Linking Angles, Radii, And Polar Graphs

Unit: Trigonometric & Polar Functions

Chapter: Linking Angles, Radii, and Polar Graphs

Reference: – Introduction, Calculus with polar functions, Polar graphing technology, Graphing Techniques, Polar functions, Symmetry, Plotting Points & Axes, Polar & Cartesian Coordinates, Conversion between polar & Cartesian Coordinates, Angle & Radius uses in AP Calculus.

After studying this chapter, you should be able to:

• Introduction to Calculus with Polar Functions
• Graphing techniques, Symmetry & Plotting Points
• Polar & Cartesian Coordinates, Uses of Angle & Radius

Introduction to Calculus with Polar Functions

The word ‘trigonometry’ is derived from the Greek words ‘trigon’ and ‘metron’ and it means ‘measuring the sides of a triangle’. The subject was originally developed to solve geometric problems involving triangles. It was studied by sea captains for navigation, surveyor to map out the new lands, by engineers and others. Currently, trigonometry is used in many areas such as the science of seismology, designing electric circuits, describing the state of an atom, predicting the heights of tides in the ocean, analysing a musical tone and in many other areas.

In earlier classes, we have studied the trigonometric ratios of acute angles as the ratio of the sides of a right angled triangle. We have also studied the trigonometric identities and application of trigonometric ratios in solving the problems related to heights and distances.

Angles

Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final position of the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex. If the direction of rotation is anticlockwise, the angle is said to be positive and if the direction of rotation is clockwise, then the angle is negative.

The definition of an angle suggests a unit, viz. one complete revolution from the position of the initial side as indicated below:

Degree measure:

The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.  One way to measure an angle is in terms of degrees.  A measure of one degree ( 1° ) is equivalent to a rotation of 1360  of a complete revolution.

To measure angles, it is convenient to mark degrees on the circumference of a circle .  Thus, a complete revolution is 360°, half a revolution is 180°, a quarter of a revolution is 90° and so forth.

If a rotation from the initial side to terminal side is 1360th  a revolution,
the angle is said to have a measure of one degree, written as 1°.
A degree is divided into 60 minutes, and a minute is divided into
60 seconds. One sixtieth of a degree is called a minute, written as 1’, and one sixtieth of a minute is called a second, written as 1”.

Thus,         1° = 60’, 1’ = 60”

Some of the angles whose measures are 360°, 180°, 270°, 420°, – 30°,
– 420° are shown below:

Radian Measure:

There is another unit for measurement of an angle, called the radian measure. Angle subtended at the centre by an arc of length 1 unit
in a unit circle (circle of radius 1 unit) is said to have a measure of
1 radian. In the figures given below, OA is the initial side and OB is
the terminal side. The figures show the angles whose measures are
1 radian, –1 radian, 112 radian and –112  radian.

We know that the circumference of a circle of radius 1 unit is 2p. Thus, one complete revolution of the initial side subtends an angle of 2pradian.

Note: If in a circle of radius r, an arc of length l subtends an angle qradian at the centre, we have θ=lror l=r θ .

Relation between Radian and Real Numbers:

Consider the unit circle with centre O. Let A be any point on the circle. Consider OA as initial side of an angle. Then the length of an arc of the circle will give the radian measure of the angle which the arc will subtend at the centre of the circle. Consider the line PAQ which is tangent to the circle at A. Let the point A represent the real number zero, AP represents positive real number and AQ represents negative real numbers. If we rope the line AP in the anticlockwise direction along the circle, and AQ in the clockwise direction, then every real number will correspond to a radian measure and conversely. Thus, radian measures and real numbers can be considered as one and the same.

Relation between Degree and Radian:

The size of a radian is determined by the requirement that there are 2  radians in a circle. Thus 2  radians equals 360 degrees. This means that 1 radian = 180/  degrees, and 1 degree =  /180 radians.

1 radian = 1800π = 57° 16’ approximately.

Also   1° = π180  radian = 0.01746 radian approximately.

Notational Convention

Since angles are measured either in degrees or in radians, we adopt the convention that whenever we write angle q°, we mean the angle whose degree measure is qand whenever we write angle b, we mean the angle whose radian measure is b.

Note that when an angle is expressed in radians, the word ‘radian’ is frequently omitted. Thus, p = 180⁰ and π4  = 45° are written with the understanding that pand π4  are radian measures. Thus, we can say that

Radian measure = π180 × Degree measure

Degree measure = 180π × Radian measure

Example: Convert 30 degrees angle to radians.

Solution: We know 180° = pradian.

               30°= π180×30  radian = π6  radian.

Example: Convert 520° degrees angle to radians.

Solution: We know that 180° = π radian

                 520^{0 =} π180×520  radian = 26π9  radian

Example: Convert 6 radians into degree measure.

Solution: We know that pradian = 180°.

       6 radians = 180π×6  degree = 1080×722  degree

    = 343711  degree = 343° + 7×6011  minute [as 1° = 60’]

    = 343° + 38’ + 211  minute                   [as 1’ = 60’’]

    = 343° + 38’ + 10.9”= 343°38’ 11”approximately.

Hence, 6 radians = 343° 38’ 11” approximately.

Example: Convert 1116  radians into degree measure.

Solution: We know that π radian = 180°

Example: Find the radius of the circle in which a central angle of
60° intercepts an arc of length 37.4 cm (use p = 227 ).

Solution: Here l= 37.4 cm and q= 60° = 60π180  radian = π3

Hence,       by r = lθ

    r = 37.4×3π=37.4×3×722  = 35.7 cm

Example: In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.

Solution: Diameter of the circle = 40 cm

∴Radius (r) of the circle = 402  cm = 20 cm

Let AB be a chord (length = 20 cm) of the circle.

In ΔOAB, OA = OB = Radius of circle = 20 cm

Also, AB = 20 cm

Thus, ΔOAB is an equilateral triangle.

∴θ = 60° = π3  radian

We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then. θ  =lr

Thus, the length of the minor arc of the chord is. 20π3  cm

Key Points

• Polar coordinates consist of an angle (θ) and a radius (r) and are used to represent points in a plane.
• The angle θ is measured counterclockwise from the positive x-axis, typically in radians.
• The radius r represents the distance from the origin to the point and can be negative.
• The conversion from polar coordinates to Cartesian coordinates is given by x = r * cos(θ) and y = r * sin(θ).
• The conversion from Cartesian coordinates to polar coordinates is given by r = √(x² + y²) and θ = arctan(y / x).
• Polar graphs are plotted using the angle θ as the independent variable and the radius r as the dependent variable.
• Symmetry in polar graphs can be determined by replacing θ with -θ or θ + π, and r with -r.
• Polar equations can have multiple representations due to periodicity, such as r = a sin(bθ) and r = a cos(bθ), where a and b are constants.
• The shape of a polar graph can be determined by analyzing the equation and identifying patterns related to the angle θ.
• Key features of polar graphs include the number of petals, the presence of loops or cusps, and the behavior at the origin.
• When graphing polar equations, it is important to choose an appropriate range of θ to capture the desired portion of the curve.
• The graphing of polar equations can be facilitated using technology like graphing calculators or computer software.
• To find the slope of a tangent line to a polar curve, the derivative must be calculated using the chain rule and trigonometric identities.
• The area bounded by a polar curve can be found using integration and the formula for the area of a sector.
• Arc lengths of polar curves can be determined by integrating a differential arc length formula based on the Pythagorean theorem.