4.4 Trigonometric Functions, Identities & Right-Triangle Applications

4.4 Trigonometric Functions, Identities & Right-Triangle Applications

Trigonometry on the Bocconi Admission Test bridges geometric intuition and algebraic manipulation. Trigonometric problems appear as standalone algebraic identity simplifications, trigonometric equation solving, or right-triangle side evaluations within composite geometric figures. Because calculators are prohibited, candidates must master exact trigonometric values at standard reference angles, unit circle symmetries, and algebraic substitutions using Pythagorean and double-angle identities.

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1. Radian Measure, Conversion & The Unit Circle

Radian Definition

A radian is the measure of a central angle that subtends an arc equal in length to the radius of the circle: $\theta = \frac{s}{r}$.

• A full revolution ($360^\circ$) equals the ratio of circumference to radius: $\frac{2\pi r}{r} = 2\pi$ radians.
• Fundamental Equivalence: $$\pi \text{ rad} = 180^\circ$$
• Conversion Formulas: $$\text{Radians} = \text{Degrees} \times \frac{\pi}{180^\circ}, \qquad \text{Degrees} = \text{Radians} \times \frac{180^\circ}{\pi}$$

The Unit Circle Definition

In the Cartesian plane, the unit circle is defined by $x^2 + y^2 = 1$ centered at the origin $O(0, 0)$. For any real number $\theta$, consider the terminal ray rotating counterclockwise from the positive $x$-axis. The terminal ray intersects the unit circle at a unique point $P(x, y)$:

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

Quadrant Signs: The ASTC Rule

The signs of the trigonometric functions across the four Cartesian quadrants follow the mnemonic ASTC ("All Students Take Calculus"):

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2. Table of Exact Trigonometric Values at Reference Angles

Instant recall of exact trigonometric values at standard angles is essential for non-calculator exam speed:

Mental Mnemonic for Sines: From $0$ to $\frac{\pi}{2}$, the values of $\sin(\theta)$ follow $\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}$. Cosine reverses this sequence.

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3. Graphs, Periodicity & Asymptotes

Sine: $y = \sin(x)$

• Domain: $\mathbb{R} = (-\infty, \infty)$.
• Range: $[-1, 1]$.
• Periodicity: Fundamental period $T = 2\pi$: $\sin(x + 2k\pi) = \sin(x)$ for all $k \in \mathbb{Z}$.
• Parity: Odd function (symmetric about origin): $\sin(-x) = -\sin(x)$.
• Zeros: $x = k\pi$ for $k \in \mathbb{Z}$.

Cosine: $y = \cos(x)$

• Domain: $\mathbb{R} = (-\infty, \infty)$.
• Range: $[-1, 1]$.
• Periodicity: Fundamental period $T = 2\pi$: $\cos(x + 2k\pi) = \cos(x)$ for all $k \in \mathbb{Z}$.
• Parity: Even function (symmetric about $y$-axis): $\cos(-x) = \cos(x)$.
• Zeros: $x = \frac{\pi}{2} + k\pi$ for $k \in \mathbb{Z}$.

Tangent: $y = \tan(x)$

• Domain: $\mathbb{R} \setminus \{\frac{\pi}{2} + k\pi \mid k \in \mathbb{Z}\}$.
• Range: $(-\infty, \infty)$.
• Periodicity: Fundamental period $T = \pi$: $\tan(x + k\pi) = \tan(x)$ for all $k \in \mathbb{Z}$.
• Parity: Odd function: $\tan(-x) = -\tan(x)$.
• Vertical Asymptotes: Occur where $\cos(x) = 0$, at $x = \frac{\pi}{2} + k\pi$.

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4. Fundamental Trigonometric Identities

The Pythagorean Identity and Its Variants

Because $(x, y) = (\cos\theta, \sin\theta)$ lies on the unit circle $x^2 + y^2 = 1$:

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

Dividing this master identity yields two secondary forms:

1. Dividing by $\cos^2(\theta)$ (for $\cos\theta \neq 0$): $$1 + \tan^2(\theta) = \frac{1}{\cos^2(\theta)} = \sec^2(\theta)$$
2. Dividing by $\sin^2(\theta)$ (for $\sin\theta \neq 0$): $$1 + \cot^2(\theta) = \frac{1}{\sin^2(\theta)} = \csc^2(\theta)$$

Cofunction and Supplementary Identities

• Cofunction (Complementary angles, sum = $\frac{\pi}{2}$): $$\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta), \qquad \cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta), \qquad \tan\left(\frac{\pi}{2} - \theta\right) = \cot(\theta)$$
• Supplementary Angles (sum = $\pi$): $$\sin(\pi - \theta) = \sin(\theta), \qquad \cos(\pi - \theta) = -\cos(\theta), \qquad \tan(\pi - \theta) = -\tan(\theta)$$

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5. Double-Angle and Addition Formulas

Angle Addition and Subtraction Formulas

$$\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$$ $$\cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)$$ $$\tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)}$$

Double-Angle Formulas

Setting $\alpha = \beta = \theta$ yields the core double-angle relations:

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

Power-Reduction Formulas

Rearranging the cosine double-angle equations expresses squared terms as linear harmonics:

$$\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}, \qquad \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$$

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6. Right-Triangle Trigonometry (SOH CAH TOA)

In a Euclidean right-angled triangle with acute angle $\theta$:

$$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}, \qquad \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}, \qquad \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$$

Oblique Triangle Laws

For any general triangle with sides $a, b, c$ and opposite angles $A, B, C$:

• Law of Sines: Relates sides to the circumradius $R$: $$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} = 2R$$
• Law of Cosines: The generalization of the Pythagorean theorem: $$c^2 = a^2 + b^2 - 2ab\cos(C)$$
• Trigonometric Area: $$\text{Area} = \frac{1}{2} a b \sin(C)$$

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7. Solving Elementary Trigonometric Equations on $[0, 2\pi)$

Trigonometric equations test your ability to factor polynomial structures and identify all angular solutions within a specified interval.

The 3-Step Solution Protocol

1. Unify the Argument: Use double-angle identities to express all terms with the same argument (e.g., convert $\sin(2x)$ to $2\sin x \cos x$).
2. Form a Polynomial in a Single Function: Use $\sin^2 x + \cos^2 x = 1$ to obtain an equation entirely in $\sin x$ or entirely in $\cos x$.
3. Substitute and Solve: Let $u = \sin x$ (or $\cos x$), solve the polynomial for $u \in [-1, 1]$, and find all corresponding angles $x$ on $[0, 2\pi)$.

Critical Bocconi Trap: Never divide both sides by a trigonometric function! In an equation like $\sin(2x) = \cos(x) \implies 2\sin x \cos x = \cos x$, dividing by $\cos x$ loses the solutions where $\cos x = 0$. Instead, factor: $\cos x (2\sin x - 1) = 0$.

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8. Worked Step-by-Step Examples

Example 1: Evaluating Functions from Quadrant Data

Given that $\sin(\theta) = -\frac{5}{13}$ and $\pi < \theta < \frac{3\pi}{2}$ (Quadrant III), calculate $\cos(\theta)$ and $\cos(2\theta)$.

1. Find $\cos(\theta)$: $$\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \left(-\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169}$$ In Quadrant III, cosine is strictly negative: $\cos(\theta) = -\sqrt{\frac{144}{169}} = -\frac{12}{13}$.
2. Calculate $\cos(2\theta)$ using $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$: $$\cos(2\theta) = \left(-\frac{12}{13}\right)^2 - \left(-\frac{5}{13}\right)^2 = \frac{144}{169} - \frac{25}{169} = \frac{119}{169}$$

Example 2: Solving a Factored Trigonometric Equation

Find all solutions to $2\sin^2(x) + 3\cos(x) - 3 = 0$ in the interval $[0, 2\pi)$.

1. Substitute $\sin^2(x) = 1 - \cos^2(x)$: $$2(1 - \cos^2 x) + 3\cos x - 3 = 0 \implies 2 - 2\cos^2 x + 3\cos x - 3 = 0 \implies -2\cos^2 x + 3\cos x - 1 = 0$$
2. Multiply by $-1$: $2\cos^2 x - 3\cos x + 1 = 0$.
3. Factor the quadratic: $(2\cos x - 1)(\cos x - 1) = 0$.
4. Set each factor to zero:
   • Branch 1: $\cos x = 1 \implies x = 0$ on $[0, 2\pi)$.
   • Branch 2: $\cos x = \frac{1}{2} \implies x = \frac{\pi}{3}$ (Quadrant I) and $x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}$ (Quadrant IV).
5. Complete solution set: $x \in \{0, \frac{\pi}{3}, \frac{5\pi}{3}\}$ (3 distinct solutions).

Example 3: Identity Simplification

Simplify $E = \frac{\sin(2\theta) - \sin(\theta)}{\cos(2\theta) - \cos(\theta) + 1}$ for all $\theta$ where defined.

1. Expand double angles: $\sin(2\theta) = 2\sin\theta\cos\theta$ and $\cos(2\theta) = 2\cos^2\theta - 1$.
2. Numerator: $2\sin\theta\cos\theta - \sin\theta = \sin\theta(2\cos\theta - 1)$.
3. Denominator: $(2\cos^2\theta - 1) - \cos\theta + 1 = 2\cos^2\theta - \cos\theta = \cos\theta(2\cos\theta - 1)$.
4. Cancel common non-zero factor $(2\cos\theta - 1)$: $$E = \frac{\sin\theta(2\cos\theta - 1)}{\cos\theta(2\cos\theta - 1)} = \frac{\sin\theta}{\cos\theta} = \tan(\theta)$$