📏Honors Pre-Calculus Unit 5 – Trigonometric Functions

Key Concepts and Definitions

• Trigonometry studies relationships between side lengths and angles in triangles
• Sine, cosine, and tangent are the primary trigonometric functions
  • Sine ($\sin$) is the ratio of the opposite side to the hypotenuse
  • Cosine ($\cos$) is the ratio of the adjacent side to the hypotenuse
  • Tangent ($\tan$) is the ratio of the opposite side to the adjacent side
• Cosecant ($\csc$), secant ($\sec$), and cotangent ($\cot$) are reciprocal functions of sine, cosine, and tangent, respectively
• Radian measure expresses angles in terms of $\pi$ and the radius of a circle
• The unit circle has a radius of 1 and is centered at the origin (0, 0)
• Trigonometric identities are equations that are true for all values of the variable for which both sides are defined

Angles and Radian Measure

• Angles can be measured in degrees or radians
• One radian is the angle subtended by an arc length equal to the radius of the circle
• To convert from degrees to radians, multiply by $\frac{\pi}{180}$
• To convert from radians to degrees, multiply by $\frac{180}{\pi}$
• Angles in the coordinate plane can be positive (counterclockwise) or negative (clockwise)
• Coterminal angles are angles that share the same terminal side (differ by multiples of 360° or 2$\pi$ radians)
• Reference angles are acute angles formed by the terminal side of an angle and the x-axis

The Unit Circle

• The unit circle is a circle with a radius of 1 centered at the origin (0, 0)
• Angles in the unit circle are measured in radians, with 0 radians at (1, 0) and increasing counterclockwise
• The coordinates of a point on the unit circle are (cos $\theta$, sin $\theta$), where $\theta$ is the angle from the positive x-axis
• The sine and cosine values for common angles (0, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$) can be derived from the unit circle
• Symmetry in the unit circle helps determine the signs of trigonometric functions in different quadrants
  • Sine is positive in quadrants I and II, negative in III and IV
  • Cosine is positive in quadrants I and IV, negative in II and III
  • Tangent is positive in quadrants I and III, negative in II and IV

Trigonometric Functions and Their Graphs

• Sine, cosine, and tangent functions are periodic, meaning they repeat at regular intervals
• The sine function ($y = \sin x$) has a period of 2$\pi$, an amplitude of 1, and a vertical shift of 0
• The cosine function ($y = \cos x$) has a period of 2$\pi$, an amplitude of 1, and a vertical shift of 0
• The tangent function ($y = \tan x$) has a period of $\pi$, undefined values at odd multiples of $\frac{\pi}{2}$, and a vertical shift of 0
• Transformations of trigonometric functions include changes in amplitude, period, phase shift, and vertical shift
  • Amplitude is controlled by a vertical stretch or compression factor $a$ in $a \sin x$ or $a \cos x$
  • Period is affected by a horizontal stretch or compression factor $b$ in $\sin bx$ or $\cos bx$
  • Phase shift is a horizontal shift of the function, as in $\sin(x - h)$ or $\cos(x - h)$
  • Vertical shift moves the function up or down, as in $\sin x + k$ or $\cos x + k$

Inverse Trigonometric Functions

• Inverse trigonometric functions "undo" the original functions and are denoted with $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$ or $\arcsin$, $\arccos$, and $\arctan$
• The domain of inverse trigonometric functions is limited to ensure they are one-to-one
  • $\sin^{-1}$ has a domain of [-1, 1] and a range of $[-\frac{\pi}{2}, \frac{\pi}{2}]$
  • $\cos^{-1}$ has a domain of [-1, 1] and a range of [0, $\pi$]
  • $\tan^{-1}$ has a domain of all real numbers and a range of $(-\frac{\pi}{2}, \frac{\pi}{2})$
• Inverse trigonometric functions can be used to solve equations like $\sin x = \frac{1}{2}$ by applying the inverse function to both sides

Trigonometric Identities and Equations

• Pythagorean identities relate the squares of trigonometric functions: $\sin^2 x + \cos^2 x = 1$, $1 + \tan^2 x = \sec^2 x$, and $1 + \cot^2 x = \csc^2 x$
• Angle addition and subtraction formulas express the sine, cosine, or tangent of a sum or difference of angles in terms of the individual angles
  • $\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$
  • $\cos(x \pm y) = \cos x \cos y \mp \sin x \sin y$
  • $\tan(x \pm y) = \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y}$
• Double-angle formulas express trigonometric functions of double an angle in terms of the original angle
  • $\sin 2x = 2 \sin x \cos x$
  • $\cos 2x = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x$
  • $\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}$
• Half-angle formulas express trigonometric functions of half an angle in terms of the original angle
  • $\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}$
  • $\cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}}$
  • $\tan \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}} = \frac{\sin x}{1 + \cos x}$

Applications and Problem Solving

• Trigonometry has applications in various fields, such as physics, engineering, and navigation
• Right triangle trigonometry can be used to solve problems involving heights, distances, and angles
  • The sine, cosine, and tangent functions relate the sides and angles of a right triangle
  • The angle of elevation is the angle between the horizontal and the line of sight to an object above the observer
  • The angle of depression is the angle between the horizontal and the line of sight to an object below the observer
• Trigonometric functions can model periodic phenomena, such as sound waves, tides, and planetary orbits
• The law of sines and the law of cosines extend trigonometry to non-right triangles
  • The law of sines states that $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ for a triangle with sides $a$, $b$, and $c$ and angles $A$, $B$, and $C$
  • The law of cosines states that $c^2 = a^2 + b^2 - 2ab \cos C$ for a triangle with sides $a$, $b$, and $c$ and angle $C$ opposite side $c$

Common Pitfalls and Study Tips

• Remember the order of operations when evaluating trigonometric expressions (PEMDAS)
• Be careful with the signs of trigonometric functions in different quadrants
• Memorize the common angles and their sine, cosine, and tangent values in the unit circle
• Practice identifying the period, amplitude, phase shift, and vertical shift of transformed trigonometric functions
• When solving trigonometric equations, consider the domain and range of the functions involved
• Sketch diagrams to visualize problems involving angles, distances, and heights
• Understand the relationships between trigonometric functions and their inverses
• Practice applying trigonometric identities to simplify expressions and solve equations
• Review the proofs of trigonometric identities to deepen your understanding of the concepts