Circle equations, arcs and unit-circle trigonometry
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| radian/ˈreɪdɪən/ | 弧度 | hú dù |
| tangent/ˈtændʒənt/ | 切线 | qiè xiàn |
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:
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:
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.
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.
Which answer fits this case?
Recover circle centre and radius by completing the square
The angle between a tangent and its radius at contact is 90 degrees.
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.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.