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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Wednesday, June 3, 2009 at 4:14 pm

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.

Sunday, October 18, 2009 at 10:08 pm

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

Saturday, April 30, 2011 at 7:58 am

TEŞEKKÜRLER 🙂

Sunday, July 3, 2011 at 8:47 am

osm man lets fool mathz hey well waiting for some more

Friday, June 22, 2012 at 10:49 am

sin=a/c

cos=b/c

tan=a/b

csc=c/a

sec=c/b

cot=b/a

Tuesday, June 26, 2012 at 1:00 am

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

Thursday, August 30, 2012 at 6:22 pm

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.

Wednesday, October 3, 2012 at 6:18 pm

Thanks for the tip, I will make an update above to reflect that.

Monday, December 30, 2013 at 8:41 pm

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!