8.4 Right-Triangle Trigonometry, the Unit Circle, and Radian Measure

Trigonometric Ratios, Unit Circle, Radian Measure & Circles

Core Advanced Mathematics: Trigonometry on the ACCUPLACER Advanced Algebra and Functions (AAF) assessment links geometric right triangle side ratios, circular radian dynamics, unit circle coordinates across all four quadrants, fundamental Pythagorean identities, and periodic sinusoidal wave models. Mastery requires computing exact trigonometric ratios, executing angle and sector conversions, applying identities, and analyzing wave parameters.

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Right Triangle Trigonometry (SOH CAH TOA) & Reciprocal Functions

In a right triangle with an acute angle $\theta$, side lengths are classified relative to $\theta$ as the Opposite leg ($O$), the Adjacent leg ($A$), and the Hypotenuse ($H$) (the side opposite the $90^\circ$ right angle).

                    Right Triangle Trigonometry Reference
                                  ▲
                                 ╱│
                                ╱ │
                Hypotenuse (H) ╱  │ Opposite (O)
                              ╱   │
                             ╱    │
                            ╱ θ   │
                           ───────┴─
                             Adjacent (A)

Primary Trigonometric Ratios (SOH CAH TOA)

$$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{O}{H} \qquad \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{A}{H} \qquad \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{O}{A}$$

Reciprocal Trigonometric Ratios

Each primary ratio has a corresponding reciprocal trigonometric function:

$$\csc\theta = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{1}{\tan\theta}$$

Co-Function Identities (Complementary Angles)

Because the two acute angles in a right triangle sum to $90^\circ$ (or $\frac{\pi}{2}$ radians), co-functions of complementary angles are equal:

• $\sin(90^\circ - \theta) = \cos\theta$ and $\cos(90^\circ - \theta) = \sin\theta$
• $\tan(90^\circ - \theta) = \cot\theta$ and $\cot(90^\circ - \theta) = \tan\theta$
• $\sec(90^\circ - \theta) = \csc\theta$ and $\csc(90^\circ - \theta) = \sec\theta$

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Special Right Triangles & Exact Trigonometric Values

Exact trigonometric values for benchmark angles ($30^\circ, 45^\circ, 60^\circ$) derive directly from two geometric special right triangles.

1. The $45^\circ-45^\circ-90^\circ$ ($\frac{\pi}{4}-\frac{\pi}{4}-\frac{\pi}{2}$) Isosceles Right Triangle

• Side Ratio: $1 : 1 : \sqrt{2}$ (Leg : Leg : Hypotenuse)
• $\sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
• $\cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
• $\tan 45^\circ = \frac{1}{1} = 1$

2. The $30^\circ-60^\circ-90^\circ$ ($\frac{\pi}{6}-\frac{\pi}{3}-\frac{\pi}{2}$) Scalene Right Triangle

• Side Ratio: $1 : \sqrt{3} : 2$ (Short Leg opposite $30^\circ$ : Long Leg opposite $60^\circ$ : Hypotenuse)
• $\sin 30^\circ = \frac{1}{2} \qquad \cos 30^\circ = \frac{\sqrt{3}}{2} \qquad \tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$
• $\sin 60^\circ = \frac{\sqrt{3}}{2} \qquad \cos 60^\circ = \frac{1}{2} \qquad \tan 60^\circ = \frac{\sqrt{3}}{1} = \sqrt{3}$

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Radian Measure, Arc Length & Sector Area

A radian is the measure of a central angle $\theta$ that subtends a circular arc equal in length to the radius $r$ of the circle.

$$1\text{ full revolution} = 360^\circ = 2\pi\text{ radians} \implies 180^\circ = \pi\text{ radians}$$

Angle Conversion Formulas

• Degrees to Radians: Multiply by $\frac{\pi}{180^\circ}$ (e.g., $135^\circ \times \frac{\pi}{180^\circ} = \frac{3\pi}{4}\text{ rad}$)
• Radians to Degrees: Multiply by $\frac{180^\circ}{\pi}$ (e.g., $\frac{2\pi}{3}\text{ rad} \times \frac{180^\circ}{\pi} = 120^\circ$)

Circle Geometry Formulas with Radians

For a circle of radius $r$ with central angle $\theta$ strictly measured in radians:

+-----------------------------------------------------------------------------------------+
|                         CIRCULAR SECTOR & ARC FORMULAS                                  |
+-----------------------+---------------------------------+-------------------------------+
| Geometric Measure     | Formula (θ in Radians)          | Formula (θ in Degrees)        |
+-----------------------+---------------------------------+-------------------------------+
| Arc Length (s)        | s = r · θ                       | s = (θ / 360°) · 2πr          |
| Sector Area (A)       | A = (1/2) · r² · θ              | A = (θ / 360°) · πr²          |
+-----------------------+---------------------------------+-------------------------------+

Worked Example: Arc Length and Sector Area

A circular sector in a circle of radius $r = 6\text{ cm}$ is subtended by a central angle of $120^\circ$. Find the arc length $s$ and sector area $A$.

• Step 1 (Convert to radians): $\theta = 120^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{3}\text{ radians}$.
• Step 2 (Calculate arc length): $s = r \cdot \theta = 6 \times \frac{2\pi}{3} = 4\pi\text{ cm} \approx 12.57\text{ cm}$.
• Step 3 (Calculate sector area): $A = \frac{1}{2} r^2 \theta = \frac{1}{2} (6)^2 \left(\frac{2\pi}{3}\right) = \frac{1}{2} (36) \left(\frac{2\pi}{3}\right) = 18 \times \frac{2\pi}{3} = 12\pi\text{ cm}^2 \approx 37.70\text{ cm}^2$.

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The Unit Circle & Quadrant Trigonometry

The Unit Circle is a circle centered at the origin $(0, 0)$ with a radius of exactly $r = 1$, governed by the algebraic equation:

$$x^2 + y^2 = 1$$

For any angle $\theta$ drawn in standard position (vertex at origin, initial ray along positive $x$-axis), the terminal ray intersects the unit circle at coordinates:

$$(x, y) = (\cos\theta, \sin\theta) \quad \text{and} \quad \tan\theta = \frac{y}{x} = \frac{\sin\theta}{\cos\theta} \; (x \neq 0)$$

                                  The Unit Circle Quadrants
                                           y ▲ (0, 1) [90°, π/2]
                                             │
                         QUADRANT II         │         QUADRANT I
                         (-x, +y)            │         (+x, +y)
                         Sine & Csc (+)      │         ALL Functions (+)
                   [180°, π]                 │                 [0°, 0 rad]
             (-1, 0) ────────────────────────┼────────────────────────► (1, 0) x
                                             │                 [360°, 2π]
                         QUADRANT III        │         QUADRANT IV
                         (-x, -y)            │         (+x, -y)
                         Tan & Cot (+)       │         Cosine & Sec (+)
                                             │
                                           ▼ (0, -1) [270°, 3π/2]

Quadrant Signs Rule (ASTC: "All Students Take Calculus")

• Quadrant I ($0 < \theta < \frac{\pi}{2}$): All trigonometric functions are positive.
• Quadrant II ($\frac{\pi}{2} < \theta < \pi$): Sine (and cosecant) are positive; cosine and tangent are negative.
• Quadrant III ($\pi < \theta < \frac{3\pi}{2}$): Tangent (and cotangent) are positive; sine and cosine are negative.
• Quadrant IV ($\frac{3\pi}{2} < \theta < 2\pi$): Cosine (and secant) are positive; sine and tangent are negative.

Reference Angles ($\theta'$)

The reference angle $\theta'$ is the positive acute angle formed between the terminal side of $\theta$ and the $x$-axis:

Master Unit Circle Exact Values Reference Table

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Fundamental Trigonometric Identities

Quotient & Reciprocal Identities

$$\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \csc\theta = \frac{1}{\sin\theta}$$

The Three Pythagorean Identities

Substituting $x = \cos\theta$ and $y = \sin\theta$ into the unit circle equation $x^2 + y^2 = 1$ establishes the primary Pythagorean Identity:

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

Dividing this equation by $\cos^2\theta$ and $\sin^2\theta$ respectively yields two secondary Pythagorean identities:

$$1 + \tan^2\theta = \sec^2\theta \qquad 1 + \cot^2\theta = \csc^2\theta$$

Worked Example: Simplifying an Identity Expression

Simplify the trigonometric expression: $\frac{1 - \cos^2\theta}{\sin\theta \cos\theta}$.

• Step 1 (Apply Pythagorean identity): Since $\sin^2\theta + \cos^2\theta = 1$, we have $1 - \cos^2\theta = \sin^2\theta$.
• Step 2 (Substitute and reduce): $$\frac{\sin^2\theta}{\sin\theta \cos\theta} = \frac{\sin\theta}{\cos\theta} = \tan\theta$$

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Sinusoidal Functions & Their Graphs (Sine & Cosine)

Sinusoidal functions model periodic, oscillating phenomena (e.g., sound waves, tides, alternating current):

$$y = A \sin(B(x - C)) + D \quad \text{or} \quad y = A \cos(B(x - C)) + D$$

Sinusoidal Parameters Explained

1. Amplitude ($|A|$): Half the distance between the maximum and minimum values: $$\text{Amplitude} = |A| = \frac{y_{\max} - y_{\min}}{2}$$ (If $A < 0$, the graph is reflected vertically across its midline).
2. Period ($T$): The horizontal distance required to complete one full cycle: $$T = \frac{2\pi}{B} \iff B = \frac{2\pi}{T}$$
3. Frequency ($f$): Number of cycles completed per unit of $x$: $$f = \frac{1}{T} = \frac{B}{2\pi}$$
4. Phase Shift ($C$): Horizontal translation (shifts right if $C > 0$, left if $C < 0$).
5. Vertical Shift / Midline ($y = D$): The horizontal centerline of oscillation: $$\text{Midline } y = D = \frac{y_{\max} + y_{\min}}{2}$$

Parent Graphs Comparison: $\sin(x)$ vs. $\cos(x)$