Trigonometry: Angles, Degrees, and Radians

Overview of Trigonometry

• Focuses on relationships between sides and angles of triangles.
• Trigonometry is derived from Greek words for triangle (trigonon) and measure (metron).
• Trig is an important piece of Geometry, Algebra, and Calculus.
• Study of angles and how to measure them in degrees and radians.
• How to find information about right triangles using trigonometric functions.
• Definitions of trig ratios and functions.
• Using definitions and fundamental Identities of trig functions.
• Understanding functions and their graphs.

Measuring Angles

• Angles have an initial side and a terminal side, in standard position with its vertex at the origin
• Measured by the rotation amount from the initial to terminal side.
• Counterclockwise rotation makes a positive angle.
• Clockwise rotation makes a negative angle.

Degrees

• A circle has 360 degrees i.e one revolution is 360°
• Used to describe angle size.

Radians

• One revolution equals 2π radians; π is ≈ 3.14.
• Radians are important for mathematical computation.
• 360° = 2π radians = 1 revolution
• 180° = π radians
• 1° = π/180 radians

Unit circle

• Useful when finding values of trigonometric ratios
• Circle centered at the origin with radius of 1
• Equation is x² + y² = 1
• Ordered pair is also known as (cos θ, sin θ)

Trigonometric functions

• Six ratios link angle measures of a right triangle to side lengths.

• A right triangle with an initial side(x) and terminal side( r), the theorem is x² + y² = r²,

• θ (theta) labels a non-right angle

• Functions help find the ratio of the side lengths of the triangle.

• Sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot)

• sin θ = y/r = opposite/hypotenuse

• cos θ = x/r = adjacent/hypotenuse

• tan θ = y/x = opposite/adjacent

• csc θ = r/y Reciprocal of sin θ

• sec θ = r/x Reciprocal of cos θ

• cot θ = x/y Reciprocal of tan θ

Finding trig values given an angle

• Can use the unit circle for angles
• Unit circle is centered around origin and has a radius of 1.
• Ordered pair (x, y) can also be known as (cos θ, sin θ),
• Can be values by memorization
• Can use a TI graphing calculator

Finding missing sides

• Use trig functions to create equations to locate missing sides of a right triangle.

Finding angle measures with trig functions

• Use a TI Graphing calculator for inverse trig functions to get the angle measure

Using Definitions (Identities)

• Trigonometric functions are
  • Reciprocal Identities
  • Quotient Identities
  • Pythagorean Identities
  • Negative Angle Identities

Functions:

• Reciprocal Identities:
  • sin θ = 1 / csc θ
  • cos θ = 1 / sec θ
  • tan θ = 1 / cot θ
  • csc θ = 1 / sin θ
  • sec θ = 1 / cos θ
  • cot θ = 1 / tan θ
• Quotient Identities:
  • tan θ = sin θ / cos θ
  • cot θ = cos θ / sin θ
• Pythagorean Identities:
  • sin² θ + cos² θ = 1
  • 1+ tan² θ = sec² θ
  • 1 + cot² θ = csc² θ
• Negative Angle Identities:
  • sin(−θ) = −sin θ
  • cos(−θ) = cos θ
  • tan(−θ) = −tan θ
• csc⁡(−θ) = −csc θ
• sec(−θ) = sec θ
• cot(−θ) = cot θ

Complementary Angle Theorem

• Two acute angles adding up to 90° are complementary.
• Co-functions of complementary angles are equivalent.
• Cofunctions: sine and cosine, tangent and cotangent, secant, and cosecant

Sum and Difference Formulas

• sin(α + β) = sin α cos β + cos α sin β
• sin(α – β) = sin α cos β - cos α sin β
• cos(α + β) = cos α cos β - sin α sin β
• cos(α – β) = cos α cos β + sin α sin β
• tan(α + β) = (tan α + tan β) / (1 - tan α tan β)
• tan(α – β) = (tan α - tan β) / (1 + tan α tan β)

Cofunction Identities

• cos(90° – θ) = sin θ
• sin(90° − θ) = cos θ
• tan(90° − θ) = cot θ
• sec(90° − θ) = csc θ
• csc(90° – θ) = sec θ
• cot(90° − θ) = tan θ

Double Angle Formulas

• cos 2θ = cos²θ – sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
• sin 2θ = 2 sin θ cos θ
• tan 2θ = (2 tan θ) / (1 - tan² θ)

Half Angle Formulas

• cos (θ/2) = ±√((1 + cos θ) / 2)
• sin (θ/2) = ±√((1 - cos θ) / 2)
• tan (θ/2) = ±√((1 - cos θ) / (1 + cos θ)) tan (θ/2) = sin θ / (1 + cos θ) tan (θ/2) = (1 - cos θ) / sin θ

Product to Sum Formulas

• cos A cos B = ½ [cos(A + B) + cos(A − B)]
• sin A sin B = ½ [cos(A − B) − cos(A + B)]
• sin A cos B = ½ [sin(A + B) + sin(A − B)]
• cos A sin B = ½ [sin(A + B) – sin(A − B)]

Sum to Product Formulas

• sin A + sin B = 2 sin ((A + B) / 2) cos ((A − B) / 2)
• sin A − sin B = 2 cos ((A + B) / 2) sin ((A − B) / 2)
• cos A + cos B = 2 cos ((A + B) / 2) cos ((A − B) / 2)
• cos A − cos B = -2 sin ((A + B) / 2) sin ((A − B) / 2)

Law of Sines and Cosines

• Help find missing oblique triangles (not right triangles) information

Law of Sines:

• (sin A) / a = (sin B) / b = (sin C) / c
• Use for two sets of angles and their opposite sides.

Law of Cosines:

• c² = a² + b² – 2ab cos C
• b² = a² + c² – 2ac cos B
• a² = b² + c² – 2bc cos A

Graphs of Trig Functions

Periodic Functions:

• Sine and cosine have equal values in regular intervals (periods).
• f is a function such that f(x) = f(x + nπ).

Key points to note:

One revolution of the unit circle is 2π radians. Circumference of the unit circle is 2π.

Graph of Sine Function (y = sin x)

• Range is [-1, 1] and domain is (−∞, ∞).
• x-intercepts are always nπ, with an integer n.
• Odd function is symmetrical relative to the origin.
• Period is 2π, in unit wave repeats every 2π.

Graph of Cosine Function (y = cos x)

• The range is [-1, 1], and the domain is (−∞, ∞).
• Even function is symmetrical relative to the y-axis cos( -x ) = cos (x)
• Period is 2π, sine the cosine wave repeats every 2π.

Key Features: Sine and Cosine Functions

• Amplitude: Measures units above and below midline. For sine, amplitude is 1.
  • Y = a sin x
• a is amplitude and y = a sin x and y = a cos x, where a≠0 have range [-|a|, |a|]

How the Graph is affected:

• Amplitude vertically stretches or shrinks sine and cosine graphs. Period still repeats every 2n units.

Period:

• When considered with a graph of y = sin x notice changes when the function turns into y = sin 2x

General Formula:

• Used to find the period (w) of sine or cosine: w = 2π / |b|

Phase Shift:

• Graphs are contracted/stretched vertically/horizontally, also shifted L/R, Up/Down. y = cos x. π If equation changes to y = cos(x − ), there is a phase shift. 2

Vertical Shift:

• Shifting wave up or down y = cos x + 3, wave shifts.
• In y = a cos (bx – c) + d, d controls shift.

Graph of Tangent Function (y = tan x)

• Range is (−∞, ∞). π Domain is {x|x ≠ nπ +, where n is any integer}. 2
• x intercepts are always nπ.
• Period is π
• Tangent will be zero wherever the numerator (sine) is zero.
• The tangent will be undefined wherever the denominator (cosine) is zero. π
• Graph has vertical asymptotes at x = nπ + 2
• Graph is symmetrical about the origin (odd function).

Vertical Asymptotes

• Tangent approaches but never crosses vertical asymptotes. bx = -π/2 and bx = π/2 find 2 consecutive asymptotes

Graphing with Technology

• It can be done with graphing calculators if put in correct mode
• Can be done with Desmos