LEARN · EXPLAIN · REVISE

Find the tangent of the acute angle between slopes 0 and t.

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

高二選擇性必修第一册(A版).pdf · 2.1 · PDF 56 / printed page 51

Revisit first: Exponents

TOPIC 01

Inclination and slope of a line

Connect slope, inclination and parallel/perpendicular directions, including vertical lines.

Connect slope, inclination and parallel/perpendicular directions, including vertical lines.
Justify the method and check the conditions in a new situation.

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

01 / Standard#Your turn

t=11

A vertical line has no finite slope; m1m2=−1 applies only when both slopes exist.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate the geometric condition into a coordinate equation.

Hint 2

Use this intermediate relation.

\tan\theta=|(m_2-m_1)/(1+m_1m_2)|

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$\tan\theta=|(11-0)/(1+0)|=11$
The denominator is nonzero here; perpendicular lines require separate handling.

The requested value is 11.

Checks and common pitfalls: The denominator is nonzero here; perpendicular lines require separate handling.

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

Teacher preparation and assessment

Connect slope, inclination and parallel/perpendicular directions, including vertical lines.
Which condition is essential in inclination and slope of a line?
Does a steeper line always have a larger slope?

Slope: For a nonvertical line, slope is rise divided by run and equals tan of the inclination.
$m=(y_2-y_1)/(x_2-x_1)$
Exceptional directions: A vertical line has no finite slope; m1m2=−1 applies only when both slopes exist.

A vertical line is perpendicular to a horizontal line even though its slope is undefined.