Angles, Angle Relationships, and Trigonometric Functions: Study Notes

Section 1.1: Angles

Basic Terminology

Understanding angles and their properties is foundational in trigonometry. Angles are formed by two rays (or line segments) sharing a common endpoint called the vertex.

• Line: Extends infinitely in both directions through two points.

• Line Segment: The portion of a line between two points, including the endpoints.

• Ray: Starts at an endpoint and extends infinitely in one direction.

• Angle: Formed by two rays with a common endpoint (vertex).

• Initial Side: The starting position of the ray.

• Terminal Side: The position after rotation.

• Positive Angle: Generated by counterclockwise rotation.

• Negative Angle: Generated by clockwise rotation.

Degree Measure and Types of Angles

• Acute Angle: $0^\circ < \theta < 90^\circ$

• Right Angle: $\theta = 90^\circ$

• Obtuse Angle: $90^\circ < \theta < 180^\circ$

• Straight Angle: $\theta = 180^\circ$

• Complementary Angles: Two angles whose measures sum to $90^\circ$.

• Supplementary Angles: Two angles whose measures sum to $180^\circ$.

Example: The complement of $55^\circ$ is $90^\circ - 55^\circ = 35^\circ$. The supplement is $180^\circ - 55^\circ = 125^\circ$.

Degrees, Minutes, and Seconds

• 1 degree ($^\circ$) = 60 minutes ($'$)

• 1 minute ($'$) = 60 seconds ($''$)

Example: $28^\circ 35' + 63^\circ 52' = 92^\circ 27'$

Converting Between Angle Measures

• To Decimal Degrees: $105^\circ 20' 32'' = 105 + \frac{20}{60} + \frac{32}{3600} = 105.342^\circ$

• To Degrees, Minutes, Seconds: $85.263^\circ = 85^\circ + 0.263 \times 60' = 85^\circ 15.78' = 85^\circ 15' 47''$ (rounded)

Standard Position and Quadrants

• Standard Position: Vertex at the origin, initial side on the positive x-axis.

• Quadrantal Angles: Terminal side lies on the x- or y-axis (e.g., $0^\circ$, $90^\circ$, $180^\circ$, $270^\circ$).

• Quadrants:

  • QI: $0^\circ < \theta < 90^\circ$

  • QII: $90^\circ < \theta < 180^\circ$

  • QIII: $180^\circ < \theta < 270^\circ$

  • QIV: $270^\circ < \theta < 360^\circ$

Coterminal Angles

Angles that share the same initial and terminal sides but differ by multiples of $360^\circ$.

• Formula: $\theta_{coterminal} = \theta + 360^\circ \times k$, where $k$ is any integer.

Example: $1106^\circ$ is coterminal with $26^\circ$ because $1106^\circ - 3 \times 360^\circ = 26^\circ$.

Application Example: Disk Drive Revolutions

To find the degrees a point moves in 5 seconds at 270 revolutions per minute:

• Degrees per revolution: $360^\circ$

• Total revolutions in 5 seconds: $270 \times \frac{5}{60} = 22.5$

• Total degrees: $22.5 \times 360^\circ = 8100^\circ$

Section 1.2: Angle Relationships and Similar Triangles

Geometric Properties

• Vertical Angles: Opposite angles formed by two intersecting lines; always equal.

• Parallel Lines and Transversal: When a transversal crosses parallel lines, several angle pairs are formed:

Angle Pair	Rule
Alternate Interior Angles	Equal
Alternate Exterior Angles	Equal
Corresponding Angles	Equal
Interior Angles on Same Side	Sum to $180^\circ$

Triangles

• Angle Sum: The sum of the angles in any triangle is $180^\circ$.

• Types of Triangles by Angles:

  • Acute: All angles less than $90^\circ$

  • Right: One angle is $90^\circ$

  • Obtuse: One angle greater than $90^\circ$

• Types by Sides:

  • Equilateral: All sides equal

  • Isosceles: Two sides equal

  • Scalene: No sides equal

Similar and Congruent Triangles

• Similar Triangles: Same shape, corresponding angles equal, sides proportional.

• Congruent Triangles: Same shape and size; all corresponding sides and angles equal.

Conditions for Similarity:

• Corresponding angles are equal.

• Corresponding sides are proportional: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$

Applications of Similar Triangles

• Finding Unknown Angles: Use the angle sum property and similarity conditions.

• Finding Unknown Sides: Set up proportions using corresponding sides.

• Indirect Measurement: Use similar triangles to find heights or distances (e.g., flagpole height using shadow lengths).

Example: If Joey (63 in. tall) casts a 42 in. shadow and a tree casts a 456 in. shadow, the tree's height is $\frac{63}{42} \times 456 = 684$ in $= 57$ ft.

Section 1.3: Trigonometric Functions

The Pythagorean Theorem and Distance Formula

• Pythagorean Theorem: In a right triangle with legs $a$, $b$ and hypotenuse $c$:

$a^2 + b^2 = c^2$

• Distance Formula: For points $P(x_1, y_1)$ and $R(x_2, y_2)$:

$d(P, R) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Definitions of Trigonometric Functions

For a point $(x, y)$ on the terminal side of angle $\theta$ in standard position, with $r = \sqrt{x^2 + y^2}$:

• $\sin \theta = \frac{y}{r}$

• $\cos \theta = \frac{x}{r}$

• $\tan \theta = \frac{y}{x}$

• $\csc \theta = \frac{r}{y}$

• $\sec \theta = \frac{r}{x}$

• $\cot \theta = \frac{x}{y}$

Note: $r > 0$ always.

Examples

• Given $(12, 5)$: $r = 13$

• $\sin \theta = \frac{5}{13}$, $\cos \theta = \frac{12}{13}$, $\tan \theta = \frac{5}{12}$, $\csc \theta = \frac{13}{5}$, $\sec \theta = \frac{13}{12}$, $\cot \theta = \frac{12}{5}$

Quadrantal Angles

• Angles whose terminal sides lie on the axes ($0^\circ$, $90^\circ$, $180^\circ$, $270^\circ$, $360^\circ$).

• Some trigonometric functions are undefined for these angles.

Angle	sin	cos	tan	cot	sec	csc
$0^\circ$	0	1	0	undefined	1	undefined
$90^\circ$	1	0	undefined	0	undefined	1
$180^\circ$	0	-1	0	undefined	-1	undefined
$270^\circ$	-1	0	undefined	0	undefined	-1
$360^\circ$	0	1	0	undefined	1	undefined

Undefined Functions: If the denominator in the definition is zero, the function is undefined.

Section 1.4: Using the Definitions of the Trigonometric Functions

Reciprocal and Quotient Identities

• Reciprocal Identities:

$\sin \theta = \frac{1}{\csc \theta}$ $\cos \theta = \frac{1}{\sec \theta}$ $\tan \theta = \frac{1}{\cot \theta}$ $\csc \theta = \frac{1}{\sin \theta}$ $\sec \theta = \frac{1}{\cos \theta}$ $\cot \theta = \frac{1}{\tan \theta}$

• Quotient Identities:

$\tan \theta = \frac{\sin \theta}{\cos \theta}$ $\cot \theta = \frac{\cos \theta}{\sin \theta}$

Pythagorean Identities

For all angles where the functions are defined:

• $\sin^2 \theta + \cos^2 \theta = 1$

• $\tan^2 \theta + 1 = \sec^2 \theta$

• $1 + \cot^2 \theta = \csc^2 \theta$

Signs of Trigonometric Functions by Quadrant

Quadrant	sin, csc	cos, sec	tan, cot
I	+	+	+
II	+	-	-
III	-	-	+
IV	-	+	-

Mnemonic: "All Students Take Calculus" (All positive in I, Sine in II, Tangent in III, Cosine in IV).

Ranges of Trigonometric Functions

Function	Range (Set Notation)	Range (Interval Notation)
sin, cos	{ y | |y| ≤ 1 }	[-1, 1]
tan, cot	All real numbers	($-\infty$, $\infty$)
sec, csc	{ y | |y| ≥ 1 }	($-\infty$, $-1$] ∪ [1, $\infty$)

Examples Using Identities and Signs

• Given $\tan \theta = 3$ in Quadrant III: $\sin \theta = -\frac{3}{\sqrt{10}}$, $\cos \theta = -\frac{1}{\sqrt{10}}$, etc.

• Given $\cot \theta = -2$ in Quadrant II: $\sin \theta > 0$, $\cos \theta < 0$; use identities to find all values.

Determining Possibility of Values

• cot $\theta = -0.999$: Possible (cotangent can take any real value).

• cos $\theta = -1.7$: Impossible (cosine range is $[-1, 1]$).

• csc $\theta = 0$: Impossible (cosecant is undefined for $0$).

Summary Table: Trigonometric Functions and Their Properties

Function	Definition	Reciprocal	Range
sin $\theta$	$\frac{y}{r}$	csc $\theta$	[-1, 1]
cos $\theta$	$\frac{x}{r}$	sec $\theta$	[-1, 1]
tan $\theta$	$\frac{y}{x}$	cot $\theta$	($-\infty$, $\infty$)
csc $\theta$	$\frac{r}{y}$	sin $\theta$	($-\infty$, $-1$] ∪ [1, $\infty$)
sec $\theta$	$\frac{r}{x}$	cos $\theta$	($-\infty$, $-1$] ∪ [1, $\infty$)
cot $\theta$	$\frac{x}{y}$	tan $\theta$	($-\infty$, $\infty$)

Additional info: Some examples and explanations were expanded for clarity and completeness, including the use of set and interval notation for ranges, and the inclusion of summary tables for quick reference.