LEARN · EXPLAIN · REVISE

The angle is in radians. What is its exact cosine?

Read the idea, work independently, then explain what changed.

TOPIC 01

Trigonometric ratios and function graphs

The positive √3/2 is the sine at this angle. Distinguish the horizontal coordinate (cosine) from the vertical coordinate (sine).

PREDICT → MOVE → EXPLAIN

See amplitude, period and phase together.

Predict the number of cycles and the horizontal shift. Switch angle units without changing the curve.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

The angle is in radians. What is its exact cosine?

\cos\frac{2\pi}{3}=?

Positive half $\tfrac12$
Negative half $-\tfrac12$
Positive square-root value $\tfrac{\sqrt3}{2}$
Negative square-root value $-\tfrac{\sqrt3}{2}$

Working and explanation

BUILD THE REASONING

Hint 1

Convert 2π/3 radians to degrees, or identify its quadrant directly.

Hint 2

It has reference angle π/3 in quadrant II. Cosine is negative in that quadrant.

Worked solution

Identify the reference angle.
$\frac{2\pi}{3}=\pi-\frac\pi3$
Reflect the horizontal coordinate on the unit circle.
$\cos\left(\pi-\frac\pi3\right)=-\cos\frac\pi3=-\frac12$

B: −1/2.

Checks and common pitfalls: The positive √3/2 is the sine at this angle. Distinguish the horizontal coordinate (cosine) from the vertical coordinate (sine).

Think first. Reveal a hint when the class is ready.