Trigonometric Functions

Ever wondered why Trigonometric functions are named the way they are?
Why is cosine named as "co"sine, "cosecant" named as "co"secant and "cotangent" named as "co"tangent?
Do these functions have any meaning outside of triangles?
Aren't terms like "secant" and "tangent" associated with circles?

The Age-old Brothers:
Sine and Cosine

Etymology: The word "sine" is derived from the latin word "Sinus" which means a "bay" or a "fold". It's Sanskrit, Arabic and Persian versions are dervided from the Greek word "khorde" which signifies a bow-string. In the below visualization, if we imagine the X-axis to act like a mirror, the image would become symmetrical. The line segments OP and OP' look like the strings of a bow, and the perpendiculars to X axis (whose length is equal to 2sin(θ)) looks like a bow

Visualization: Consider a unit circle (a circle of radius 1). Consider a line drawn from the origin O, that makes an angle θ with the X-axis in the anti-clockwise direction. This line intersects the circle at a point P. The length of the projection of the line segment OP on the Y-axis is equal to the Sine of θ


Adjust angle using the following slider:

Or, type the angle here:


Relationship between Sine and Cosine:

Cosine has a serendipitous co-existence with Sine. The length of projection of the same line segment OP on the X-axis is equal to the Cosine of θ



Deductions

The basic trigonometric identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \]

Just using pythagoras theorem, we see that sin(θ)^2 + cos(θ)^2 = 1 for all angles θ

Value of Sines and Cosines of multiples of 90 degrees:
  1. The Sine of 0 degrees is 0, since a line coinciding with the X-axis has no projection on the Y-axis
  2. The Cosine of 0 degrees is 1, since a line drawn within a unit circle and coinciding with the X-axis has a projection of length 1 on the X-axis
  3. The Sine of 90 degrees is 1, since a line drawn within a unit circle and coinciding with the Y-axis has a projection of length 1 on the Y-axis
  4. The Cosine of 90 degees is 0, since a line coinciding with the Y-axis has no projection on the X-axis
  5. The same logic holds good for 180 degrees and 270 degrees
  6. This visualization also allows us to extend the definition of trigonometric functions beyond 90 degrees, since we are no longer restricted to a triangle
Signs of Sine and Cosine functions:
  1. Any angle from 0 to 180 degrees has a positive Sine, since the length of projection on the Y-axis is measured along the positive Y-direction
  2. Sine is negative for rest of the angles because the length of projection on the Y-axis is measured along the negative Y-direction
  3. Any angle from 0 to 90 degrees and 270 to 360 degrees has a positive Cosine, since the length of projection on the X-axis is measured along the positive X-direction
  4. Cosine is negative for rest of the angles because the length of projection on the X-axis is measured along the negative X-direction
Sine is an odd function: \[ \sin (-\theta) = - \sin (\theta) \]
  1. Consider an angle θ, having magnitude between 0 and 90 degrees
  2. The Sine of +θ is always positive since the projection on the Y-axis is always along the positive Y-direction
  3. The Sine of -θ is always negative since the projection on the Y-axis is always along the negative Y-direction
  4. Hence, sin(-θ) = -sin(θ): Sine is an odd function
  5. The same also holds good for an angle θ having magnitude between 90 and 180 degrees: Sin(θ) is always positive whereas sin(-θ) is always negative
Cosine is an even function: \[ \cos (-\theta) = \cos (\theta) \]
  1. Consider an angle θ, having magnitude between 0 and 90 degrees
  2. The Cosine of +θ is always positive since the projection on the X-axis is always along the positive X-direction
  3. The Cosine of -θ is always positive since the projection on the X-axis is always along the positive X-direction
  4. Hence, cos(-θ) = cos(θ): Cosine is even odd function
  5. The same also holds good for an angle θ having magnitude between 90 and 180 degrees: Cos(θ) and cos(-θ) are both negative

The Circle Grazers:
Tangent and Cotangent

Etymology: The word "tangent" comes from the Latin verb "tangere" which means "to touch"

Visualization: Consider the same unit cirle with the same line segment OP. Consider a tangent drawn to the circle at point P. The length between the point P and the point where this tangent intersects the X-axis is equal to the tangent of the angle θ

Adjust angle using the following slider:

Or, type the angle here:


Relationship between Tangent and Cotangentt:

The length of the same tangent, but from the point P to its point of intersection with the Y-axis is equal to the Cotangent of θ

Deductions

Values of Tangents and Cotangents of multiples of 90 degrees
  1. The Tangent of 0 degrees is 0, since the point P is already intersecting with the X-axis
  2. The Cotangent of 0 degrees is infinite, since the tangent never meets the Y-axis
  3. The Tangent of 90 degrees is infinite, since the tangent never meets the X-axis
  4. The Cotangent of 90 degrees is 0, since the point P is already intersecting with the Y-axis
  5. The same logic holds good for 180 degrees and 270 degrees

The Inner Perpetrators:
Secant and Cosecant

Etymology: The word "secant" is derived from the French verb "secanter" which means "to cut"

Visualization: Consider the same unit cirle with the same line segment OP and the tangent drawn to the circle at point P. The length between the Origin and the point where the tangent intersects the X-axis is equal to the Secant of θ

Adjust angle using the following slider:

Or, type the angle here:


Relationship between Secant and Cosecant:

The distance between the Origin and the point where the same tangent intersects the Y-axis is equal to the Cosecant of θ

Deductions

Trigonometric identities:

On applying Pythagoras theorem to the two traingles, we obtain:

\[ 1 + \tan^2 \theta = \sec^2 \theta \] \[ 1 + \cot^2 \theta = \csc^2 \theta \] Values of Secant and Cosecant of multiples of 90 degrees
  1. The Secant of 0 degrees is 1, since the point P coincides with the X-axis
  2. The Cosecant of 0 degrees is infinite, since the tangent never meets the Y-axis
  3. The Secant of 90 degrees is infinite, since the tangent never meets the X-axis
  4. The Cosecant of 0 degrees is 1, since the point P coincides with the Y-axis
  5. The same logic holds good for 180 degrees and 270 degrees


Developed by ChanRT | Fork me at Github