Measurement of an Angle

An angle is the measurement of rotation between two lines, including the type of rotation, which can be clockwise or anticlockwise (for two dimensions).

Angle comes into existence when two lines are joined at a single endpoint, called vertex, from which the angle is measured. One line, which is kept steady, is called the initial side. The other line that rotates with the pivot being the vertex is called the terminal side.

Measurement units used for angle measurement

Usually, there are two units used for angle measurement.

1. Degree

2. Radian

Degree:

Full rotation is of 360°, and then that is divided proportionally to the parts the terminal side is rotated to full rotation.

For example, if the terminal side rotated half of the full rotation, its measure would be half of 360°, which is 180°, and so on. 

Radian:

Full rotation is of 2πᶜ

That superscript ‘c’ is usually ignored or not written, and the angle’s unit is understood from the context.

Similar to degree measurement, the angle measurement in radians is what part of the full rotation is the given rotation.

Converting from radians to degrees and vice versa

Suppose that an angle is of  D degrees   and you want to convert it to radians.

We can use the fact that the core thing is full rotation. 

Full rotation = 360 degrees = 2πᶜ

Thus 1 degree = πᶜ/180

And therefore, conversion of D degrees  will be done as:

1 degree = πᶜ/180

D degrees = ( πᶜ/180 ) × D

This is the same method that will be used for conversion from radians to degrees.

One thing to note is that πᶜ ≠ π as the former one is a measurement of angle and later one is just a quantity. πᶜ is 3.14159… radians, and π is a mathematical constant 3.14159…

Use of angles in trigonometry

A triangle is characterised by its angles and sides. In trigonometry, the trigonometric ratios are defined from the perspective of specific angles.

A triangle has three angles. 

A right-angled triangle has one of its three angles as of $90^\circ$ or $\pi}{2}$ 

Thus, two of its angles are remaining, which are used as angles for finding trigonometric ratios.

A right-angled triangle consists of a slant side, and two sides, which are perpendicular to each other.

The slant side of a right-angled triangle is called the Hypotenuse of that triangle.

From the perspective of a non-right angle of a right-angled triangle, the opposite side to that angle is called Perpendicular,  and the rest of the side is called Base of that triangle from that considered angle’s view point.

Then, we get the six trigonometric ratios defined as:

sin(θ) = Length of Perpendicular/Length of   Hypotenuse

cos(θ) = Length of  Base/Length of  Hypotenuse

tan(θ) = Length of  Perpendicular/Length of  Base

cot(θ) = Length of  Base/Length of  Perpendicular

sec(θ) = Length of Hypotenuse/Length of  Base

csc(θ) = Length of Hypotenuse/Length of  Perpendicular

For theta = alpha, we get the first three ratios as:

sin(α) = |BC|/|AC|

cos(α) = |AB|/|AC|

tan(α) = |BC|/|AB|

Trigonometric ratios are ratios of sides of a right-angled triangle from the perspective of one of its non-right angles.

One can easily figure out that as only angles of a right-angled triangle are used for calculating the trigonometric ratios, thus, the length of the sides of a right-angled triangle doesn’t matter.

As there are 3 sides, and trigonometric ratio is defined for 2 of them, where order of them matters(which comes in numerator and which comes in denominator), thus, there are in total ³P₂ = 6 trigonometric ratios.

Pythagorean identities

Using the Pythagoras theorem for a right-angled triangle, if one of its angle is measured to be of θ, then these three basic identities(called Pythagorean identities)  are true for that triangle:

sin²(θ) + cos²(θ) = 1

1 + tan²(θ) = sec²(θ)

1 + cot²(θ) = csc²(θ)

Signs of angle measurement:

If the angle between initial side and terminal side is formed by rotating the terminal side in an anticlockwise rotation with respect to the initial side, then that angle is measured with positive sign, and if there was clockwise rotation, then the angle is measured with negative sign.

Majorly used angles for trigonometric ratios, and ratios’ values:

Some standard values of trigonometric ratios are worth memorising, or at least having them is useful.

For angle 0, 30, 45, 60, and 90 (in degrees), three of the basic trigonometric ratios are:

Relation between trigonometric ratios

By their definitions, one can easily see that sin is inverse of csc (cosecant), cos is inverse of sec(secant), and tan is inverse of cot (cotangent)

Also, just by the information of any one ratio, we can derive all the rest of the 5 trigonometric ratios.

For example, if we know the value of sin ratio for an angle, then the other ratios are calculated as:

cos(θ) = √1-sin²(θ)

tan(θ) = sin(θ)/cos(θ)

cot(θ) = 1/tan(θ)

sec(θ)} = 1/cos(θ)

csc(θ) = 1/sin(θ)

Angles lying in a quadrant and effect on trigonometric ratios

Take a point on a 2 dimensional plane.

That point is the pivot point. Now, get a fixed line as the initial side, and get a movable side, and call it the terminal side.

Laying down two axes, one vertical, and one horizontal, will divide the plane in four parts, called four quadrants.

• The top-right quadrant is called the first quadrant.

• The top-left quadrant is called the second quadrant.

• The bottom-left quadrant is called the third quadrant.

• The bottom-right quadrant is called the fourth quadrant.

Angles’ quadrant will decide what will be the sign of which trigonometric ratio.

They are listed below. 

• If the angle measured falls in the first quadrant, then trigonometric ratios will evaluate to positive sign.

• If the angle measured falls in the second quadrant, then only the sine and cosecant (csc) ratio will be positive, rest will be negative.

• If the angle measured falls in the third quadrant, then only the tangent and cotangent (tan and cot) ratios for that angle will be positive, rest will be negative.

• If the angle measured falls in the fourth quadrant, then cos and sec ratios for that angle will be positive, rest will be negative.

Conclusion

Angles are an important concept for measurement of rotation. Without them, we would lose data of how a thing is rotated. Angles store two important pieces of data. The degree of rotation, and the method of rotation(clockwise or counterclockwise).

Trigonometric ratios are defined for each angle. Some trigonometric ratios are not defined for some specific angles.