Circular measure

1. Circular measure Definition of p Definition of radians

2. Unit 4:Mathematics Aims • Introduce radians and circular theorem. Objectives • Identify parts of a circle and calculate triangles within a circle. • Calculate circular and segment measures.

3. Re-Call • sine x = (side opposite x)/hypotenuse cosine x = (side adjacent x)/hypotenuse tangent x =(side opposite x)/(side adjacent x) • sin A = a/c, cosine A = b/c, & tangent A = a/b.

4. Re-Call • The reciprocal ratios are trigonometric ratios, too.  They are outlined below. • cotangent x = 1/tan x = (adjacent side)/(opposite side) • secant x = 1/cos x = (hypotenuse)/(adjacent side) • cosecant x = 1/sin x = (hypotenuse)/(opposite side)

5. Definition of  • Take any size of circle. Cis the Circumference, the distance around the outside. c d d is the Diameter.

6. Cis the Circumference, the distance around the outside. • d is the Diameter C 1 2 3 d C ynymwneud3gwaithd Cis about 3 times d

7. C 1 2 3 d C = 3.1415927 x d =3.1415927 (pi) This is true for any size of circle

8. Definition of radians s  = r + s  r

9. The picture below illustrates the relationship between the radius, and the central angle in radians. The formula is s = rθ where s represents the arc length, θ represents the central angle in radians and r is the length of the radius.

10. What is the value of the arc length s in the circle pictured below?

11. Calculate the measure of the arc length s in the circle pictured below?

12. Definition radians S  = r  = 2p radians r S = 2pr

13. radians _Degrees

15. Circumference • The circumference of a circle is the perimeter of the circle GylcheddCircumference

16. The diameter of a circle is a line across the circle which passes through the centre. • Radius • The radius of a circle is the distance from the centre of the circle to any point on the circumference. The radius is half the length of the diameter

17. Segment • A chord divides a circle into two segments: a minor segment and a major segment. Chord minor Segment bach Tant chord major Segment mawr

18. Tangent • A tangent is a line which touches the circumference of a circle at one point only and is parallel to the circumference at that point. • An arc is part of the circumference of a circle. • A sector is formed between 2 radii and the circumference arc tangent sector

19. Properties of a Circle • The angle in a semi-circle is always a right angle • If 2 chords are drawn from a point on the circumference of a circle to each end of a diameter the angle between the two chords is always a right angle.

20. The angle at the centre of a circle = twice the angle at the circumference If lines are drawn from each end of a chord to a point on the circumference of a circle and to the centre, the angle at the centre is twice the angle at the circumference.

21. Angles in the same segment are equal If two chords are drawn from a point on the circumference of a circle to each end of a third chord the intersecting angle is the same no matter where the point is providing the points are in the same segment of the circle.

22. Opposite angles in a cyclic quadrilateral are supplementary A cyclic quadrilateral is a quadrilateral whose vertices lie on the circumference of a circle. The opposite angles of a cyclic quadrilateral add up to 180

23. Tangents • A radius drawn from the point where a tangent touches a circle is perpendicular (at 90) to the tangent.

24. Tangents drawn to a circle from the same point outside the circle are equal in length. PA = PB • OP bisects angle APB A P O B

25. B is the centre of the circle. Find angles BPC, BCP, ABP and PAB. • BP = BC,  BPC = BCP • (isosceles triangle) • BPC + BCP = 180 – 66 = 114 • BPC = BCP = 57 • ABP = 180 – 66 = 114 • (angles on a straight line = 180) • PAB = ½ × 66 = 33 • (angle at centre of a circle = twice that at the circumference) P 66 A C B

26. Calculate angles p, q and r. p + 85 = 180 (opposite angles in a cyclic quadrilateral) • p = 180 – 85 = 95 • q + 101 = 180 (opposite angles in a cyclic quadrilateral) • q = 180 – 101 = 79 • r + q = 180 (angles on a • straight line) • r = 180 – 79 = 101 r q p 85 101

27. 3. Work out angles a, b, c and d. • a + 41 + 101 = 180 (angles in a triangle) • a = 180 – 41 – 101 = 38 • b = 101 (opposite angles) • d = 41 (angles in the same segment) • c = 38 • (angles in thesamesegment) a 41 101 b c d

28. 4. Work out angles a, b and c. • c = 90 (angles in a semicircle) • a + 59 + 90 = 180 (angles in a triangle) • a = 180 – 59 – 90 = 31 • a + b = 90 (radius is • perpendicular to the tangent) • b = 90 – 31 = 59 c 59 b a

29. P • 5. Calculate angles XPY and OXY . • PX = PY  PYX =  PXY • (isosceles triangle) • PYX = 75 • PXO = 90 (radius is • perpendicular to the tangent) • OXY = 90 – 75 = 15 • XPY = 180 – 75 – 75 = 30 • (angles in a triangle) 75 X Y O

30. 6. XTY is a tangent to the circle, centre O. P and Q are points on the circumference. OQ is parallel to PT. Angle QOT = 37. Find angles OPT and PTY. • OTP = 37 (alternate angles) • OP = OT  OPT = OTP = 37(isosceles triangle) • OTY = 90 (radius is • perpendicular to the tangent) • PTY = OTY - OTP • PTY = 90 – 37 = 53 O 37 Q P T Y X

31. 7. PTR is a tangent to the circle, centre O. The chord AB is parallel to PR. X is a point on the circumference. Angle ORT = 18. • Work out angle AXB. • OTR = 90 (radius is perpendicular to the tangent) • TOR = 180 – 90 – 18 = 72 (angles in a triangle) • TAB = ½ × 72 = 36 (angle at circumference = ½ angle at centre). • ATP = 36 (alternate angles) • ATO = 90 – 36 = 54 X O B A 18 R T P

32. OTR = 90, TOR = 72, TAB = 36, • ATP = 36, ATO = 54 • OT = OB  OTB = OBT • (isosceles triangle) • OTB + OBT = 180 - TOR = 180 – 72 = 108 • OTB = ½ × 108 = 54 • ATB = ATO + OTB • ATB = 54 + 54 = 108 • AXB = 180 – 108 = 72 • (angles in a cyclic • quadrilateral) X O B A 18 R T P