NULL ANGLE: DEFINITION AND CHARACTERISTICS, EXAMPLES, EXERCISES - MATHS - 2024

The null angle is one whose measure is 0, both in degrees and in radians or another system of angle measurement. Therefore it lacks width or opening, like that formed between two parallel lines.

Although its definition sounds simple enough, the null angle is very useful in many physics and engineering applications, as well as in navigation and design.

Figure 1. Between the speed and the acceleration of the car there is a zero angle, therefore the car goes faster and faster. Source: Wikimedia Commons.

There are physical quantities that must be aligned in parallel to achieve certain effects: if a car moves in a straight line along a highway and between its velocity vector v and its acceleration vector a there is 0º, the car moves faster and faster, but if the car brakes, its acceleration is opposite to its speed (see figure 1).

The following figure shows different types of angle including the null angle to the right. As can be seen, the 0º angle lacks width or opening.

Figure 2. Angle types, including the null angle. Source: Wikimedia Commons. Orias.

Examples of null angles

Parallel lines are known to form a zero angle with each other. When you have a horizontal line, it is parallel to the x axis of the Cartesian coordinate system, therefore its inclination with respect to it is 0. In other words, horizontal lines have zero slope.

Figure 3. The horizontal lines have zero slope. Source: F. Zapata.

Also the trigonometric ratios of the null angle are 0, 1, or infinity. Therefore the null angle is present in many physical situations that involve operations with vectors. These reasons are:

-sin 0º = 0

-cos 0º = 1

-tg 0º = 0

-sec 0º = 1

-cosec 0º → ∞

-ctg 0º → ∞

And they will be useful to analyze some examples of situations in which the presence of the null angle plays a fundamental role:

- Effects of the null angle on physical magnitudes

Vector addition

When two vectors are parallel, the angle between them is zero, as seen in Figure 4a above. In this case, the sum of both is carried out by placing one after the other and the magnitude of the sum vector is the sum of the magnitudes of the addends (figure 4b).

Figure 4. Sum of parallel vectors, in this case the angle between them is a null angle. Source: F. Zapata.

When two vectors are parallel, the angle between them is zero, as seen in Figure 4a above. In this case, the sum of both is carried out by placing one after the other and the magnitude of the sum vector is the sum of the magnitudes of the addends (figure 4b)

The torque or torque

The torque or torque causes the rotation of a body. It depends on the magnitude of the applied force and how it is applied. A very representative example is the wrench in the figure.

For best turning effect, force is applied perpendicular to the wrench handle, either up or down, but no rotation is expected if the force is parallel to the handle.

Figure 5. When the angle between the position and force vectors is zero, no torque is produced and therefore there is no spin effect. Source: F. Zapata.

Mathematically the torque τ is defined as the vector product or cross product between the vectors r (position vector) and F (force vector) of figure 5:

τ = r x F

The magnitude of the torque is:

τ = r F sin θ

Θ being the angle between r and F. When sin θ = 0 the torque is zero, in this case θ = 0º (or also 180º).

Electric field flow

Electric field flux is a scalar quantity that depends on the intensity of the electric field as well as the orientation of the surface through which it passes.

In Figure 6 there is a circular surface of area A through which the electric field lines E pass. The orientation of the surface is given by the normal vector n. On the left the field and the normal vector form an arbitrary acute angle θ, in the center they form a null angle with each other, and on the right they are perpendicular.

When E and n are perpendicular, the field lines do not cross the surface and therefore the flux is zero, while when the angle between E and n is zero, the lines completely cross the surface.

Denoting the electric field flux by the Greek letter Φ (read “fi”), its definition for a uniform field as in the figure, looks like this:

Φ = E • n A

The point in the middle of both vectors denotes the dot product or scalar product, which is alternatively defined as follows:

Φ = E • n A = EAcosθ

The bold and the arrows above the letter are resources to differentiate between a vector and its magnitude, which is denoted by normal letters. Since cos 0 = 1, the flux is maximum when E and n are parallel.

Figure 6. The electric field flux depends on the orientation between the surface and the electric field. Source: F. Zapata.

Exercises

- Exercise 1

Two forces P and Q act simultaneously on a point object X, both forces initially form an angle θ between them. What happens to the magnitude of the resultant force as θ decreases to zero?

Figure 7. The angle between two forces that act on a body decreases until it is canceled, in which case the magnitude of the resulting force acquires its maximum value. Source: F. Zapata.

Solution

The magnitude of the resultant force Q + P increases gradually until it is maximum when Q and P are completely parallel (figure 7 right).

- Exercise 2

Indicate if the null angle is a solution of the following trigonometric equation:

Solution

A trigonometric equation is one in which the unknown is part of the argument of a trigonometric ratio. To solve the proposed equation, it is convenient to use the formula for the cosine of the double angle:

cos 2x = cos ² x - sin ² x

Because in this way, the argument on the left side becomes x instead of 2x. So:

cos ² x - sin ² x = 1 + 4 sin x

On the other hand cos ² x + sin ² x = 1, so:

cos ² x - sin ² x = cos ² x + sin ² x + 4 sin x

The term cos ² x cancels and remains:

- sin ² x = sin ² x + 4 sin x → - ² sin ² x - 4 sinx = 0 → ² sin ² x + 4 sinx = 0

Now the following variable change is made: sinx = u and the equation becomes:

2u ² + 4u = 0

2u (u + 4) = 0

Whose solutions are: u = 0 and u = -4. Returning the change we would have two possibilities: sin x = 0 and sinx = -4. This last solution is not viable, because the sine of any angle is between -1 and 1, so we are left with the first alternative:

sin x = 0

Therefore x = 0º is a solution, but any angle whose sine is 0 also works, which can also be 180º (π radians), 360º (2 π radians) and the respective negatives as well.

The most general solution of the trigonometric equation is: x = kπ where k = 0, ± 1, ± 2, ± 3,…. k an integer.

References

1. Baldor, A. 2004. Plane and Space Geometry with Trigonometry. Publicaciones Cultural SA de CV México.
2. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 3. Particle Systems. Edited by Douglas Figueroa (USB).
3. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 5. Electrical Interaction. Edited by Douglas Figueroa (USB).
4. OnlineMathLearning. Types of angles. Recovered from: onlinemathlearning.com.
5. Zill, D. 2012. Algebra, Trigonometry and Analytical Geometry. McGraw Hill Interamericana.