Chapter 4: Trigonometric Functions – Comprehensive Study Notes

4.1: Angles and Radian Measures

Definition of Angles

An angle is formed by two rays (sides) with a common endpoint (vertex). The initial side is where the angle starts, and the terminal side is where it ends after rotation.

• Standard Position: An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis.

• Positive Angle: Formed by rotating the initial side counterclockwise.

• Negative Angle: Formed by rotating the initial side clockwise.

Types of Angles:

• Acute Angle: Less than 90°

• Right Angle: Exactly 90°

• Obtuse Angle: Between 90° and 180°

• Straight Angle: Exactly 180°

Example: Draw angles in standard position for 60°, 225°, and -120°.

Co-terminal Angles

• Angles with the same initial and terminal sides are called co-terminal angles.

• To find a co-terminal angle, add or subtract multiples of 360° (or $2\pi$ radians): $\theta_{coterminal} = \theta + 360k$ (degrees) or $\theta + 2\pi k$ (radians), where $k$ is any integer.

Example: Find a positive angle less than 360° that is co-terminal with 480°.

Complementary and Supplementary Angles

• Complementary Angles: Add up to 90° ($\frac{\pi}{2}$ radians).

• Supplementary Angles: Add up to 180° ($\pi$ radians).

Example: Find the complement and supplement of 70°.

Radian Measure

• One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius.

• Formula: If a central angle $\theta$ intercepts an arc of length $s$ in a circle of radius $r$, then $\theta = \frac{s}{r}$ radians.

• There are $2\pi$ radians in a full circle (360°).

Converting Between Degrees and Radians

• To convert degrees to radians: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$

• To convert radians to degrees: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$

Example: Convert 30° to radians; convert $\frac{2\pi}{3}$ radians to degrees.

Arc Length and Sector Area

• Arc length: $s = r\theta$ (where $\theta$ is in radians)

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

Example: Find the length of the arc intercepted by a central angle of 45° in a circle of radius 6.

Angular Speed vs. Linear Speed

• Linear Speed (v): $v = \frac{s}{t}$, where $s$ is arc length and $t$ is time.

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

• Relationship: $v = r\omega$

Example: A ferris wheel with radius 25 ft rotates at 3 revolutions per minute. Find the linear speed.

4.2 & 4.3: Right Triangle Trigonometry and the Unit Circle

Trigonometric Ratios

For a right triangle with angle $\theta$:

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

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

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

The reciprocal functions are:

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

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

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

Example: Find all six trig values for a triangle with sides 8, 15, and 17.

Special Triangles and Exact Values

• 30°-60°-90° triangle: sides in ratio 1 : $\sqrt{3}$ : 2

• 45°-45°-90° triangle: sides in ratio 1 : 1 : $\sqrt{2}$

Example: Find $\sin 30^\circ$, $\cos 45^\circ$, $\tan 60^\circ$ using special triangles.

The Unit Circle

• The unit circle is a circle of radius 1 centered at the origin.

• For any angle $\theta$, the coordinates $(x, y)$ on the unit circle correspond to $(\cos \theta, \sin \theta)$.

• Common angles (in degrees and radians) and their coordinates are used to find exact trig values.

Example: Use the unit circle to find $\sin 90^\circ$, $\cos 180^\circ$, $\tan 270^\circ$.

Reference Angles and Signs

• The reference angle is the acute angle formed by the terminal side of $\theta$ and the x-axis.

• Signs of trig functions depend on the quadrant:

Quadrant	sin	cos	tan
I	+	+	+
II	+	-	-
III	-	-	+
IV	-	+	-

4.4: Trigonometric Functions of Any Angle

Trig Functions for Points on the Plane

• If a point $(x, y)$ is on the terminal side of angle $\theta$ (with distance $r = \sqrt{x^2 + y^2}$ from the origin):

• $\sin \theta = \frac{y}{r}$

• $\cos \theta = \frac{x}{r}$

• $\tan \theta = \frac{y}{x}$

• Reciprocal functions as above.

Example: If $(x, y) = (-7, 5)$, find all six trig functions for $\theta$.

Finding Other Trig Values

• If one trig value and the quadrant are known, use the Pythagorean identity to find others:

• $\sin^2 \theta + \cos^2 \theta = 1$

Example: If $\sin \theta = -\frac{12}{13}$ and $\theta$ is in quadrant III, find the other five trig functions.

4.5: Graphs of Sine and Cosine Functions

Basic Graphs

• The graph of $y = \sin x$ and $y = \cos x$ are periodic with period $2\pi$.

• Amplitude: The maximum value from the midline, $|a|$ in $y = a \sin x$ or $y = a \cos x$.

• Period: $\frac{2\pi}{b}$ for $y = a \sin(bx)$ or $y = a \cos(bx)$.

• Vertical Shift: $y = a \sin(bx) + d$ shifts the graph up or down by $d$.

• Horizontal Shift: $y = a \sin(b(x - c))$ shifts the graph right by $c$ units.

Example: Graph $y = 2 \sin x$ and $y = \sin 2x$; compare amplitude and period.

4.6: Graphs of Other Trigonometric Functions

• Tangent: $y = \tan x$ has period $\pi$; vertical asymptotes at $x = \frac{\pi}{2} + k\pi$.

• Cotangent: $y = \cot x$ has period $\pi$; vertical asymptotes at $x = k\pi$.

• Secant: $y = \sec x$ and Cosecant: $y = \csc x$ have period $2\pi$; vertical asymptotes where $\cos x = 0$ or $\sin x = 0$ respectively.

Example: Graph $y = \tan x$, $y = \sec x$, $y = \csc x$.

4.7: Inverse Trigonometric Functions

• Inverse functions "undo" the original trig functions, but are restricted to certain domains for uniqueness.

• Notation: $y = \sin^{-1} x$, $y = \cos^{-1} x$, $y = \tan^{-1} x$.

• Ranges:

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

Example: Find $\sin^{-1}(0.5)$, $\cos^{-1}(-\frac{\sqrt{2}}{2})$, $\tan^{-1}(1)$.

4.8: Trigonometric Applications

Solving Right Triangles

• Given two pieces of information (sides or angles), use trig ratios and the Pythagorean theorem to solve for unknowns.

• Label sides relative to the angle of interest: opposite, adjacent, hypotenuse.

Example: Solve triangle ABC where $A = 41.5^\circ$, $b = 20$.

Angle of Elevation and Depression

• Angle of Elevation: Angle above horizontal from observer's eye to an object.

• Angle of Depression: Angle below horizontal from observer's eye to an object.

Example: A flagpole 14 m tall casts a 10 m shadow. Find the angle of elevation of the sun.

Real-Life Applications

• Draw a diagram and label all known and unknown values.

• Use trigonometric ratios to set up equations and solve for unknowns.

Example: A police helicopter is flying at 800 ft. A stolen car is sighted at an angle of depression of 72°. Find the distance to the car.

Additional info: These notes cover the main concepts, definitions, and applications from Chapter 4: Trigonometric Functions, including angle measurement, right triangle trigonometry, the unit circle, graphs of trig functions, inverse trig functions, and real-world applications.