, to $(\cos a,\sin a)$ and $(-\sin a,\cos a)$. The second is perpendicular to the first, in the positive direction. The point corresponding to $b$ is the sum of $\cos b$ times the horizontal vector with $\sin b$ times the vertical. Rotation preserves this decomposition. After turning $a$: $$\begin{aligned} (\cos(a+b),\sin(a+b)) &=\cos b(\cos a,\sin a)\\ &\quad+\sin b(-\sin a,\cos a). \end{aligned}$$Demonstrate and assemble the formulas Comparing the coordinates, we obtain the sum formulas. Replacing $b$ with $-b$ and using parity gives the difference:

October 20, 2026 Read

Complements, reduction identities, and parity

Reduce the table further In a right-angled triangle, the acute angles are complementary. The side opposite to one is adjacent to the other. Like this: $$\sin\left(\frac\pi2-x\right)=\cos x,\qquad \cos\left(\frac\pi2-x\right)=\sin x.$$For $\pi/4\le x\le\pi/2$, the complement $\pi/2-x$ belongs to $[0,\pi/4]$. In this way, values in the second half of the first quadrant can be calculated from those in the first half, exchanging sine for cosine. Also: $$\tan\left(\frac\pi2-x\right)=\cot x,$$ $$\sec\left(\frac\pi2-x\right)=\csc x,$$in their respective domains. The geometric interpretation by exchanging the legs demonstrates the relationships for acute angles; reading the cycle coordinates extends the identities to the other allowed values.

October 20, 2026 Read

Reduction of the second, third and fourth quadrants

A point and its reflections Choose $0<\alpha<\pi/2$, whose image is $(\cos\alpha,\sin\alpha)$. A reflection on the vertical axis changes the sign of the first coordinate; a reflection on the horizontal axis exchanges that of the second; symmetry about the origin exchanges both. These three operations generate the points of $\pi-\alpha$, $\pi+\alpha$ and $2\pi-\alpha$. Logo: Angle Sine Cosine Tangent $\pi-\alpha$ $\sin\alpha$ $-\cos\alpha$ $-\tan\alpha$ $\pi+\alpha$ $-\sin\alpha$ $-\cos\alpha$ $\tan\alpha$ $2\pi-\alpha$ $-\sin\alpha$ $\cos\alpha$ $-\tan\alpha$ Demonstration for the second quadrant The reflection of $P=(u,v)$ on the vertical axis is $P'=(-u,v)$. The height remains the same, but the horizontal position becomes negative. Since $P'$ corresponds to angle $\pi-\alpha$:

October 20, 2026 Read

Identities and three demonstration paths

An equality between functions A trigonometric identity is an equality valid for all entries of a declared domain. It is necessary to check both the values and the points at which the expressions exist. For example, $\sec^2x-\tan^2x=1$ is valid when $\cos x\ne0$. The right side also exists in the excluded points, but this does not extend the domain of the expression to the left. First way: transform one of the members To demonstrate $\sin x\tan x+\cos x=\sec x$, with $\cos x\ne0$, we start from the left:

October 20, 2026 Read

Fundamental relationships demonstrated

Pythagoras in the circle The point $P=(\cos x,\sin x)$ is one unit away from the origin. The projections on the axes form a right-angled triangle with sides $|\cos x|$ and $|\sin x|$. Applying Pythagoras, we obtain: $$\sin^2x+\cos^2x=1.$$On the axes, equality is verified directly: one coordinate is zero and the other has modulus 1. Therefore, the identity holds for every real $x$. Bring relationships together The definitions and previous identity provide:

October 20, 2026 Read

Cotangent, secant and cosecant

Three quotients, three domains The reciprocal functions expand the circular family: $$\cot x=\frac{\cos x}{\sin x},\qquad \sec x=\frac1{\cos x},\qquad\csc x=\frac1{\sin x}.$$ Function Points excluded Image Fundamental period $\cot x$ $x=k\pi$ $\mathbb R$ $\pi$ $\sec x$ $x=\pi/2+k\pi$ $(-\infty,-1]\cup[1,\infty)$ $2\pi$ $\csc x$ $x=k\pi$ $(-\infty,-1]\cup[1,\infty)$ $2\pi$ In the entire table, $k\in\mathbb Z$. The cotangent is not defined by $1/\tan x$ in its entire domain: in $x=\pi/2$, the cotangent is worth 0, while the tangent does not exist. The identity $\cot x=1/\tan x$ requires that both expressions exist.

October 20, 2026 Read

Tangent function: slope, domain and graph

Compare coordinates In the first quadrant, the straight line of the radius that passes through $P=(\cos x,\sin x)$ has a slope equal to the height divided by the horizontal displacement. We extend this ratio to all angles where the denominator is not zero: $$\tan x=\frac{\sin x}{\cos x}.$$Logo: $$\operatorname{Dom}(\tan)=\mathbb R\setminus \left\{\frac\pi2+k\pi:k\in\mathbb Z\right\}.$$The image is $\mathbb R$. The tangent is positive in quadrants I and III and negative in quadrants II and IV. Its zeros are $x=k\pi$.

October 20, 2026 Read

Cosine function: horizontal coordinate and graph

The width with sign For $P=(u,v)$ in the cycle, we define $\cos x=u$. The cosine is positive in the right semicircle and negative in the left. Its domain is $\mathbb R$ and its image is $[-1,1]$. The axis points provide: $x$ $0$ $\pi/2$ $\pi$ $3\pi/2$ $2\pi$ $\cos x$ $1$ $0$ $-1$ $0$ $1$ Zeros are $x=\pi/2+k\pi$. The maxima occur at $x=2k\pi$ and the minima at $x=\pi+2k\pi$, with $k\in\mathbb Z$. Demonstration of parity and period The points of $x$ and $-x$ are reflections on the horizontal axis. Its horizontal coordinate is equal:

October 20, 2026 Read

Sine function: definition, signs and graph

The height of the point If the image of the real $x$ in the cycle is $P=(u,v)$, we define $\sin x=v$. The sine is the vertical, signed coordinate, not the unsigned distance to the horizontal axis. Any real has an image in the cycle. The height varies from $-1$ to 1, including the extremes. Therefore: $$\operatorname{Dom}(\sin)=\mathbb R,\qquad\operatorname{Im}(\sin)=[-1,1].$$In each turn, the sine is positive in the upper semicircle and negative in the lower one. Its zeros are $x=k\pi$, to $k\in\mathbb Z$.

October 20, 2026 Read

Circular functions and periodicity

From movement to a function A function associates a single output with each input of the domain. When making a point go through the trigonometric cycle, we can observe its height, its horizontal position or other geometrically constructed quantities. Each choice produces a circular function. The input will generally be a real number representing the spin in radians. Even if two different inputs lead to the same point, each has a unique image per function. A function can repeat outputs.

October 20, 2026 Read