g10-07-06 · Core topics
Where does tangent break?
Can a tangent graph pass through90°?
1 MIN · PREDICT ON PAPER
Predict tanθ at85°,90°,95°.
Discuss or write first. No typed answer is required to continue.
The quotient is defined at this input.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
Tangent is sinθ/cosθ. At90°+180°k its denominator is0, so these inputs are excluded, not assigned a very large value.
Conditions: θ≠90°+180°k for integer k. The basic period is180°.
Mathematical justification: Both sine and cosine reverse sign after180°, so their quotient is unchanged where defined. Cosθ=0 makes division invalid.
1 MIN · EXIT CHECK
State whether tan270° exists.
Check after answering
No: cos270°=0.
Teacher guidance and extension
Before: Angles, coordinates and basic function graphs.
Ask: Is a large finite table value the same as undefined?
Watch for: Joining the two sides of a vertical asymptote with a straight line.
Challenge: Compare tan30° and tan210° without calculating each separately.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
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