Trigonometric Functions from the Unit Circle and Right Triangles

Trigonometric Functions: Definitions from the Unit Circle and Right Triangles

Introduction to the Unit Circle

The unit circle is a circle of radius 1 centered at the origin (0,0) in the Cartesian plane. It is fundamental in defining the trigonometric functions for all real numbers and provides a geometric interpretation of sine, cosine, and tangent.

• Standard position of an angle: The vertex is at the origin, and the initial side lies along the positive x-axis.

• Terminal side: The ray that forms the angle with the initial side.

• Any point (x, y) on the terminal side of angle θ at distance r from the origin satisfies $r = \sqrt{x^2 + y^2}$.

Trigonometric Functions in Terms of x, y, and r

For any angle θ in standard position, and a point (x, y) on its terminal side (not necessarily on the unit circle):

• $\sin \theta = \dfrac{y}{r}$

• $\cos \theta = \dfrac{x}{r}$

• $\tan \theta = \dfrac{y}{x}$ (x ≠ 0)

• $\csc \theta = \dfrac{r}{y}$ (y ≠ 0)

• $\sec \theta = \dfrac{r}{x}$ (x ≠ 0)

• $\cot \theta = \dfrac{x}{y}$ (y ≠ 0)

On the unit circle ($r = 1$):

• $\sin \theta = y$

• $\cos \theta = x$

Geometric Interpretation Using Right Triangles

By drawing a perpendicular from the point (x, y) to the x-axis, a right triangle is formed with:

• Adjacent side: x

• Opposite side: y

• Hypotenuse: r

Using similar triangles and the Pythagorean theorem ($x^2 + y^2 = r^2$), the trigonometric functions can be related to the sides of the triangle.

Example: Finding Trigonometric Values from a Point

Given: Point (3, -4) on the terminal side of θ. Find $\sin \theta$, $\cos \theta$, $\tan \theta$.

• Calculate $r = \sqrt{3^2 + (-4)^2} = 5$

• $\sin \theta = \dfrac{-4}{5}$

• $\cos \theta = \dfrac{3}{5}$

• $\tan \theta = \dfrac{-4}{3}$

Signs of Trigonometric Functions in Each Quadrant

The sign of each trigonometric function depends on the quadrant in which the terminal side of θ lies:

Quadrant	sin θ	cos θ	tan θ	Mnemonic
I	+	+	+	All
II	+	-	-	Students
III	-	-	+	Take
IV	-	+	-	Calculus

Mnemonic: "All Students Take Calculus" helps remember which functions are positive in each quadrant.

Example: Determining Signs and Values in a Quadrant

Given: $\sin \theta = \dfrac{5}{13}$ in Quadrant II. Find $\cos \theta$ and $\tan \theta$.

• Since $\sin \theta = \dfrac{y}{r}$, $y = 5$, $r = 13$.

• Use the Pythagorean theorem: $x^2 + 5^2 = 13^2 \implies x^2 = 169 - 25 = 144 \implies x = -12$ (negative in QII).

• $\cos \theta = \dfrac{x}{r} = \dfrac{-12}{13}$

• $\tan \theta = \dfrac{y}{x} = \dfrac{5}{-12} = -\dfrac{5}{12}$

Summary Table: Trigonometric Functions in Terms of x, y, r

Function	Expression
$\sin \theta$	$\dfrac{y}{r}$
$\cos \theta$	$\dfrac{x}{r}$
$\tan \theta$	$\dfrac{y}{x}$
$\csc \theta$	$\dfrac{r}{y}$
$\sec \theta$	$\dfrac{r}{x}$
$\cot \theta$	$\dfrac{x}{y}$

Key Concepts and Applications

• Reference angle: The acute angle formed by the terminal side of θ and the x-axis. Used to determine the values of trigonometric functions in different quadrants.

• Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$

• Applications: These definitions allow calculation of trigonometric values for any angle, not just those in right triangles, and are foundational for graphing and solving trigonometric equations.

Example: Using the Pythagorean Theorem

Given a right triangle with sides x and y, and hypotenuse r:

• $x^2 + y^2 = r^2$

• If two sides are known, the third can be found using this relationship.

Summary

• The unit circle provides a geometric basis for defining trigonometric functions for all real numbers.

• Trigonometric functions can be expressed in terms of the coordinates (x, y) of a point on the terminal side of an angle and the distance r from the origin.

• The signs of trigonometric functions depend on the quadrant in which the angle's terminal side lies.

• These concepts are essential for further study in trigonometry, calculus, and their applications.