Trigonometry (advanced)

Welcome to *the only place where the trigonometric values on unit circle is properly connected to the trigonometric ratios of right-triangles*.

• The trigonometric ratios of right-triangles are defined as -- "The right-triangles having a given angle are similar. The ratio of sides for those right-triangles is a known constant".

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The trigonometric values on unit-circle are defined as -- "The representative similar triangle is taken in unit-circle with hypotenuse $1$. The trigonometric ratios become into the horizontal and vertical projections.".

The details explained are *ingenious and found nowhere else*.

The trigonometric values for all quadrants and for compound angles are also covered.

Trigonometric Values: Unit Circle Form

In this page, the trigonometric values in unit circle form is introduced and explained.

Trigonometric Values in Four Quadrants

This page explains the quadrants, various angles in different quadrants, trigonometric ratios of such angles.

Trigonometric values for Any Angle: First Principles

This page explains the first principles to calculate a trigonometric ratio for a given angle. The rest of the knowledge extends the first principles to arrive at standard formulas or results.

First Quadrant Equivalent of Trigonometric Values

For angles in 2nd, 3rd, and 4th quadrants, can trigonometric ratios be equivalently given as a trigonometric ratio of acute angle in 1st quadrant? In doing so, the sign and the complementary trigonometric ratios are to be appropriately matched.

Angles in 2nd Quadrant

The results of representing trigonometric ratios of angles in the 2nd quadrant equivalently as trigonometric ratios of acute angle in the 1st quadrant are explained.

Compound Angles: Geometrical Proof for $\mathrm{sin}(A+B)$

In this page, a simple and intuitive geometrical proof is explained for expressing $\mathrm{sin}(A+B)$ in terms of $\mathrm{sin}A$, $\mathrm{sin}B$, etc.

Compound Angles: cos(A+B), sin(A-B), cos(A-B)

In this page, a simple proof, based on the earlier result of $\mathrm{sin}(A+B)$, is explained for expressing $\mathrm{cos}(A+B)$, $\mathrm{sin}(A-B)$, $\mathrm{cos}(A-B)$ in terms of $\mathrm{sin}A$, $\mathrm{sin}B$, $\mathrm{cos}A$ , and $\mathrm{cos}B$.

Compound Angles: tan(A+B) and tan(A-B)

In this page, expressing $\mathrm{tan}(A+B)$ and $\mathrm{tan}(A-B)$ in terms of $\mathrm{tan}A$ and $\mathrm{tan}B$ is explained.

More results for trigonometric values of Compound Angles

In this page, some additional results involving trigonometric function of compound angle is discussed.