3.5 Sin, Cos, Tan on the Unit Circle & Identities

1. Sine and Cosine on the Unit Circle

The trigonometric functions can be fundamentally defined using a unit circle (radius $r=1$) mapped onto a Cartesian coordinate system.

Given a point $P(x,y)$ residing on the unit circle perimeter, where the radius forms an angle $\theta$ measured from the positive x-axis:

The sine function evaluates strictly to the vertical displacement: $\sin\theta = \frac{y}{1} = y$.
The cosine function evaluates strictly to the horizontal displacement: $\cos\theta = \frac{x}{1} = x$.

Consequently, any angle $\theta$ correlates directly to a coordinate point on the circle where $\mathbf{(\cos\theta, \sin\theta) = (x, y)}$.

Quadrant Analysis

Because the unit circle maps across all four Cartesian quadrants, the signs of trigonometric outputs fluctuate predictably based on spatial position.

1st Quadrant ($0^\circ < \theta < 90^\circ$): Both sine and cosine are positive.
2nd Quadrant ($90^\circ < \theta < 180^\circ$): Sine is positive (y-axis is positive), cosine is negative.
3rd Quadrant ($180^\circ < \theta < 270^\circ$): Both sine and cosine are negative.
4th Quadrant ($270^\circ < \theta < 360^\circ$): Sine is negative, cosine is positive.

Geometric Note: Due to circular periodicity, infinitely many angles map to the same geometric point. Adding integer multiples of a full rotation ($360^\circ k$ or $2k\pi$) yields identical sine and cosine values.

2. Tangent on the Unit Circle

The tangent function is geometrically modeled by constructing a supplementary vertical axis acting as a tangent line touching the unit circle exactly at $(1, 0)$.

Extending the radius line associated with angle $\theta$ until it physically intersects this exterior vertical tangent axis reveals that the resulting y-coordinate intersection precisely equals $\tan\theta$.

Critical Properties of the Tangent Function:

Output values span all real numbers: $-\infty < \tan\theta < +\infty$.
The function becomes mathematically undefined when the extended radius runs perfectly parallel to the vertical tangent axis. This failure state occurs at strictly $90^\circ$ and $270^\circ$ (or $\frac{\pi}{2} + k\pi$).
Diametrically opposite angles share identical tangent values: $\tan(\theta + 180^\circ) = \tan\theta$.

3. Trigonometric Identities

Mathematical identities provide rigid algebraic bridges between distinct trigonometric functions.

Fundamental Pythagorean Identity:

$\sin^2\theta + \cos^2\theta = 1$

Double Angle Identities (HL focus): These expansions convert compound arguments into base variable ratios.

$\sin 2\theta = 2\sin\theta\cos\theta$

$\cos 2\theta = \cos^2\theta - \sin^2\theta$

$\cos 2\theta = 2\cos^2\theta - 1$

$\cos 2\theta = 1 - 2\sin^2\theta$

$\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$

EXAMPLE 1

Given $\sin\theta = \frac{3}{5}$, derive the evaluations for $\cos\theta$, $\tan\theta$, $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ under specific domain bounds.

Solution: Apply the foundational Pythagorean identity to isolate the unknown ratio.

$\cos^2\theta = 1 - \sin^2\theta = 1 - \left(\frac{3}{5}\right)^2 = \frac{16}{25}$. Evaluating the root establishes $\cos\theta = \pm\frac{4}{5}$.

(a) Assuming an acute domain ($\theta < 90^\circ$): The cosine ratio remains positive.

$\cos\theta = \frac{4}{5}$.

$\tan\theta = \frac{3/5}{4/5} = \frac{3}{4}$.

$\sin 2\theta = 2\left(\frac{3}{5}\right)\left(\frac{4}{5}\right) = \mathbf{\frac{24}{25}}$.

$\cos 2\theta = \left(\frac{4}{5}\right)^2 - \left(\frac{3}{5}\right)^2 = \mathbf{\frac{7}{25}}$.

$\tan 2\theta = \frac{24/25}{7/25} = \mathbf{\frac{24}{7}}$.

(b) Assuming an obtuse domain ($90^\circ < \theta < 180^\circ$): The cosine ratio flips to negative.

$\cos\theta = -\frac{4}{5}$.

$\tan\theta = -\frac{3}{4}$.

$\sin 2\theta = 2\left(\frac{3}{5}\right)\left(-\frac{4}{5}\right) = \mathbf{-\frac{24}{25}}$.

$\cos 2\theta = \left(-\frac{4}{5}\right)^2 - \left(\frac{3}{5}\right)^2 = \mathbf{\frac{7}{25}}$.

$\tan 2\theta = \mathbf{-\frac{24}{7}}$.

Notice on Right-Angled Triangle Visualization

Rather than relying purely on algebraic identities, isolating missing ratios can be accomplished rapidly by sketching a reference right triangle mapping the known ratio (e.g., opposite=2, hypotenuse=3), calculating the adjacent side via Pythagoras ($\sqrt{5}$), and adjusting the final numerical signs according to the target quadrant.