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Trigonometric Functions and the Unit Circle

  • Grade 12
  • Mathematics
  • F-TF.A.2
  • Functions
  • Free lesson

Aligned to F-TF.A.2 — Common Core State Standards for Mathematics.

What this lesson teaches

On the unit circle (radius 1), the cosine of an angle is the x-coordinate and the sine is the y-coordinate. This extends trig to all angles, not just those in right triangles.

Worked examples

Example 1

At 90° the point is (0, 1), so cos(90°) = 0 and sin(90°) = 1. At 0° the point is (1, 0).

Example 2

At 180° the unit-circle point is (−1, 0), so cos(180°) = −1 and sin(180°) = 0. At 270° the point is (0, −1), giving cos(270°) = 0 and sin(270°) = −1. Reading coordinates off the circle replaces memorizing a table: the x-coordinate IS the cosine and the y-coordinate IS the sine.

Example 3

Common mistake: swapping which coordinate is which. At 30° the point is (√3/2, 1/2), so cos(30°) = √3/2 and sin(30°) = 1/2 — not the other way round. The anchor: cosine comes first alphabetically and x comes first in a coordinate pair. Sketch the angle and ask whether the point is far right (large cosine) or high up (large sine) before writing anything down.

Practice questions

  1. What are sine and cosine of 0°?
  2. On the unit circle, what coordinate gives the cosine?
  3. Find sin(180°) using the unit circle.

Watch the lesson

Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.

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En español

Funciones Trigonométricas y el Círculo Unitario

En el círculo unitario (radio 1), el coseno de un ángulo es la coordenada x y el seno es la coordenada y. Esto extiende la trigonometría a todos los ángulos, no solo a los de los triángulos rectángulos.

Ejemplo 1: En 90° el punto es (0, 1), entonces cos(90°) = 0 y sin(90°) = 1. En 0° el punto es (1, 0).

Ejemplo 2: En 180° el punto del círculo unitario es (−1, 0), así que cos(180°) = −1 y sen(180°) = 0. En 270° el punto es (0, −1), lo que da cos(270°) = 0 y sen(270°) = −1. Leer coordenadas en el círculo reemplaza memorizar una tabla: la coordenada x ES el coseno y la coordenada y ES el seno.

Ejemplo 3: Error común: intercambiar qué coordenada es cuál. En 30° el punto es (√3/2, 1/2), así que cos(30°) = √3/2 y sen(30°) = 1/2 — no al revés. El anclaje: coseno va primero alfabéticamente y x va primero en un par de coordenadas. Dibuja el ángulo y pregúntate si el punto está muy a la derecha (coseno grande) o muy arriba (seno grande) antes de escribir nada.

More Functions lessons

Equivalent Forms of a FunctionF-IF.C.8Comparing Functions in Different FormsF-IF.C.9Building Functions from Two QuantitiesF-BF.A.1Transformations of Function GraphsF-BF.B.3Inverse FunctionsF-BF.B.4Radian Measure and Arc LengthF-TF.A.1

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