Trigonometry: Identities and Equations

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Trigonometry Identities — Key Facts

Fundamental Identities:

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

Reciprocal Identities: $$\sin\theta = \frac{1}{\csc\theta},\quad \cos\theta = \frac{1}{\sec\theta},\quad \tan\theta = \frac{\sin\theta}{\cos\theta}$$ $$\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{1}{\tan\theta}$$

Signs of Trig Functions by Quadrant:

Quadrant	sin	cos	tan
I (0° to 90°)	+	+	+
II (90° to 180°)	+	-	-
III (180° to 270°)	-	-	+
IV (270° to 360°)	-	+	-

ASTC Rule: All Students Take Calculus (positive in QI, sin positive in QII, tan positive in QIII, cos positive in QIV)

Radians and Degrees: $$180° = \pi \text{ radians}$$ $$1° = \frac{\pi}{180} \text{ rad},\quad 1 \text{ rad} = \frac{180°}{\pi}$$

⚡ ECAT Exam Tip: For solving equations, find the principal value first, then use periodicity (sin and cos: 360° or 2π; tan: 180° or π) to find all solutions.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Compound Angle Formulas:

$$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$$ $$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$$ $$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$$

Double Angle Formulas:

From compound angles with A = B: $$\sin 2A = 2\sin A \cos A = \frac{2\tan A}{1+\tan^2 A}$$ $$\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A = \frac{1-\tan^2 A}{1+\tan^2 A}$$ $$\tan 2A = \frac{2\tan A}{1-\tan^2 A}$$

Half Angle Formulas:

$$\sin\frac{A}{2} = \pm\sqrt{\frac{1-\cos A}{2}}$$ $$\cos\frac{A}{2} = \pm\sqrt{\frac{1+\cos A}{2}}$$ $$\tan\frac{A}{2} = \pm\sqrt{\frac{1-\cos A}{1+\cos A}} = \frac{\sin A}{1+\cos A} = \frac{1-\cos A}{\sin A}$$

Solving Trigonometric Equations:

Example 1: Solve $\sin x = \frac{\sqrt{3}}{2}$ for $0° ≤ x ≤ 360°$

• $\sin x = \sqrt{3}/2$ at x = 60° and x = 180° - 60° = 120°
• General solution: x = nπ + (-1)ⁿ(π/3) or x = 60° + 360°n, 120° + 360°n

Example 2: Solve $\tan 2x = 1$ for $0 ≤ x < π$

• $\tan 2x = 1$ at 2x = π/4, 5π/4
• So x = π/8, 5π/8

⚡ ECAT Exam Tip: Always check for extraneous solutions when squaring both sides. Also verify solutions don’t make any denominator zero.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Sum and Product Identities:

$$\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}$$ $$\sin A - \sin B = 2\cos\frac{A+B}{2}\sin\frac{A-B}{2}$$ $$\cos A + \cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2}$$ $$\cos A - \cos B = -2\sin\frac{A+B}{2}\sin\frac{A-B}{2}$$

Product to Sum: $$\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$$ $$\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$$ $$\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$$

Triple Angle Formulas:

$$\sin 3A = 3\sin A - 4\sin^3 A$$ $$\cos 3A = 4\cos^3 A - 3\cos A$$ $$\tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A}$$

Inverse Trigonometric Functions:

Function	Domain	Range
$\sin^{-1}x$	$[-1, 1]$	$[-\frac{\pi}{2}, \frac{\pi}{2}]$
$\cos^{-1}x$	$[-1, 1]$	$[0, \pi]$
$\tan^{-1}x$	$\mathbb{R}$	$(-\frac{\pi}{2}, \frac{\pi}{2})$

Important Properties: $$\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \text{ for } x \in [-1,1]$$ $$\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \text{ for } x \in \mathbb{R}$$ $$\tan^{-1}a + \tan^{-1}b = \tan^{-1}\frac{a+b}{1-ab} \text{ (when } ab<1\text{)}$$

Submultiple Angle Formulas:

From $\cos 3A = 4\cos^3 A - 3\cos A$: If $3A = \theta$, then $\cos\theta = 4\cos^3\frac{\theta}{3} - 3\cos\frac{\theta}{3}$

General Solutions:

Equation	General Solution
$\sin x = \sin\alpha$	$x = n\pi + (-1)^n\alpha$
$\cos x = \cos\alpha$	$x = 2n\pi \pm \alpha$
$\tan x = \tan\alpha$	$x = n\pi + \alpha$

Solving Systems of Trigonometric Equations:

Example: Solve $\sin x + \sin y = \sqrt{3}$ and $x + y = \pi/3$ Using sum-to-product: $2\sin\frac{x+y}{2}\cos\frac{x-y}{2} = \sqrt{3}$ $2\sin\frac{\pi}{6}\cos\frac{x-y}{2} = \sqrt{3}$ $2 \times \frac{1}{2} \times \cos\frac{x-y}{2} = \sqrt{3}$ $\cos\frac{x-y}{2} = \sqrt{3}$ — no solution since $\cos\theta ≤ 1$

⚡ ECAT 2023 Question Analysis: Questions involving $\tan^{-1}x + \tan^{-1}y$ formulas and general solutions appeared in recent papers. Also watch for equations requiring the identity $\sin 2x = 2\tan x/(1+\tan^2x)$ for solving in terms of $\tan x$.

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