Comprehensive Trigonometry Reference for Precalculus

Trigonometric Functions: Definitions and Properties

Right Triangle Definition

Trigonometric functions can be defined using the sides of a right triangle, where θ is an acute angle (0 < θ < π/2 or 0° < θ < 90°). The sides are labeled as opposite (to θ), adjacent (to θ), and hypotenuse (the longest side).

• sin(θ) = opposite / hypotenuse

• cos(θ) = adjacent / hypotenuse

• tan(θ) = opposite / adjacent

• csc(θ) = hypotenuse / opposite

• sec(θ) = hypotenuse / adjacent

• cot(θ) = adjacent / opposite

Unit Circle Definition

For any angle θ, the trigonometric functions can also be defined using the unit circle (a circle of radius 1 centered at the origin). If a point (x, y) lies on the unit circle and forms an angle θ with the positive x-axis, then:

• sin(θ) = y

• cos(θ) = x

• tan(θ) = y / x

• csc(θ) = 1 / y

• sec(θ) = 1 / x

• cot(θ) = x / y

Key Properties of Trigonometric Functions

• Domain:

  • sin(θ), cos(θ): θ can be any angle

  • tan(θ), sec(θ): θ ≠ (n + 1/2)π, n ∈ ℤ

  • csc(θ), cot(θ): θ ≠ nπ, n ∈ ℤ

• Range:

  • sin(θ), cos(θ): −1 ≤ value ≤ 1

  • tan(θ), cot(θ): (−∞, ∞)

  • sec(θ), csc(θ): value ≤ −1 or value ≥ 1

• Period:

  • sin(ωθ), cos(ωθ), csc(ωθ), sec(ωθ): T = 2π/ω

  • tan(ωθ), cot(ωθ): T = π/ω

Trigonometric Identities and Formulas

Basic Identities

• Tangent and Cotangent:

  • $tan(\theta) = \frac{sin(\theta)}{cos(\theta)}$

  • $cot(\theta) = \frac{cos(\theta)}{sin(\theta)}$

• Reciprocal Identities:

  • $csc(\theta) = \frac{1}{sin(\theta)}$

  • $sec(\theta) = \frac{1}{cos(\theta)}$

  • $cot(\theta) = \frac{1}{tan(\theta)}$

• Pythagorean Identities:

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

  • $tan^2(\theta) + 1 = sec^2(\theta)$

  • $1 + cot^2(\theta) = csc^2(\theta)$

Even/Odd Properties

• $sin(-\theta) = -sin(\theta)$

• $cos(-\theta) = cos(\theta)$

• $tan(-\theta) = -tan(\theta)$

• Similar properties hold for csc, sec, and cot.

Periodicity

• $sin(\theta + 2\pi n) = sin(\theta)$

• $cos(\theta + 2\pi n) = cos(\theta)$

• $tan(\theta + \pi n) = tan(\theta)$

• where n is any integer.

Angle Conversion

• Degrees to radians: $t = \frac{\pi x}{180}$

• Radians to degrees: $x = \frac{180t}{\pi}$

Double Angle Formulas

• $sin(2\theta) = 2 sin(\theta) cos(\theta)$

• $cos(2\theta) = cos^2(\theta) - sin^2(\theta) = 2 cos^2(\theta) - 1 = 1 - 2 sin^2(\theta)$

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

Half Angle Formulas

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

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

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

Sum and Difference Formulas

• $sin(\alpha \pm \beta) = sin(\alpha) cos(\beta) \pm cos(\alpha) sin(\beta)$

• $cos(\alpha \pm \beta) = cos(\alpha) cos(\beta) \mp sin(\alpha) sin(\beta)$

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

Product-to-Sum and Sum-to-Product Formulas

• $sin(\alpha) sin(\beta) = \frac{1}{2} [cos(\alpha - \beta) - cos(\alpha + \beta)]$

• $cos(\alpha) cos(\beta) = \frac{1}{2} [cos(\alpha - \beta) + cos(\alpha + \beta)]$

• $sin(\alpha) + sin(\beta) = 2 sin\left(\frac{\alpha + \beta}{2}\right) cos\left(\frac{\alpha - \beta}{2}\right)$

• $cos(\alpha) + cos(\beta) = 2 cos\left(\frac{\alpha + \beta}{2}\right) cos\left(\frac{\alpha - \beta}{2}\right)$

Cofunction Formulas

• $sin\left(\frac{\pi}{2} - \theta\right) = cos(\theta)$

• $cos\left(\frac{\pi}{2} - \theta\right) = sin(\theta)$

• $tan\left(\frac{\pi}{2} - \theta\right) = cot(\theta)$

• Similar relationships hold for sec and csc.

The Unit Circle and Special Angles

Coordinates and Values

The unit circle provides the values of sine and cosine for common angles. For any ordered pair (x, y) on the unit circle, $cos(\theta) = x$ and $sin(\theta) = y$.

• Example: $cos\left(\frac{5\pi}{3}\right) = \frac{1}{2}$, $sin\left(\frac{5\pi}{3}\right) = -\frac{\sqrt{3}}{2}$

Inverse Trigonometric Functions

Definitions and Properties

• $y = sin^{-1}(x)$ is equivalent to $x = sin(y)$

• $y = cos^{-1}(x)$ is equivalent to $x = cos(y)$

• $y = tan^{-1}(x)$ is equivalent to $x = tan(y)$

• Alternate notation: $sin^{-1}(x) = arcsin(x)$, $cos^{-1}(x) = arccos(x)$, $tan^{-1}(x) = arctan(x)$

Domain and Range Table

Function	Domain	Range
$y = sin^{-1}(x)$	$-1 \leq x \leq 1$	$-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$
$y = cos^{-1}(x)$	$-1 \leq x \leq 1$	$0 \leq y \leq \pi$
$y = tan^{-1}(x)$	$-\infty < x < \infty$	$-\frac{\pi}{2} < y < \frac{\pi}{2}$

Inverse Properties

• $sin(sin^{-1}(x)) = x$

• $sin^{-1}(sin(\theta)) = \theta$

• Similar properties hold for cosine and tangent.

Law of Sines, Cosines, and Tangents

Law of Sines

• $\frac{sin(\alpha)}{a} = \frac{sin(\beta)}{b} = \frac{sin(\gamma)}{c}$

Law of Cosines

• $a^2 = b^2 + c^2 - 2bc \cos(\alpha)$

• $b^2 = a^2 + c^2 - 2ac \cos(\beta)$

• $c^2 = a^2 + b^2 - 2ab \cos(\gamma)$

Law of Tangents

• $\frac{a-b}{a+b} = \frac{tan\left(\frac{1}{2}(\alpha-\beta)\right)}{tan\left(\frac{1}{2}(\alpha+\beta)\right)}$

• Similar formulas apply for other pairs of sides and angles.

Mollweide’s Formula

• $\frac{a+b}{c} = \frac{cos\left(\frac{1}{2}(\alpha-\beta)\right)}{sin\left(\frac{1}{2}\gamma\right)}$

Additional info: This guide summarizes the essential trigonometric definitions, identities, and laws relevant to precalculus, including both right triangle and unit circle perspectives, as well as inverse functions and key formulas for solving triangles.