円の方程式、弧と単位円三角関数
導入済み| English | 意味 |
|---|---|
| radian/ˈreɪdɪən/ | ラジアン |
| tangent/ˈtændʒənt/ | 接線 |
A decision before an answer
- A radius squared of 25 means radius 5. The signs in (x-2)²+(y+3)² locate the centre at (2,-3).
- Your goal: Recover circle centre and radius by completing the square.
Read the relationship
- A circle with centre (h,k) and radius r satisfies (x-h)²+(y-k)²=r². For an expanded equation, complete the square in x and y and add the balancing constants to the other side. The resulting right side must be positive for a real nondegenerate circle. Keep the radius distinct from its square.
- Convert central angles and calculate arc length and sector area.
Centre and radius of (x+1)²+(y-4)²=9:
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The centre signs follow x-h,y-k and radius is √9.
Use the defining rule
- A central angle θ in radians gives arc length rθ and sector area r²θ/2. In degrees, use the fraction θ/360 of the full circumference or area. A full turn is 2π radians=360°. The formulas rθ and r²θ/2 require radians; inserting degrees without conversion gives a wrong scale.
- Use tangent, inscribed-angle and unit-circle relationships.
Radius 4 with central angle π/2 gives arc length:
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rθ=4·π/2=2π.
Check the conditions
- A tangent at a circle point is perpendicular to the radius there. An inscribed angle is half the central angle subtending the same arc; an angle subtending a diameter is a right angle. Identify the actual shared arc and centre before applying these statements. A chord is not a tangent and need not be perpendicular to a radius unless further conditions establish it.
- Use tangent, inscribed-angle and unit-circle relationships.
x²+y²-4x+6y-12=0 becomes (x-2)²+(y+3)²=25, giving centre (2,-3), radius 5. For radius 6 and central angle π/3, arc length is 2π and sector area 6π. An inscribed angle on that same 60° arc is 30°. At 150°, cosine is -√3/2 and sine 1/2 because the point is in quadrant II.
180 degrees equals ____ radians.
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A half-turn is π radians.
Apply the task format
- On the unit circle, a point at angle θ has coordinates (cos θ,sin θ), so cos²θ+sin²θ=1. Signs depend on quadrant. For acute complementary angles, sin θ=cos(90°-θ). Use special triangles for exact values: at 30° the sine is 1/2 and cosine √3/2; at 45° both are √2/2.
- Use tangent, inscribed-angle and unit-circle relationships.
Check degree/radian units, centre signs and whether an angle is central or inscribed before applying the formula.
このケースに当てはまる答えはどれか？
平方完成により円の中心と半径を復元する
The angle between a tangent and its radius at contact is 90 degrees.
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The tangent-radius perpendicular theorem applies at the contact point.
Keep the distinctions
- radian 弧度 — An angle measure equal to arc length divided by radius.
- tangent 切线 — A line meeting a circle at a point and perpendicular to its radius there.
- Recover circle centre and radius by completing the square.
- Convert central angles and calculate arc length and sector area.
- Use tangent, inscribed-angle and unit-circle relationships.
Match each term with its precise meaning in this lesson.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.