Trigonometric Circle: Identities, Tangent, and Cotangent

Vocabulary flashcards covering the trigonometric circle, Pythagorean identity, and the geometric definitions of tangent and cotangent using similar triangles.

Pythagorean Trigonometric Identity

The identity derived by applying the Pythagorean theorem in right triangle $OAC$, stating that $(\text{sin}(\alpha))^2 + (\text{cos}(\alpha))^2 = 1$ (or $\text{sin}^2(\alpha) + \text{cos}^2(\alpha) = 1$).

Cosine Axis

The horizontal axis in the trigonometric circle along which cosine values are measured.

Sine Axis

The vertical axis in the trigonometric circle along which sine values are measured.

Geometric Definition of Tangent ($\text{tan}(\alpha)$)

The trigonometric ratio defined geometrically by the directed segment length $\overline{ET}$ on the vertical tangent line $t$ drawn at point $E$, established via similar triangles $OAA'$ and $OET$ such that $\text{tan}(\alpha) = \frac{\text{sin}(\alpha)}{\text{cos}(\alpha)} = \overline{ET}$.

Geometric Definition of Cotangent ($\text{cot}(\alpha)$)

The trigonometric ratio defined geometrically by the directed segment length $\overline{BU}$ on the horizontal tangent line $m$ drawn at point $B$, established via similar triangles $OAA'$ and $OBU$ such that $\text{cot}(\alpha) = \frac{\text{cos}(\alpha)}{\text{sin}(\alpha)} = \overline{BU}$.

Line $t$ (Tangent Construction)

The line tangent to the trigonometric circle at point $E$, which intersects the line $(OA)$ at point $T$.

Line $m$ (Cotangent Construction)

The line tangent to the trigonometric circle at point $B$, which intersects the line $(OA)$ at point $U$.