Graphical Sketch - Three Leaf Rose

Category: Analytic Geometry, Trigonometry, Algebra

"Published in Newark, California, USA"

Sketch the graph for a Three Leaf Rose

Solution:

The given equation above is also another type of Polar Equation where the parameters are r and θ. The purpose of converting the equations of Rectangular Coordinate to Polar Coordinate is to simplify the equation. Most of the equations of Polar Equations are complicated if they are written in Rectangular Coordinate. The graphs of Polar Equations are mostly leaves, flowers, and spiral shapes. Let's consider the above equation again

We need to assign the values of θ from 30° to 360° in order to get the values of r as well. All angles must be expressed in radians because radians are unitless values. You can use calculator in order to get the values of r as follows

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Next, you need a polar paper in order to plot the graph. If you don't have a polar paper, you can use an ordinary graphing paper but you need a compass, protractor, and ruler. As you noticed at the above table that some values of r are negative. We will not use the parameters that have negative values of r. If you plot the values of r and θ and then connect all the points, the graph of the given equation will be like this

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Rectangular - Polar Coordinate

Category: Analytic Geometry, Algebra, Trigonometry

"Published in Newark, California, USA"

Convert the equation of the hyperbola from Rectangular Coordinate to Polar Coordinate for

Solution:

There are two types of coordinate system in Analytic Geometry. The first type is the Rectangular Coordinate System where the coordinates are x and y. This is a very commonly used in plotting the points and sketching the graphs. The other type is the Polar Coordinate System where the coordinates are r and θ. The radius, r is always positive in the equation unlike the angle, θ can be a positive or a negative. If θ is positive, then the angle is measured in counterclockwise direction and if θ is negative, then the angle is measured in clockwise direction.

To convert the Rectangular Coordinate System to Polar Coordinate System, let's consider a point P(x, y) in a Rectangular Coordinate System. From point P, draw a vertical line perpendicular to x-axis. The resulting figure is a right triangle with r as the hypotenuse or a distance from a point to the origin and θ is the adjacent angle of the hypotenuse from the x-axis. Use Pythagorean Theorem and trigonometric functions to convert the values of x and y into their equivalent value of r and θ as follows

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Consider the given equation above

Substitute the values of x and y as follows

Graphical Sketch - Trigonometric Functions

Category: Trigonometry

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Draw the graphs of y₁ = Sin x and y₂ = Cos x for ranging from 0 to 2π, and then by the addition of ordinates, obtain the graph of y = Sin x + Cos x. What is the period of Sin x + Cos x and also the amplitude?

Solution:

The first thing that we have to do is to create the table so that it is easier for us to compute their values of y. By the way, all the units are unitless and so their angles must be expressed in radians. You can use calculator so that you can get the accurate value of y.

Note: The conversion of degrees to radians is given by the formula

Next, use your scientific calculator to compute for their values of trigonometric functions as follows

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Since we have now the values of x and y, we can start now the graphing of the given trigonometric functions. First, let's start with the Sine function. As you notice that the trend of the graph is a wave because of the periodically repeating the values of y but the highest and the lowest values are always 1 and -1. 

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For the Cosine function, the trend of the graph is almost the same but it didn't pass through the origin and the highest and the lowest values are always 1 and -1 also.

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To draw the Sin x + Cos x function, you need a compass or divider for this. Using a divider, measure the height or the value of y of the cosine graph from the x-axis and then transfer it or plot the point to the sine function from the curve proper. Connect all the plotted points and the trend of the graph will be like this. 

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If you don't have a compass or a divider, you can use the values of y = Sin x + Cos x from the table to plot the points directly and then connect all the points. As you noticed that the resulted graph is also a repeated wave because of the periodically repeating the values of y also. 

From the trend of a graph and the table, the amplitude or the height of the curve is 1.4142 (+ and -) which is also equal to √2. 

To compute for the period, you need the graph proper to see the trend of a curve. The trend of a curve that we need is a one complete wave or one cycle which is started from y = 0 and then ends at y = 0 that passed the minimum and maximum amplitudes. By looking at the graph, the approximate value of a period is 2.3 - (- 3.9) = 2.3 + 3.9 = 6.2 radians. 

There's another way to compute for the period. You can also use the table and look for the value of y = Sin x + Cos x. As you can see that if y = 0 then x = 2.3562 radians and if y = 0 then x = 5.4978 radians. Therefore, the value of a period is 2(5.4978 - 2.3562) = 6.2852 radians which also equal to 360°.