Comprehensive Study Notes: Trigonometric Functions and Their Applications

Angles, Arc Length, and Circular Motion

Angle Measurement and Conversions

Understanding angle measurement is fundamental in trigonometry. Angles can be measured in degrees, minutes, seconds (DMS), or radians. Converting between these units is essential for solving trigonometric problems.

• DMS Conversions: 1 degree (1º) = 60 minutes (60'), 1 minute (1') = 60 seconds (60'').

• Degree to Radian Conversion: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$

• Radian to Degree Conversion: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$

Arc Length and Area of a Sector

For a circle of radius r and a central angle $\theta$ (in radians):

• Arc Length: $s = r\theta$

• Area of a Sector: $A = \frac{1}{2} r^2 \theta$

Circular Motion

Circular motion describes how a point moves along a circle. Two key quantities are linear speed and angular speed.

• Linear Speed: $v = \frac{s}{t} = r\omega$

• Angular Speed: $\omega = \frac{\theta}{t}$ (where $\theta$ is in radians)

Right Triangle Trigonometry

Trigonometric Functions

Trigonometric functions relate the angles of a right triangle to the ratios of its sides.

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

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

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

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

Pythagorean Theorem: $h^2 = a^2 + b^2$

Complementary Angles

• $\sin \theta = \cos (90^\circ - \theta)$

• $\cos \theta = \sin (90^\circ - \theta)$

• $\tan \theta = \cot (90^\circ - \theta)$

• Similar relationships hold for secant and cosecant.

Graphs of Trigonometric Functions

General Equation and Parameters

The general form for the sine and cosine functions is:

• $y = A \sin(\omega x - \varphi) + \beta$

• $y = A \cos(\omega x - \varphi) + \beta$

• Amplitude (A): The maximum displacement from the midline.

• Period (P): $P = \frac{2\pi}{\omega}$

• Horizontal Shift (\varphi): The phase shift of the graph.

• Vertical Shift (\beta): The upward or downward translation of the graph.

Asymptotes for Reciprocal Functions

Reciprocal trigonometric functions (secant, cosecant, tangent, cotangent) have vertical asymptotes where their base functions are undefined.

• Tangent: $\theta \neq \frac{\pi}{2} + n\pi$

• Cotangent: $\theta \neq n\pi$

• Secant: $\theta \neq \frac{\pi}{2} + n\pi$

• Cosecant: $\theta \neq n\pi$

Asymptotes repeat every $\pi$ radians due to periodicity.

Inverse Trigonometric Functions

Definition and Properties

Inverse trigonometric functions allow us to find angles given function values. Each function has a restricted range to ensure it is one-to-one.

• If $\theta = \sin(y)$, then $y = \sin^{-1}(\theta)$

• $\sin^{-1}(\sin(\theta)) = \theta$ (within restricted interval)

Restriction Intervals

Function	Principal Interval for y	Undefined at
sin / csc	$-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$	$y = 0$ (csc)
cos / sec	$0 \leq y \leq \pi$	$y = \frac{\pi}{2}$ (sec)
tan	$-\frac{\pi}{2} < y < \frac{\pi}{2}$	–
cot	$0 < y < \pi$	–

Note: [ ] includes endpoint, ( ) excludes endpoint.

Trigonometric Identities

Pythagorean Identities

• $\sin^2(x) + \cos^2(x) = 1$

• $\tan^2(x) + 1 = \sec^2(x)$

• $\cot^2(x) + 1 = \csc^2(x)$

Even-Odd Identities

• $\sin(-\theta) = -\sin(\theta)$

• $\cos(-\theta) = \cos(\theta)$

• $\tan(-\theta) = -\tan(\theta)$

• Similar relationships hold for secant, cosecant, and cotangent.

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

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

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

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

Half Angle Formulas

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

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

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

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

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

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

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

Applications of Trigonometric Functions

Law of Sines

• $\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$

• For any triangle, $A + B + C = 180^\circ$

Law of Cosines

• $c^2 = a^2 + b^2 - 2ab\cos(C)$

• $b^2 = a^2 + c^2 - 2ac\cos(B)$

• $a^2 = b^2 + c^2 - 2bc\cos(A)$

Simple Harmonic Motion

• $d(t) = a\cos(\omega t)$ or $d(t) = a\sin(\omega t)$

• Frequency: $f = \frac{1}{T} = \frac{\omega}{2\pi}$

• Damped motion: $d(t) = ae^{-bt}\cos(\omega t)$ (amplitude decreases over time)

Polar Coordinates

Conversion Between Polar and Rectangular Coordinates

• $x = r\cos(\theta)$

• $y = r\sin(\theta)$

• $r^2 = x^2 + y^2$

• $\tan(\theta) = \frac{y}{x}$ (if $x \neq 0$)

Polar Equations

• $r = 2a\sin(\theta)$: Circle, center at $(0, a)$

• $r = 2a\cos(\theta)$: Circle, center at $(a, 0)$

• $a = r\sin(\theta)$: Horizontal line $a$ units above/below the pole

• $a = r\cos(\theta)$: Vertical line $a$ units right/left of the pole