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Trigonometric Functions

Table of Contents


Introduction

Most students are familiar with the drawing in Figure 1 used to define sinθ and cosθ for an angle θ.

Figure 1. The sine and cosine of a given angle θ.

From this diagram, we may deduce, using the Pythagorean theorem, that sin2θ + cos2θ = 1.

Unfortunately, at this point, high-school teachers will then proceed to simply define the other trigonometric functions using algebra formulae (e.g., tan(θ) = sin(θ)/cos(θ) ) without any reference to the geometry.

Tangent (tan) and Secant (sec)

To begin, in Figure 2 we draw a line (red) which is tangent to the point (1, 0) (the point labeled 1) and we extend the line 0a to intersect this tangent line at b. The length of the line 1b is by definition tanθ and the length of the line 0b is secθ.

Figure 2. The tangent and secant of a given angle θ.

With these definitions, we may use similar triangles and the Pythagorean theorem to come up with a full set of identities:

  • tan2θ + 1 = sec2θ.
  • tanθ:1 = sinθ:cosθ ⇒ tanθ = sinθ/cosθ.
  • secθ:1 = 1:cosθ ⇒ secθ = 1/cosθ.
  • secθ:tanθ = 1:sinθ ⇒ sinθ = tanθ/secθ.
  • (tanθ - sinθ)2 + (1 - sinθ)2 = (secθ - 1)2.

To figure out where the last identity came from, look for a small right-angled triangle.

Complementary Tangent (cot) and Complementary Secant (csc)

Similarly, we can draw a line (in blue) which is tangent to the point (0,1) (the point labeled 1'), as shown in Figure 3 and extend the line 0a until it intersects the complementary tangent line at c. The length of the line 1'c is by definition the complementary tangent, or cotangent of θ and the length of the line 0c is the complementary secant of θ, or secθ.

Figure 3. The cotangent and cosecant of a given angle θ.

Yet again, we may derive all the various identities:

  • cot2θ + 1 = csc2θ.
  • cotθ:1 = cosθ:sinθ ⇒ cotθ = cosθ/sinθ.
  • cscθ:1 = 1:sinθ ⇒ cscθ = 1/sinθ.
  • cscθ:cotθ = 1:cosθ ⇒ cosθ = cotθ/cscθ.
  • (cotθ - cosθ)2 + (1 - cosθ)2 = (cscθ - 1)2.

In addition to these identities, we may also come up with the following identities between the various functions:

  • cotθ:1 = 1:tanθ ⇒ cotθ = 1/tanθ.
  • cscθ:1 = secθ:tanθ ⇒ cscθ = secθ/tanθ.
  • cscθ:cotθ = secθ:1 ⇒ secθ = cscθ/cotθ.

½π - θ Formula

If you take a look at Figure 4, you may also observe that the line which is the tangent for θ is the cotangent line for π/2 - θ, and therefore we see that the definitions for tangent and secant for θ are equal to the definitions of cotangent and cosecant for ½π - θ, and vice versa.

Figure 4. Relationship between θ and π/2 - θ.

From this comes the relationships:

  • sin(π/2 - θ) = cosθ.
  • cos(π/2 - θ) = sinθ.
  • tan(π/2 - θ) = cotθ.
  • sec(π/2 - θ) = cscθ.
  • csc(π/2 - θ) = secθ.
  • cot(π/2 - θ) = tanθ.

Approximations

Additionally, we can see why for small values of θ (in radians):

  • sinθθ,
  • cosθ ≈ 1,
  • tanθθ,
  • secθ ≈ 1,
  • cscθ ≈ ∞, and
  • cotθ ≈ ∞.

Recall that the value θ in radians is the length of the curve 1a shown in Figure 1.