Trigonometric Functions

Trigonometric Functions

Trigonometric functions relate angles to the ratios of sides in a right triangle and are fundamental to mathematics. Here's what you need to know:

Basic Definitions

In Right Triangles

For an angle θ in a right triangle:

• $$\sin(\theta) = \frac{opposite}{hypotenuse}$$
• $$\cos(\theta) = \frac{adjacent}{hypotenuse}$$
• $$\tan(\theta) = \frac{opposite}{adjacent} = \frac{\sin(\theta)}{\cos(\theta)}$$
• $$\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{hypotenuse}{opposite}$$
• $$\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{hypotenuse}{adjacent}$$
• $$\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{adjacent}{opposite}$$

Unit Circle Definition

For a point (x, y) on the unit circle corresponding to angle θ:

• $$\sin(\theta) = y$$
• $$\cos(\theta) = x$$
• $$\tan(\theta) = \frac{y}{x}$$ (when x ≠ 0)

Domains and Ranges

• Sine: Domain: All real numbers; Range: [-1, 1]
• Cosine: Domain: All real numbers; Range: [-1, 1]
• Tangent: Domain: All real numbers except $$\frac{\pi}{2} + n\pi$$ (where n is an integer); Range: All real numbers
• Cosecant: Domain: All real numbers except $$n\pi$$ (where n is an integer); Range: (-∞, -1] ∪ [1, ∞)
• Secant: Domain: All real numbers except $$\frac{\pi}{2} + n\pi$$ (where n is an integer); Range: (-∞, -1] ∪ [1, ∞)
• Cotangent: Domain: All real numbers except $$n\pi$$ (where n is an integer); Range: All real numbers

Special Angles

Values of trigonometric functions at common angles:

Angle	0°	30°	45°	60°	90°
Radians	0	$$\frac{\pi}{6}$$	$$\frac{\pi}{4}$$	$$\frac{\pi}{3}$$	$$\frac{\pi}{2}$$
$$\sin(\theta)$$	0	$$\frac{1}{2}$$	$$\frac{\sqrt{2}}{2}$$	$$\frac{\sqrt{3}}{2}$$	1
$$\cos(\theta)$$	1	$$\frac{\sqrt{3}}{2}$$	$$\frac{\sqrt{2}}{2}$$	$$\frac{1}{2}$$	0
$$\tan(\theta)$$	0	$$\frac{\sqrt{3}}{3}$$	1	$$\sqrt{3}$$	undefined

Graphs of Trigonometric Functions

Sine Function

$$y = \sin(x)$$

• Period: $$2\pi$$
• Amplitude: 1
• x-intercepts: $$x = n\pi$$ (where n is an integer)

Cosine Function

$$y = \cos(x)$$

• Period: $$2\pi$$
• Amplitude: 1
• x-intercepts: $$x = \frac{\pi}{2} + n\pi$$ (where n is an integer)

Tangent Function

$$y = \tan(x)$$

• Period: $$\pi$$
• Vertical asymptotes: $$x = \frac{\pi}{2} + n\pi$$ (where n is an integer)

Transformations

General Form

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

• $$A$$ affects the amplitude (|A|)
• $$B$$ affects the period ($$\frac{2\pi}{|B|}$$)
• $$C$$ affects the phase shift (horizontal shift)
• $$D$$ affects the vertical shift

Inverse Trigonometric Functions

Definitions

• $$\sin^{-1}(x)$$ or $$\arcsin(x)$$: The angle whose sine is x
• $$\cos^{-1}(x)$$ or $$\arccos(x)$$: The angle whose cosine is x
• $$\tan^{-1}(x)$$ or $$\arctan(x)$$: The angle whose tangent is x

Domains and Ranges

• Arcsine: Domain: [-1, 1]; Range: [$$-\frac{\pi}{2}, \frac{\pi}{2}$$]
• Arccosine: Domain: [-1, 1]; Range: [0, π]
• Arctangent: Domain: All real numbers; Range: ($$-\frac{\pi}{2}, \frac{\pi}{2}$$)

Trigonometric Identities

Fundamental Identities

• Reciprocal Identities:
  • $$\csc(\theta) = \frac{1}{\sin(\theta)}$$
  • $$\sec(\theta) = \frac{1}{\cos(\theta)}$$
  • $$\cot(\theta) = \frac{1}{\tan(\theta)}$$
• Pythagorean Identities:
  • $$\sin^2(\theta) + \cos^2(\theta) = 1$$
  • $$1 + \tan^2(\theta) = \sec^2(\theta)$$
  • $$1 + \cot^2(\theta) = \csc^2(\theta)$$
• Quotient Identities:
  • $$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$
  • $$\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$$

Sum and Difference Formulas

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

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

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

Applications

Solving Triangles

• Law of Sines: $$\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$$
• Law of Cosines: $$c^2 = a^2 + b^2 - 2ab\cos(C)$$

Area of a Triangle

$$Area = \frac{1}{2}ab\sin(C)$$

Real-World Applications

• Navigation and surveying
• Periodic phenomena (sound waves, light waves, etc.)
• Circular motion
• Architecture and engineering

Common Pitfalls and Tips

• Remember that angles can be measured in degrees or radians (most calculus uses radians)
• Be careful with the domains of trigonometric functions, especially tangent and secant
• When using the Law of Sines, be aware of the ambiguous case (SSA)
• Memorize the values of trigonometric functions at special angles
• Practice converting between different forms of trigonometric expressions using identities
• When graphing, pay attention to period, amplitude, phase shift, and vertical shift