2.2-2.3 Notes: Standard Position, Coterminal & Reference Angles, Minutes/Seconds, Sine/Cosine, and Periodic Functions

Standard Position of an Angle

• An angle is in standard position if:

  • The vertex is at the origin.

  • The initial side lies along the positive x-axis.

  • The angle is measured counterclockwise from the initial side as positive; clockwise is negative.

• Terminology:

  • The initial side is where the angle starts.

  • The terminal side is where the angle ends.

• Examples from the video:

  • If the angle to some terminal line is labeled as θ = 60°, the line is the terminal side for that angle.

  • If an angle is opened clockwise by 120°, it can be written as θ = -120°.

• The standard position reference frame:

  • 0° is on the positive x-axis.

  • 90° is on the positive y-axis.

  • 180° is on the negative x-axis.

  • 270° is on the negative y-axis.

  • 360° (and multiples) bring you back to the positive x-axis.

• Quadrants and angle types:

  • First quadrant (0° to 90°): acute angles.

  • Second quadrant (90° to 180°): obtuse angles.

• Notes on notation:

  • In the textbook, the author sometimes uses θ for degrees and x, y for axes. To avoid confusion, he sometimes uses U and V for axes when θ is the angle measure. If you prefer, you can still use x and y for axes and θ for the angle; context will guide you.

• Visual idea:

  • If you rotate from the initial side (positive x-axis) by a small CCW amount, you get a small positive θ in the first quadrant. A clockwise rotation yields a negative θ.

Angles Greater than 360° and Coterminal Angles

• Definition:

  • Coterminal angles terminate at the same position after rotation.

  • Infinitely many coterminal angles exist for any given θ, obtained by adding or subtracting whole revolutions of 360° (or 2π radians).

General formula (degrees):

$ext{coterminal}( heta) = heta + 360^{\, ext{o}} \, n,</p><p>ext{ for any } n \,\in\, \mathbb{Z}.$

(You can write only the plus 360° and rely on n to be negative for subtraction; the integer set handles both directions.)

• Example reductions to the 0–360° interval:

  • If θ = 133°, coterminal by adding 360° gives 493°, by subtracting 360° gives -227°. Infinitely many such angles exist.

• Example problem from the video:

  • Given θ = 3,723°, find the coterminal angle between 0° and 360°.

  • Subtract 3,600° (which is 10 × 360°) to get 123°.

  • Given θ = -600°, find the coterminal angle between 0° and 360°.

  • Add 720° (2 × 360°) to get 120°.

• Radians note (brief):

  • The coterminal angle form in radians is
    $ext{coterminal}( heta) = heta + 2\,\pi\, n, \ n \in \mathbb{Z}.$

Reference Angles

• Definition:

  • A reference angle is a positive acute angle formed with the x-axis by the terminal side of θ.

  • It is the smallest positive angle between the terminal side and the x-axis (positive or negative x direction).

• How to find it by quadrant:

  • Quadrant I: ref(θ) = θ.

  • Quadrant II: ref(θ) = 180° - θ.

  • Quadrant III: ref(θ) = θ - 180°.

  • Quadrant IV: ref(θ) = 360° - θ.

• Examples from the video:

  • θ = 71° (QI) → ref(θ) = 71°.

  • θ = 133° (QII) → ref(θ) = 180° - 133° = 47°.

  • θ = 254° (QIII) → ref(θ) = 254° - 180° = 74°.

  • θ = 317° (QIV) → ref(θ) = 360° - 317° = 43°.

• Negative angle example:

  • θ = -62° is equivalent to θ ≡ 298° (mod 360°), which lies in QIV; ref(θ) = 62°.

• Practical note:

  • The reference angle is always a positive acute angle 0° < ref(θ) < 90°.

  • Signs of sine, cosine, and tangent in a given quadrant follow the ASTC rule (All Students Take Calculus).

• ASTC mnemonic (signs by quadrant):

  • Quadrant I: All functions positive.

  • Quadrant II: Sine and Cosecant positive.

  • Quadrant III: Tangent and Cotangent positive.

  • Quadrant IV: Cosine and Secant positive.

• Worked example with a point on the terminal side:

  • Terminal point (8, -5) lies in Quadrant IV.

  • Hypotenuse:
    $r = \sqrt{8^2 + (-5)^2} = \sqrt{89}.$

  • Sine and cosine:
    $\ ext{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{8}{\sqrt{89}},$
    $\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{-5}{\sqrt{89}}.$

  • Sign pattern: sine negative in QIV; cosine positive in QIV.

  • Rationalizing the denominator: not required here; leaving as (\frac{8}{\sqrt{89}}) is acceptable.

• Reference angle with coterminal points:

  • The same triangle (ratios) applies in all coterminal positions; only the signs change according to the quadrant.

  • Example quadrants for signs (given the same reference angle for a given θ):

  • QII: x negative, y positive → sine positive, cosine negative.

  • QIII: both x and y negative → sine and cosine negative.

  • QIV: x positive, y negative → sine negative, cosine positive.

• Minutes and seconds (degrees) in angles:

  • Angles can include minutes (′) and seconds (′′): e.g., 73° 28′, 153° 27′′.

  • 1 degree = 60 minutes; 1 minute = 60 seconds.

  • Borrowing for subtraction when minutes are involved:

  • Example: to compute ref(153°27′) in QII with an original angle like 73°28′ is handled by borrowing 1 degree to obtain 69? or 70? and 60′ into minutes to subtract 27′ from 60′, yielding a ref angle of 26°33′.

  • Decimal conversion for minutes:

  • If a calculation yields a decimal degree like 26.5500°, convert the fractional part to minutes by multiplying by 60: 0.55 × 60 = 33′, giving 26°33′.

  • Calculator tips (degrees mode):

  • Use the calculator’s angle function (blue angle key) to input degrees and minutes for reference angle calculations.

  • If you accidentally use radian mode, switch to degree mode to get correct minute conversion.

  • Seconds input on calculator:

  • Seconds can be input with a double prime symbol (″). On many calculators you enter this by pressing a sequence like Alpha and then the double quote or a specific key (often shown as a green key near the plus sign).

Sine and Cosine Basics; How to Read the Right Triangle

• SOH CAH TOA reminder (for sine and cosine):

  • Sine:
    $ext{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{v}{r}.$

  • Cosine:
    $\text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{u}{r}.$

  • Hypotenuse: (r = \sqrt{u^2 + v^2}).

• Quadrant signs (revisiting):

  • In Quadrant I, sine and cosine are positive (and thus tangent as well).

  • In Quadrant II, sine is positive, cosine is negative, tangent negative.

  • In Quadrant III, sine and cosine are negative, tangent positive.

  • In Quadrant IV, cosine is positive, sine is negative, tangent negative.

• Example with coordinates:

  • If the terminal point is (8, -5): as above, r = (\sqrt{89}), cos θ = 8/(\sqrt{89}), sin θ = -5/(\sqrt{89}).

• Graphical intuition (not required to graph by hand yet):

  • Sine and cosine waves are periodic; vertical shifts affect amplitude but not period; horizontal dilations affect period (e.g., sin(2θ) has period halved).

  • Horizontal shifts move the graph left/right but do not change the fact that the function repeats after its period.

  • The course will cover graphing in Chapter 3; you can rely on this conceptual framework now.

Periodic Functions; Transformations and the Idea of a Period

• Definition:

  • A periodic function is one that repeats its values at regular intervals. Formally, there exists a number (p > 0) such that
    $f(x + p) = f(x) \text{for all } x.$

• Ferris wheel analogy:

  • A person on a Ferris wheel repeats their height as the wheel turns, oscillating between a maximum and a minimum height at a constant rate.

  • If the wheel slows down, the period increases; if it speeds up, the period decreases.

• Period and transformations:

  • Horizontal translations (shifting left/right) do not change the period; they only change the starting point of a cycle.

  • Vertical translations (shifting up/down) and vertical dilations change amplitude but not the period.

  • Horizontal dilations (like sin(2θ)) change the period: the period becomes shorter by the factor of the dilation.

  • For example, sin(2θ) has a period of
    $\text{Period} = \dfrac{2\pi}{2} = \pi,$
    which means two cycles occur in the interval [0, 2π].

• Relevance to trigonometric functions:

  • The sine and cosine functions are periodic with period 2π in radians (or 360° in degrees).

  • Understanding periodicity helps with solving equations like sin(x) = a, cos(x) = b, etc., across all coterminal angles.

Quick Reference: Putting It All Together

• Key takeaways:

  • Standard position gives a consistent reference frame for measuring angles and determining quadrant signs.

  • Coterminal angles differ by multiples of 360° (or 2π radians); you can generate infinite families via θ + 360°n.

  • Reference angles provide a consistent way to relate all quadrants back to an acute angle with the x-axis; they determine the sine/c cosine values’ magnitudes and, combined with quadrant signs, their actual signs.

  • For sine and cosine, the basic right-triangle definitions apply:
    $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad r = \sqrt{\text{adjacent}^2 + \text{opposite}^2}.$

  • The ASTC mnemonic helps quickly determine signs in each quadrant:

  • Quadrant I: All positive; Quadrant II: Sine positive; Quadrant III: Tangent positive; Quadrant IV: Cosine positive.

  • When dealing with degrees and minutes/seconds, remember: 1° = 60′, 1′ = 60″; convert as needed for calculations and for calculator input.

• Practical calculation flow (examples to practice):

  • Given θ = 254°, find ref(θ) and sign pattern for sine/cosine. Ref(θ) = 254° − 180° = 74°; Quadrant III -> sine and cosine negative.

  • Given a coterminal task: find the angle between 0° and 360° for θ = 3,723°; remainder after subtracting 3,600° is 123°.

  • Given a vector (8, -5) in standard position, compute r, sin θ, cos θ, and note signs based on Quadrant IV.

• Calculator etiquette (recap):

  • Use degree mode for angle calculations and for minutes/seconds conversions.

  • The “angle” function (often colored blue) helps input degrees/minutes; minutes are entered as minutes (′) and seconds (″) using appropriate keys (Alpha, then the green keys for minutes/seconds on many calculators).

Note: The transcript contains a few minor typographical inconsistencies (e.g., some comments about “between 90” in the first quadrant). The standard mathematical conventions are used above, with corrections noted for clarity where helpful.