Unit circle & exact values
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Unit-circle coordinates at angle θ?
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All Flashcards in Topic 3.5
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3.5.19 cards
Unit-circle coordinates at angle θ?
(cos θ, sin θ): cos is x, sin is y.
Exact sin/cos of 30°?
sin 30° = ½, cos 30° = √3/2.
Exact sin/cos of 45°?
sin 45° = cos 45° = √2/2 (= 1/√2).
Exact sin/cos of 60°?
sin 60° = √3/2, cos 60° = ½.
tan of 30°, 45°, 60°?
1/√3, 1, √3.
What is CAST?
Positive ratios by quadrant: All (Q1), Sin (Q2), Tan (Q3), Cos (Q4).
sin(180° − θ) = ?
sin θ — supplementary angles share the same sine.
cos(180° − θ) = ?
−cos θ.
Given cos θ = 2/3 (acute), find sin θ.
sin²θ = 1 − 4/9 = 5/9 ⇒ sin θ = √5/3.
3.5.28 cards
When does the ambiguous case occur?
Using the sine rule to find an ANGLE (two sides + a non-included angle, SSA).
Why are there two possible angles?
Because sin θ = sin(180° − θ) — an acute and an obtuse angle share the same sine.
How do you get the second angle?
Subtract the acute sin⁻¹ value from 180°.
Is finding a side ambiguous?
No — only finding an angle with the sine rule can give two answers.
Is the cosine rule ambiguous for angles?
No — cos⁻¹ gives a single angle in 0°–180°.
How do you check if the obtuse triangle is valid?
Add it to the known angle; keep it only if the total is under 180°.
sin B = 0.6 — find both angles.
B ≈ 36.9° or 180° − 36.9° = 143.1°.
A = 70°, B = 50° or 130° — which are valid?
Only 50°, since 70° + 130° = 200° > 180°.
Topic 3.5 study notes
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Math AA SL exam skills
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