LEARN · EXPLAIN · REVISE

Explain why tangent is undefined at 90°.

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

高一必修第一册(A版).pdf · 5.2 · PDF 184 / printed page 177

TOPIC 01

Concept of trigonometric functions

Build understanding of concept of trigonometric functions through definitions, contrasting cases and justified applications.

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

01 / Foundation#Your turn

\tan\theta=\sin\theta/\cos\theta

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use coordinates on the terminal ray and a positive radius.

Hint 2

Locate the point at the top of the unit circle.

Worked solution

cos 90° is zero, so the ratio divides by zero.

Checks and common pitfalls: Undefined is different from an extremely large real value.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

What must be true before using the main rule for concept of trigonometric functions?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Use concept of trigonometric functions with explicit angle units and domains.
Coordinate definitions: Use a positive radius and signed coordinates.
$\sin\theta=y/r,\quad\cos\theta=x/r$
Basic identity: The unit-circle equation gives sine squared plus cosine squared equals one.
$\sin^2\theta+\cos^2\theta=1$
Close with: conditions → representation → reasoning → check.

Expected reasoning: Use a positive radius and signed coordinates.
Expected reasoning: The unit-circle equation gives sine squared plus cosine squared equals one.
Expected correction: Select signs using the quadrant after taking a square root.