Unit Circle and Trig Identities on a TI-84 Plus CE

How it works The button opens this sheet in the CiNotes notes editor, ready to change. It’s 729 words, so it’s a one-time $3 (typed notes are free up to 300 words). You get a ZIP by email to send to your calculator. Works on the TI-84 Plus CE and CE Python. On OS 5.5 or newer, do the free one-time setup first.

Unit circle and trig identities sheet

Each unit circle point is (cos θ, sin θ). On the calculator version, √2 and √3 are written r2 and r3 to keep lines short.

Unit circle, first and second quadrants

0° = 0	(1, 0)
30° = π/6	(√3/2, 1/2)
45° = π/4	(√2/2, √2/2)
60° = π/3	(1/2, √3/2)
90° = π/2	(0, 1)
120° = 2π/3	(−1/2, √3/2)
135° = 3π/4	(−√2/2, √2/2)
150° = 5π/6	(−√3/2, 1/2)

Unit circle, third and fourth quadrants

180° = π	(−1, 0)
210° = 7π/6	(−√3/2, −1/2)
225° = 5π/4	(−√2/2, −√2/2)
240° = 4π/3	(−1/2, −√3/2)
270° = 3π/2	(0, −1)
300° = 5π/3	(1/2, −√3/2)
315° = 7π/4	(√2/2, −√2/2)
330° = 11π/6	(√3/2, −1/2)

Tangent and signs

Tangent values	tan 0° = 0; tan 30° = √3/3; tan 45° = 1; tan 60° = √3; tan 90° undefined
ASTC	Q1 all positive; Q2 sin and csc; Q3 tan and cot; Q4 cos and sec (“All Students Take Calculus”)

Angles and triangles

Conversions	degrees × π/180 = radians; radians × 180/π = degrees
Reference angle	Q2: 180° − θ; Q3: θ − 180°; Q4: 360° − θ
Arc length and sector area	s = rθ; A = ½r²θ (θ in radians)
SOH CAH TOA	sin = opp/hyp; cos = adj/hyp; tan = opp/adj
Special triangles	30-60-90 sides 1 : √3 : 2; 45-45-90 sides 1 : 1 : √2

Reciprocal and quotient identities

Reciprocal	csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ
Quotient	tan θ = sin θ/cos θ; cot θ = cos θ/sin θ

Pythagorean identities

Main	sin² θ + cos² θ = 1
Divide by cos²	1 + tan² θ = sec² θ
Divide by sin²	1 + cot² θ = csc² θ

Even/odd and cofunction identities

Even	cos(−θ) = cos θ; sec(−θ) = sec θ
Odd	sin(−θ) = −sin θ; tan(−θ) = −tan θ; csc(−θ) = −csc θ; cot(−θ) = −cot θ
Cofunction	sin(π/2 − θ) = cos θ; cos(π/2 − θ) = sin θ; tan(π/2 − θ) = cot θ

Sum and difference

Sine	sin(a ± b) = sin a cos b ± cos a sin b
Cosine	cos(a ± b) = cos a cos b ∓ sin a sin b
Tangent	tan(a ± b) = (tan a ± tan b)/(1 ∓ tan a tan b)

Double angle

Sine	sin 2θ = 2 sin θ cos θ
Cosine	cos 2θ = cos² θ − sin² θ = 2cos² θ − 1 = 1 − 2sin² θ
Tangent	tan 2θ = 2 tan θ/(1 − tan² θ)

Half angle and power reducing

Sine	sin(θ/2) = ±√[(1 − cos θ)/2]
Cosine	cos(θ/2) = ±√[(1 + cos θ)/2]
Tangent	tan(θ/2) = (1 − cos θ)/sin θ = sin θ/(1 + cos θ)
Which sign	pick it from the quadrant of θ/2
Power reducing	sin² θ = (1 − cos 2θ)/2; cos² θ = (1 + cos 2θ)/2

Product to sum

sin a cos b	½[sin(a + b) + sin(a − b)]
cos a cos b	½[cos(a − b) + cos(a + b)]
sin a sin b	½[cos(a − b) − cos(a + b)]

Inverse trig and graphs

Ranges	arcsin: [−π/2, π/2]; arccos: [0, π]; arctan: (−π/2, π/2)
Domains	arcsin and arccos need −1 ≤ x ≤ 1
Sinusoid	y = A sin(B(x − C)) + D: amplitude \|A\|, period 2π/\|B\|, phase shift C, midline y = D
Tangent and cotangent period	π/\|B\|

Any triangle

Law of sines	a/sin A = b/sin B = c/sin C (SSA can give 0, 1 or 2 triangles)
Law of cosines	c² = a² + b² − 2ab cos C
Area	½ab sin C

On the TI-84 Plus CE

Mode	MODE: RADIAN or DEGREE before you solve
Inverse trig	2ND SIN, 2ND COS, 2ND TAN
csc, sec, cot	no keys; type 1/sin( and so on
Trig window	ZOOM ▸ ZTrig

What it looks like on the calculator

The calculator version is the same sheet in plain text with short lines, so it fits the screen. Square roots are written as sqrt( ), powers with ^ and Greek letters as words. On the calculator, search an angle like “225” or a word like “double” to jump straight to it. The notes are stored in Archive, so a RAM clear won’t erase them.

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Common questions

How much does it cost to put this on my calculator?

This sheet is 729 words, so it’s a one-time $3. Typed notes are free up to 300 words and $3 from 301 to 10,000 words. If you cut it down to 300 words or fewer, it’s free. You can also add your own notes to the same order, up to 10,000 words in total (about 9,271 more words).

Why put the unit circle on a calculator instead of memorizing it?

You should still learn it. Having it in your notes helps while you practice: you can check a value in a second instead of digging through your binder, and the patterns stick faster when you see them often.

Can I change the notes before I pay?

Yes. The button opens the sheet in the CiNotes notes editor. Delete what you don’t need, add your teacher’s examples, and check the calculator preview before you check out.

Will it work on my calculator?

It works on the TI-84 Plus CE and the TI-84 Plus CE Python. It doesn’t work on the TI-84 Evo or on black-and-white TI-84s. If your calculator is on OS 5.5 or newer, it needs a free one-time setup first. The setup guide walks you through it.

Will the notes disappear if my calculator resets?

No. The notes are stored in Archive, so a RAM clear won’t erase them. On OS 5.5 or newer, open AsmHook2 from the APPS menu once after a RAM clear, then open your notes as usual.

More subject sheets: AP Chemistry, Chemistry, AP Physics 1, AP Calculus AB, AP Statistics, Precalculus, Algebra 2. Or see all formula notes, every way to put notes on a TI-84, and what CiNotes is.

Unit circle and trig identities for your TI-84 Plus CE