Sectors and Circles trigonometry problems are presented along with their detailed solutions.

When solving the problems below, we will make use of the definition of the trigonometric functions of angle ? in terms of the coordinates of point P(x , y) that is on the terminal side of an angle ?.

cos(?) = x / R tan(?) = y / x sec(?) = R / x

sin(?) = y / R cot(?) = x / y CSC(?) = R / y

R = √(x^{2} + y^{2})

We will also use the relationship between the arc length S, the radius R and the central angle ? of a
sector: S = R ? (? in RADIANS)

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If the coordinates of point A, in the circle below, are (8, 0) and arc s has a length of 20 units, find the coordinates of point P in the diagram below.

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__Solution__

Radius R of the given circle is equal to 8. We first use the formula for the arc s in terms of the radius R and the angle ? to find ? in radians.

s = R ?

2 0 = 8 ?

? = 20 / 8 = 2.5 radians

We now use the definitions for sine and cosine to find the coordinates x and y of P.

cos(?) = x / R and sin(?) = y / R

cos(?) = x / R gives x = R cos(?) = 8 cos(2.5) = -6.40

sin(?) = y / R gives y = R sin(?) = 8 sin(2.5) = 4.79

Point P has the coordinates (-6.40 , 4.79).

In a rectangular coordinate system, a circle with its center at the origin passes through the point(4√2 , 5√2).

What is the length of the arc S?

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__Solution__

We use the definiton for tangent to find angle ?.

tan(?) = y / x = 5√2 / 4√2 = 5 / 4

? = arctan(5 / 4)

Use fromula for s in terms of ? and the radius R.

S = R ? = √ (4√2)^{2} + (5√2)^{2} arctan(5 / 4) ? 8.11

The length of the minutes hand in a clock is 4.5 cm. Find the length of the arc traced by the end of the minutes hand between 11:10 pm and 11:50 pm.

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__Solution__

From 11:10 to 11:50, there

11:50 - 11:10 = 40 minutes

One complete rotation of the minutes hand corresponds to 60 minutes and to an angle of 360°. We therefore can write that there are

360 ° / 60 minutes = 6 ° / minutes

If ? is the central angle corresponding to the arc traced by the minutes hand, then it is given by

? = (6 °/ minute) × 40 minutes = 240 °

The arc S traced is then given by

S = R ? , (? in radians).

S = R ? = 4.5 cm 240 × ? / 180 = 6?

The points P(a , b), Q(5 , 0) and M(c , -1) are located on the circle with the center O(0 , 0) and radius R of 5 units as shown below. Calculate the:

a) The coordinates (a,b) of point P.

b) The arc length between Q and P in counterclockwise direction.

c) The x-coordinate c of point M.

d) Angle ?.

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__Solution__

a) Point P is the terminal side of an angle in standard position of size 35°. Hence

a = R cos (35°) = 5 cos (35°) and b = R sin(35°) = 5 sin (35°)

b) The central angle QOP is known, therefore the length of the arc PQ is given by

arc PQ = r×35 × ? / 180 = 5×35 × ? / 180 ? 3.05 units

c) Point C is on the circle, therefore the distance from the center of the circle (0,0) to point C is equal to the radius.

√ (c - 0)^{2} + (-1 - 0)^{2} = 5

Square both sides of the above equation and solve for c.

(c - 0)^{2} + (-1 - 0)^{2} = 25

c = ± √ 24 = ± 2√ 6

Point M is in quadrant III and therefore c is negative. Hence point M has the coordinates

M(- 2√ 6, -1)

d) Point M is on the terminal side of angle ?+35°. Hence using the definition of the tangent of an angle in standard position in terms of the coordinates of a point on the terminal side, we have

tan(? + 35 °) = 1 / 2√ 6

Point M is in quadarnt III, therefore ? + 35 ° is greater than 180° and therefore

? + 35 ° = 180 + arctan(1 / 2√ 6)

? = 180 + arctan(1 / 2√ 6) - 35 ° = 156.54°

Circle of radius R = 4 units and center O is shown below.

a) Find the length of the arc between D and P.

b) Find the coordinates (a,b) of P.

c) Find the length of segment PD,.

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__Solution__

a) The central angle DOP is known, therefore length of the arc DP is given by (do not forget to convert degrees to radians)

arc PQ = R 130 ? / 180 = 4 × 130 ? / 180 ? 9.08 units

b) Point P is on the terminal side of an angle in standard position. Therefore the coordinates are given by:

a = R cos(130°) = 4 cos(130°) and b = R sin(130°) = 4 sin(130°)

c) Use the distance formula between two points P and D to find the length of the line segment PD.

PD = √ (a - 4)^{2} + (b - 0)^{2}

= √ (4 cos(130°) - 4)^{2} + (4 sin(130°) - 0)^{2} ? 7.25

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__Solution__

We first need to find the size ? of the central angle AOB.

S = R ?

? = S / R = 12.5 / 3 radians

We now calculate the coordinates a and b of point B.

a = R cos? = 3 cos(12.5 / 3)

b = R cos? = 3 sin(12.5 / 3)

Point P has coordinates (4.1 , b) and is located on the circle, of center O and radius R = 5, in quadrant I. Find the length of arc S.

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__Solution__

We first need to calculate the size ? of the central angle DOP using the x coordinate of point P and the radius.

4.1 = 5 cos(?)

cos(?) = 4.1 / 5

? = arccos(4.1 / 5)

We calculate the arc S.

S = R ? = 5 arccos(4.1 / 5)

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