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Old School Trigonometry Tricks & Tips

9 Comments

Note to self:

I was cleaning out some old notebooks and file and found my old TI-83 Graphing Calculator from high school (circa 1994-ish).  Inside its slip cover I found some old sticky notes with my trig tricks and tips.

The trig triangle of choice:

This triangle will be referenced by most of the tips below.

Radians to Degrees and back again:

  • radians = degrees * π / 180
  • degress = radians * 180 / π

SOH CAH TOA

  • sin ϴ = opposite / hypotenuse
  • cos ϴ = adjacent / hypotenuse
  • tan ϴ = opposite / adjacent
  • cot ϴ = adjacent / opposite (reciprocal of tan ϴ)
  • sec ϴ = hypotenuse / adjacent (reciprocal of cos ϴ)
  • csc ϴ = hypotenuse / opposite (reciprocal of sin ϴ)

Subdivision of cirle in 30º increments:

  • 30º = π / 6
  • 60º = 2π / 6
  • 90º = 3π / 6
  • 120º = 4π / 6
  • 150º = 5π / 6
  • 180º = π
  • 210º = 7π / 6
  • 240º = 8π / 6
  • 270º = 9π / 6
  • 300º = 10π / 6
  • 330º = 11π / 6
  • 360º = 2π

Subdivision of cirle in 45º increments:

  • 45º = π / 4
  • 90º = 2π / 4
  • 135º = 3π / 4
  • 180º = π
  • 225º = 5π / 4
  • 270º = 6π / 4
  • 315º = 7π / 4
  • 360º = 2π

Cosine constants for common values:

  • cos 0º = 1
  • cos 30º = √3 / 2
  • cos 45º = √2 / 2
  • cos 60º = 1 / 2
  • cos 90º = 0
  • cos 180º = -1
  • cos 270º = 0

Sine constants for common values:

  • sin 0º = 0
  • sin 30º = 1 / 2
  • sin 45º = √2 / 2
  • sin 60º = √3 / 2
  • sin 90º = 1
  • sin 180º = 0
  • sin 270º = -1

Tangent constants for common values:

  • tan 0º = 0
  • tan 30º = √3 / 3
  • tan 45º = 1
  • tan 60º = √3
  • tan 90º = ∞
  • tan 180º = 0
  • tan 270º = ∞

Secant constants for common values:

  • sec 0º = 1
  • sec 30º = 2√3 / 3
  • sec 45º = √2
  • sec 60º = 2
  • sec 90º = ∞
  • sec 180º = -1
  • sec 270º = ∞

Cosecant constants for common values:

  • sec 0º = ∞
  • sec 30º = 2
  • sec 45º = √2
  • sec 60º = 2√3 / 3
  • sec 90º = 1
  • sec 180º = ∞
  • sec 270º = -1

Cotangent constants for common values:

  • sec 0º = ∞
  • sec 30º = √3
  • sec 45º = 1
  • sec 60º = √3 / 3
  • sec 90º = 0
  • sec 180º = ∞
  • sec 270º = 0

Law of Cosines:

  • a² = b² + c² – 2bc * cos α
  • b² = a² + c² – 2ac * cos β
  • c² = a² + b² – 2ab * cos γ

….formulated to solve for the cosines:

  • cos α = (b² + c² – a²) / 2bc
  • cos β = (a² + c² – b²) / 2ac
  • cos γ = (a² + b² – c²) / 2ab

Law of Sines:

Given a circle with a radius = r circumscribed around a triangle,

  • sin α / a = 2r
  • sin β / b = 2r
  • sin γ / c = 2r

and the rest that I couldn’t make sense of:

Some inversions of the advanced stuff I never use:

  • arcsec ϴ = arccos (1 / ϴ)
  • arccsc ϴ = arcsin (1 / ϴ)

Some domain, range and quadrant stuff regarding f(x) = y:

  • y = tan^-1 x:
    • Domain: all real numbers
    • Range: -π / 2 – π / 2
    • Quadrants: I, IV
  • y = cot^-1 x:
    • Domain: all real numbers
    • Range: 0 – π
    • Quadrants: I, II
  • y = sec^-1 x:
    • Domain: x ≥ 1, x ≤ -1
    • Range: 0 – π
    • Quadrants: I, II
  • y = csc^-1 x:
    • Domain: x ≥ 1, x ≤ -1
    • Range: -π / 2 – π / 2
    • Quadrants: I, IV
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9 thoughts on “Old School Trigonometry Tricks & Tips

  1. Those are neat trig tips!

    On the AM of a H S Trig test, I had not done much homework, & hadn’t memorized the Sin Cos Tan Cot Sec Csc x,y,& r formulas.

    I stared at them a few mins before the test, & saw a pattern in the numerator: Y X Y X R R, then starting at CSC, & working up, the denominator had the same pattern.

    Thus:
    TAN = y/r
    COT = x/r
    SIN = y/x
    COS = x/y
    SEC = r/x
    CSC = r/y

    This ‘trick’ served me well through H S Trig, college Trig, physics, surveying, & EE classes.

    At no time could I have answered “What is SEC” – w/o writing or visualizing my memory ‘trick’. I made sure that I wrote it on the test paper that was handed out, rather than on my scratch paper – in case the proff walked by & saw it.

    When required to use the “opposite over adjacent” terminology, I had to sketch or visualize a right triangle, with my y, x, & r, then figure it out.

    The “Unit Circle” concept one proff taught us helped a LOT in visualizing which functions went from 1-0, or 0-1, or 0-infinity, while reinforcing my system.

    A quick query from the proff gave me a problem, but that seldom happened.

    In college trig, I realized I could derive or prove the Identities using my system.

    My sons had minors in math, & used this ‘trick’ I’d taught them during HS. One is an EE, & he also taught it to his elder son, in H S Trig – now entering Aero Engrg.

    • Your tips about the yxyxrr are really great except it should be sin cos tan cot sec csc. this is really gonna help me in my high school class i appriciate it greatly that you took the time to post this!!!!! thksssss

  2. TEŞEKKÜRLER 🙂

  3. osm man lets fool mathz hey well waiting for some more

  4. sin=a/c
    cos=b/c
    tan=a/b
    csc=c/a
    sec=c/b
    cot=b/a

  5. i think the best and simple trick to get answer is go through the formulae……..

  6. think of a native princess.
    SOH CAH TOA……..
    Sin=Opposite/Hypthenuse
    Cos=Adjacent/Hypothenuse
    Tan=Opposite/Adjacent
    then go in opposite orders throught the last 3 trig functions. the cotangent is opposite of the tangent and so forth. the last and first are opposites.
    Cot=adjacent/opposite (reciprocal of tan)
    Sec=Hypothenuse/adjacent (reciprocal of cos)
    Csc= Hypothenuse/opposite (reciprocal of sin)
    HOPE IT HELPS.

  7. This is a very convenient page with many of the most common trigonometry concepts and conversions. I had something similar to this back when I was in university. Thanks for sharing this – it’s a great resource that I hope many people will bookmark for reference!

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