Trigonometric Functions: Definitions, Properties, and Applications

Trigonometric Functions and Related Concepts

The Pythagorean Theorem

The Pythagorean Theorem is a fundamental result in geometry, especially relevant to trigonometry. It states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.

• Formula: $a^2 + b^2 = c^2$ where a and b are the legs, and c is the hypotenuse.

• Application: Used to find missing side lengths in right triangles, which is essential for defining trigonometric functions.

• Example: If a triangle has legs of length 3 and 4, the hypotenuse is $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

The Distance Formula

The Distance Formula is used to calculate the distance between two points in a rectangular coordinate system. This formula is derived from the Pythagorean Theorem and is crucial for determining the radius in trigonometric function definitions.

• Formula: $d(P, R) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Application: Used to find the length of the segment connecting two points, which is often the radius in trigonometric contexts.

• Example: The distance between points (1, 2) and (4, 6) is $\sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

The Six Trigonometric Functions

There are six fundamental trigonometric functions associated with an angle θ in standard position. These functions relate the coordinates of a point on the terminal side of the angle to the radius (distance from the origin).

• Sine (sin θ): $\sin\theta = \frac{y}{r}$

• Cosine (cos θ): $\cos\theta = \frac{x}{r}$

• Tangent (tan θ): $\tan\theta = \frac{y}{x}$

• Cotangent (cot θ): $\cot\theta = \frac{x}{y}$

• Secant (sec θ): $\sec\theta = \frac{r}{x}$

• Cosecant (csc θ): $\csc\theta = \frac{r}{y}$

Where r is the distance from the origin to the point (x, y), calculated as $r = \sqrt{x^2 + y^2}$.

Examples

• Example 1: For the point (12, 5) on the terminal side of θ, $r = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13$. The six functions are:

  • $\sin\theta = \frac{5}{13}$

  • $\cos\theta = \frac{12}{13}$

  • $\tan\theta = \frac{5}{12}$

  • $\cot\theta = \frac{12}{5}$

  • $\sec\theta = \frac{13}{12}$

  • $\csc\theta = \frac{13}{5}$

• Example 2: For the point (8, -6), $r = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10$. The six functions are:

  • $\sin\theta = \frac{-6}{10} = -0.6$

  • $\cos\theta = \frac{8}{10} = 0.8$

  • $\tan\theta = \frac{-6}{8} = -0.75$

  • $\cot\theta = \frac{8}{-6} = -1.33$

  • $\sec\theta = \frac{10}{8} = 1.25$

  • $\csc\theta = \frac{10}{-6} = -1.67$

Signs of Trigonometric Functions in Quadrants

The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies.

• Quadrant II: $\frac{y}{r}$ is positive (since y > 0, r > 0)

• Quadrant III: $\frac{y}{x}$ is positive (since both y < 0 and x < 0, their ratio is positive)

Quadrantal Angles

A quadrantal angle is an angle whose terminal side lies along the x-axis or y-axis. Common quadrantal angles include 0º, 90º, 180º, 270º, and 360º. For these angles, it is often convenient to select r = 1.

• Definition: Quadrantal angles are multiples of 90º.

• Application: Used to quickly determine trigonometric function values without calculation.

Function Values of Quadrantal Angles

θ	sin θ	cos θ	tan θ	cot θ	sec θ	csc θ
0º	0	1	0	undefined	1	undefined
90º	1	0	undefined	0	undefined	1
180º	0	-1	0	undefined	-1	undefined
270º	-1	0	undefined	0	undefined	-1
360º	0	1	0	undefined	1	undefined

Examples

• Example 5: Find the values of the six trigonometric functions for 360º and for an angle whose terminal side passes through (0, -5):

  • For 360º: sin = 0, cos = 1, tan = 0, cot = undefined, sec = 1, csc = undefined

  • For (0, -5): r = 5, sin = -1, cos = 0, tan = undefined, cot = 0, sec = undefined, csc = -1

• Example 6: Find the indicated function values, if undefined say so:

  • $\sin 270^\circ = -1$

  • $\tan -900^\circ$ is undefined