4.1 Fundamental Trigonometric Identities & Angle Addition Formulas

Fundamental Trigonometric Identities & Angle Addition Formulas

In the Intermediate Mathematics Part I curriculum and the University of Engineering and Technology (UET) ECAT entry test, trigonometry constitutes one of the most heavily weighted domains. Because the ECAT strictly prohibits handheld calculators while imposing an average pace of 60 seconds per question across 100 multiple-choice questions, candidates cannot rely on numeric approximations. Success requires instantaneous recognition of structural identities, fluency with circular coordinate definitions, and rapid deployment of algebraic shortcuts for angle reductions and product-to-sum conversions.

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1. Circular Angle Measurement, Radians & Sector Geometry

In circular geometry, an angle $\theta$ is formed by rotating a directed ray from an initial side to a terminal side around a vertex. While degree measure divides a full circular rotation into $360^\circ$, advanced engineering mathematics and calculus rely fundamentally on the radian.

The Radian Definition

A radian is defined as the measure of the central angle subtended by an arc of a circle whose length equals the radius of the circle:

$\theta = \frac{s}{r}$

where $s$ is the arc length and $r$ is the radius. Since the circumference of a circle is $C = 2\pi r$, a full revolution subtends $2\pi$ radians:

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

$1^\circ = \frac{\pi}{180} \text{ rad} \approx 0.01745 \text{ rad}, \quad 1 \text{ rad} = \left(\frac{180}{\pi}\right)^\circ \approx 57^\circ 17' 45''$

Core Metric Formulas for Circular Sectors

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

1. Arc Length: $s = r\theta$
2. Area of a Circular Sector: $A = \frac{1}{2}r^2 \theta = \frac{1}{2}rs$

ECAT Trap Alert: Applying $s = r\theta$ or $A = \frac{1}{2}r^2\theta$ with an angle in degrees produces completely incorrect results. Always convert $\theta^\circ$ to radians by multiplying by $\frac{\pi}{180^\circ}$ before computing metric lengths or sector areas.

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2. Signs of Trigonometric Ratios and the ASTC Rule

Let $P(x, y)$ be a point on the terminal side of an angle $\theta$ in standard position on the Cartesian plane, at a distance $r = \sqrt{x^2 + y^2} > 0$ from the origin. The six fundamental circular functions are defined as:

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

$\csc\theta = \frac{r}{y} \quad (y \neq 0), \quad \sec\theta = \frac{r}{x} \quad (x \neq 0), \quad \cot\theta = \frac{x}{y} \quad (y \neq 0)$

Because $r > 0$ is unconditionally positive, the algebraic signs of the trigonometric functions depend entirely on the coordinates $(x, y)$ of the point $P$.

The ASTC Quadrant Distribution

The quadrant signs are memorized via the mnemonic ASTC ("All Students Take Calculus" or "Add Sugar To Coffee"):

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3. Allied Angles and Systematic Reduction Formulas

Angles of the form $\left(n \cdot \frac{\pi}{2} \pm \theta\right)$ where $n \in \mathbb{Z}$ and $\theta$ is an acute angle are termed allied angles. In ECAT problems, simplifying trigonometric expressions with massive positive or negative arguments requires the systematic Allied Angle Reduction Algorithm.

The Two-Step Reduction Algorithm

1. Step 1: Function Identity Check (Parity of $n$):

   • If $n$ is even ($n = 0, \pm 2, \pm 4, \dots$, corresponding to rotations off the horizontal x-axis: $k\pi \pm \theta$): $\text{Trig}\left(n\frac{\pi}{2} \pm \theta\right) = \pm \text{Trig}(\theta)$ (The base function remains unchanged: $\sin \to \sin$, $\cos \to \cos$, $\tan \to \tan$).
   • If $n$ is odd ($n = \pm 1, \pm 3, \pm 5, \dots$, corresponding to rotations off the vertical y-axis: $\frac{(2k+1)\pi}{2} \pm \theta$): $\text{Trig}\left(n\frac{\pi}{2} \pm \theta\right) = \pm \text{Co-Trig}(\theta)$ (The function converts into its co-function: $\sin \leftrightarrow \cos$, $\tan \leftrightarrow \cot$, $\sec \leftrightarrow \csc$).

2. Step 2: Sign Determination:

   • Treat $\theta$ as an acute angle ($0 < \theta < \frac{\pi}{2}$) to determine which quadrant the angle $\left(n\frac{\pi}{2} \pm \theta\right)$ falls in.
   • Apply the algebraic sign ($+$ or $-$) of the original (input) trigonometric function in that specific quadrant.

Common Allied Angle Equalities

• Quadrant II Reductions:
  • $\sin(\pi - \theta) = \sin\theta, \quad \cos(\pi - \theta) = -\cos\theta, \quad \tan(\pi - \theta) = -\tan\theta$
  • $\sin\left(\frac{\pi}{2} + \theta\right) = \cos\theta, \quad \cos\left(\frac{\pi}{2} + \theta\right) = -\sin\theta, \quad \tan\left(\frac{\pi}{2} + \theta\right) = -\cot\theta$
• Quadrant III Reductions:
  • $\sin(\pi + \theta) = -\sin\theta, \quad \cos(\pi + \theta) = -\cos\theta, \quad \tan(\pi + \theta) = \tan\theta$
  • $\sin\left(\frac{3\pi}{2} - \theta\right) = -\cos\theta, \quad \cos\left(\frac{3\pi}{2} - \theta\right) = -\sin\theta$
• Quadrant IV Reductions & Parity:
  • $\sin(2\pi - \theta) = \sin(-\theta) = -\sin\theta$ (Odd function)
  • $\cos(2\pi - \theta) = \cos(-\theta) = \cos\theta$ (Even function)
  • $\tan(2\pi - \theta) = \tan(-\theta) = -\tan\theta$ (Odd function)

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4. Fundamental Pythagorean and Reciprocal Identities

From the Pythagorean theorem on the Cartesian plane $x^2 + y^2 = r^2$, dividing by $r^2$, $x^2$, and $y^2$ respectively establishes the three fundamental Pythagorean identities:

1. Primary Pythagorean Identity: $\sin^2\theta + \cos^2\theta = 1$
   • Derived rearrangements: $\sin^2\theta = 1 - \cos^2\theta$, $\cos^2\theta = 1 - \sin^2\theta$.
2. Secant-Tangent Identity: $1 + \tan^2\theta = \sec^2\theta \implies \sec^2\theta - \tan^2\theta = 1$
3. Cosecant-Cotangent Identity: $1 + \cot^2\theta = \csc^2\theta \implies \csc^2\theta - \cot^2\theta = 1$

The Conjugate Difference-of-Squares Shortcut

Factoring $\sec^2\theta - \tan^2\theta = 1$ and $\csc^2\theta - \cot^2\theta = 1$ yields two powerful reciprocal relations frequently tested on the ECAT:

$(\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1 \implies \sec\theta - \tan\theta = \frac{1}{\sec\theta + \tan\theta}$

$(\csc\theta - \cot\theta)(\csc\theta + \cot\theta) = 1 \implies \csc\theta - \cot\theta = \frac{1}{\csc\theta + \cot\theta}$

ECAT Speed Example: If $\sec\theta + \tan\theta = p$, then instantly $\sec\theta - \tan\theta = \frac{1}{p}$. Adding both expressions yields $2\sec\theta = p + \frac{1}{p} = \frac{p^2 + 1}{p} \implies \sec\theta = \frac{p^2 + 1}{2p}$, and subtracting yields $\tan\theta = \frac{p^2 - 1}{2p}$. Consequently, $\sin\theta = \frac{\tan\theta}{\sec\theta} = \frac{p^2 - 1}{p^2 + 1}$, obtained entirely without drawing triangles.

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5. The Fundamental Law of Trigonometry & Compound Angle Formulas

The Fundamental Law of Trigonometry governs the cosine of the difference between two angles. In the unit circle, calculating the Euclidean distance between points $A(\cos\alpha, \sin\alpha)$ and $B(\cos\beta, \sin\beta)$ and rotating the chord to the initial position $(1, 0)$ proves:

$\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta$

Deductions from the Fundamental Law

By substituting allied angles, all remaining compound angle formulas are systematically derived:

1. Cosine Addition: Replace $\beta$ with $-\beta$: $\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$
2. Sine Addition & Subtraction: Using $\sin\phi = \cos\left(\frac{\pi}{2} - \phi\right)$: $\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$ $\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta$
3. Tangent Addition & Subtraction: Dividing $\sin(\alpha \pm \beta)$ by $\cos(\alpha \pm \beta)$ and dividing numerator and denominator by $\cos\alpha\cos\beta$: $\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}$ $\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}$
4. Cotangent Addition & Subtraction: $\cot(\alpha \pm \beta) = \frac{\cot\alpha\cot\beta \mp 1}{\cot\beta \pm \cot\alpha}$

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6. Double-Angle, Half-Angle, and Triple-Angle Identities

Setting $\alpha = \beta = \theta$ in the compound angle formulas yields the double-angle identities:

Double-Angle Formulas

1. Sine Double-Angle: $\sin(2\theta) = 2\sin\theta\cos\theta = \frac{2\tan\theta}{1 + \tan^2\theta}$
2. Cosine Double-Angle: $\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta = \frac{1 - \tan^2\theta}{1 + \tan^2\theta}$
3. Tangent Double-Angle: $\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$

Half-Angle Identities (Power Reduction)

Isolating $\sin^2\theta$ and $\cos^2\theta$ from the cosine double-angle forms establishes the primary power-reduction identities:

$\sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{2} \implies \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{2}}$

$\cos^2\left(\frac{\theta}{2}\right) = \frac{1 + \cos\theta}{2} \implies \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos\theta}{2}}$

$\tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} = \frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}$

Triple-Angle Formulas

Expanding $\sin(2\theta + \theta)$ and $\cos(2\theta + \theta)$ yields:

1. Sine Triple-Angle: $\sin(3\theta) = 3\sin\theta - 4\sin^3\theta$
2. Cosine Triple-Angle: $\cos(3\theta) = 4\cos^3\theta - 3\cos\theta$
3. Tangent Triple-Angle: $\tan(3\theta) = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$

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7. Product-to-Sum and Sum-to-Product Transformations

In ECAT integration and algebraic simplification, transforming products into sums or sums into factorized products is essential.

Product-to-Sum Formulas

Adding and subtracting the sine and cosine compound angle expansions produces:

$2\sin\alpha\cos\beta = \sin(\alpha + \beta) + \sin(\alpha - \beta)$

$2\cos\alpha\sin\beta = \sin(\alpha + \beta) - \sin(\alpha - \beta)$

$2\cos\alpha\cos\beta = \cos(\alpha + \beta) + \cos(\alpha - \beta)$

$-2\sin\alpha\sin\beta = \cos(\alpha + \beta) - \cos(\alpha - \beta) \iff 2\sin\alpha\sin\beta = \cos(\alpha - \beta) - \cos(\alpha + \beta)$

Sum-to-Product Formulas (Factorization Formulas)

Setting $\alpha + \beta = C$ and $\alpha - \beta = D$ (so $\alpha = \frac{C+D}{2}$ and $\beta = \frac{C-D}{2}$) yields:

$\sin C + \sin D = 2\sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right) \quad [S + S = 2SC]$

$\sin C - \sin D = 2\cos\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right) \quad [S - S = 2CS]$

$\cos C + \cos D = 2\cos\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right) \quad [C + C = 2CC]$

$\cos C - \cos D = -2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right) = 2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{D-C}{2}\right) \quad [C - C = -2SS]$

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8. High-Speed ECAT Special Angle Shortcuts & Triple Products

Rapid Values for $15^\circ$ and $75^\circ$

Using $\sin(45^\circ - 30^\circ)$ and $\cos(45^\circ - 30^\circ)$:

• $\sin(15^\circ) = \cos(75^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}$
• $\cos(15^\circ) = \sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}$
• $\tan(15^\circ) = \cot(75^\circ) = \frac{\sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}} = 2 - \sqrt{3}$
• $\tan(75^\circ) = \cot(15^\circ) = 2 + \sqrt{3}$

Rapid Values for $18^\circ$ and $36^\circ$

Let $\theta = 18^\circ$, so $5\theta = 90^\circ \implies 2\theta = 90^\circ - 3\theta$. Taking the sine of both sides yields $\sin 2\theta = \cos 3\theta$, which expands to $2\sin\theta\cos\theta = 4\cos^3\theta - 3\cos\theta$. Dividing by $\cos\theta \neq 0$ and using $\cos^2\theta = 1 - \sin^2\theta$ produces the quadratic $4\sin^2\theta + 2\sin\theta - 1 = 0$. Solving for acute $\theta$ establishes:

• $\sin(18^\circ) = \cos(72^\circ) = \frac{\sqrt{5} - 1}{4}$
• $\cos(36^\circ) = \sin(54^\circ) = 1 - 2\sin^2(18^\circ) = \frac{\sqrt{5} + 1}{4}$
• $\cos(18^\circ) = \sin(72^\circ) = \frac{\sqrt{10 + 2\sqrt{5}}}{4}$
• $\sin(36^\circ) = \cos(54^\circ) = \frac{\sqrt{10 - 2\sqrt{5}}}{4}$

The 60-Degree Symmetrical Triple Product Identity

A classic ECAT problem structure asks for products of three related trigonometric terms:

$\sin\theta \cdot \sin(60^\circ - \theta) \cdot \sin(60^\circ + \theta) = \frac{1}{4}\sin(3\theta)$

$\cos\theta \cdot \cos(60^\circ - \theta) \cdot \cos(60^\circ + \theta) = \frac{1}{4}\cos(3\theta)$

$\tan\theta \cdot \tan(60^\circ - \theta) \cdot \tan(60^\circ + \theta) = \tan(3\theta)$

Worked Example: Evaluate $\sin 20^\circ \sin 40^\circ \sin 80^\circ$ in 10 seconds:

• Recognize $\theta = 20^\circ$, with $60^\circ - 20^\circ = 40^\circ$ and $60^\circ + 20^\circ = 80^\circ$.
• Apply the identity: $\frac{1}{4}\sin(3 \times 20^\circ) = \frac{1}{4}\sin(60^\circ) = \frac{1}{4}\left(\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{3}}{8}$.