Trigonometric Functions by Unit Circle

For secondary school students, I define cosine and sine as the x and y-components of the point A (cos θ, sin θ) on the unit circle x² + y² = 1, and the tangent function as the quotient of sine over cosine.

\coordinate (O) at (0,0);
\coordinate (H) at (0.6,0);
\coordinate (A) at (0.6,0.8);
\coordinate (E) at (1,0);
\coordinate (T) at (1,0.8/0.6);
\draw (O) circle (1);
\draw[->] (-1.3,0) -- (1.3,0) node [right]{$x$};
\draw[->] (0,-1.3) -- (0,1.3) node [above]{$y$};
\draw (O) node [below left] {$O$}
    -- (H) node [below right] {$H$}
    node [below, midway] {$\cos \theta$}
    -- (A) node [below, midway, sloped] {$\sin \theta$}
    node [above=5pt] {$A$}
    -- cycle node [above left, midway] {$1$};
    \clip (O) -- (A) -- (H) -- cycle;
    \draw (O) circle (0.1) node[right=7pt, above=5pt, anchor=west] {\small $\theta$};
\draw (H) rectangle ++(-0.1,0.1);
\draw (E) rectangle ++(-0.1,0.1);
\draw (E) node [below right] {$E$}
    -- (T) node [below, midway, sloped] {$\tan \theta$}
    node [above, right] {$T$}
    -- (A);

sine cosine unit circle

This definition allows us to retrieve the famous “sohcahtoa” mnemonics and some elementary trigonometry identies like the Pythagorean identity and the complementary angle identity sin(90° − θ) = cos θ for all real-valued θ.

To see the previous identity for non-acute θ, it suffices to swap the role of x and y. That means a reflection along the line y = x on the graph. That might be a bit hard to imagine, so let’s consider a simpler reflection along the real line:

x---→ -----x----- ←---x
A          M          B

Points A and B are reflection of each other with respect to their midpoint M. One reflection flips the orientation of an arrow ‘→’/‘←’ once.

Getting back to the complementary angle identity, observe that the average of θ and 90° − θ is 45°, and the sign of θ on both sides are different. “cos(something)” represents the length of a horizontal line segment. After a reflection along y = x, the horizontal line segment becomes vertical.

axis x y
orientation horizontal vertical
neighbouring angle θ 90° − θ
chosen trigo funct sin cos

Remarks: In the last row of the above table, “sin” and “cos” can be swapped.


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