SOH: Sine=Opposite÷Hypotenuse. CAH: Cosine=Adjacent÷Hypotenuse. TOA: Tangent=Opposite÷Adjacent. The hypotenuse is always opposite the right angle — the longest side.
Know which trig functions are positive in each quadrant instantly
Quadrant I: All positive. Quadrant II: only Sine (and csc) positive. Quadrant III: only Tangent (and cot) positive. Quadrant IV: only Cosine (and sec) positive.
A — Q1
All positive: sin, cos, tan, csc, sec, cot all positive (0°–90°)
S — Q2
Sine positive: sin and csc positive; cos and tan negative (90°–180°)
T — Q3
Tangent positive: tan and cot positive; sin and cos negative (180°–270°)
C — Q4
Cosine positive: cos and sec positive; sin and tan negative (270°–360°)
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🃏 Trigonometry · Unit Circle
All Students Take Calculus
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🃏 Answer
A — Q1All positive: sin, cos, tan, csc, sec, cot all positive (0°–90°)
S — Q2Sine positive: sin and csc positive; cos and tan negative (90°–180°)
T — Q3Tangent positive: tan and cot positive; sin and cos negative (180°–270°)
C — Q4Cosine positive: cos and sec positive; sin and tan negative (270°–360°)
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📐 Trigonometry · Identities
sin²θ + cos²θ = 1
Pythagorean Identity — the most important trig identity; derives two more: tan²θ+1=sec²θ and 1+cot²θ=csc²θ
The single most used trig identity on every exam
sin²θ+cos²θ=1. Rearrange: sin²θ=1−cos²θ or cos²θ=1−sin²θ. Divide by cos²θ→tan²θ+1=sec²θ. Divide by sin²θ→1+cot²θ=csc²θ. Three powerful identities from one!
sin²+cos²=1
The root identity — from the Pythagorean theorem on the unit circle (x²+y²=1)
tan²+1=sec²
Divide sin²+cos²=1 by cos² — useful with tangent and secant
1+cot²=csc²
Divide sin²+cos²=1 by sin² — useful with cotangent and cosecant
Special Right Triangles — memorize these two for exact trig values without a calculator
Find exact trig values at 30°, 45°, 60° without a calculator
30-60-90: short leg=1, long leg=√3, hypotenuse=2. So sin30°=½, cos30°=√3/2, sin60°=√3/2, cos60°=½. 45-45-90: legs=1, hypotenuse=√2. So sin45°=cos45°=√2/2, tan45°=1.
Special Right Triangles — memorize these two for exact trig values without a calculator — 30-60-90: short leg=1, long leg=√3, hypotenuse=2. So sin30°=½, cos30°=√3/2, sin60°=√3/2, cos60°=½. 45-45-90: legs=1, hypotenuse=√2. So sin45°=cos45°=√2/2, tan45°=1.
Double Angle Formulas — sin(2θ) and cos(2θ) — cos(2θ) has three equivalent forms — sin(2θ)=2sin(θ)cos(θ). cos(2θ)=cos²(θ)−sin²(θ)=2cos²(θ)−1=1−2sin²(θ). Three forms of cos(2θ) — choose the most convenient based on what is known.
Amplitude=|A|, Period=2π/B, Phase shift=C (right if positive), Vertical shift=D. Always identify all four before graphing. The midline is y=D; the graph oscillates |A| units above and below it.
D
D (vertical shift) — moves the midline up or down from y=0
A
A (amplitude) — |A| is the distance from midline to peak or trough
B
B (frequency) — period = 2π/B; larger B means faster oscillation
C
C (phase shift) — horizontal shift; positive C shifts the graph right
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🃏 Trigonometry · Graphing
Sinusoid transformations (DABC) — what does each letter do?
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🃏 Answer
DABC: y = D + A·sin(B(x − C))
DD (vertical shift) — moves the midline up or down from y=0
AA (amplitude) — |A| is the distance from midline to peak or trough
BB (frequency) — period = 2π/B; larger B means faster oscillation
CC (phase shift) — horizontal shift; positive C shifts the graph right
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📐 Trigonometry · Laws
Law of Sines: a/sinA = b/sinB = c/sinC
Law of Sines — use for AAS, ASA, or SSA (the ambiguous case)
Solve any non-right triangle with a known angle-opposite-side pair
Use when you know: two angles and any side (AAS or ASA) or two sides and a non-included angle (SSA — the ambiguous case: could have 0, 1, or 2 valid triangles!). Set up ratios: side ÷ sin(opposite angle).
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🃏 Trigonometry · Laws
The Law of Sines?
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🃏 Answer
Law of Sines: a/sinA = b/sinB = c/sinC
Law of Sines — use for AAS, ASA, or SSA (the ambiguous case) — Use when you know: two angles and any side (AAS or ASA) or two sides and a non-included angle (SSA — the ambiguous case: could have 0, 1, or 2 valid triangles!). Set up ratios: side ÷ sin(opposite angle).
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📐 Trigonometry · Laws
Law of Cosines: c² = a² + b² − 2ab·cosC
Law of Cosines — use for SAS (two sides + included angle) or SSS (all three sides)
Solve any non-right triangle when you know two sides and the included angle
Use for SAS (two sides + included angle) or SSS (all three sides). It is the generalized Pythagorean theorem — when C=90°, cos90°=0 and it reduces to a²+b²=c².
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🃏 Trigonometry · Laws
The Law of Cosines?
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🃏 Answer
Law of Cosines: c² = a² + b² − 2ab·cosC
Law of Cosines — use for SAS (two sides + included angle) or SSS (all three sides) — Use for SAS (two sides + included angle) or SSS (all three sides). It is the generalized Pythagorean theorem — when C=90°, cos90°=0 and it reduces to a²+b²=c².
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📐 Trigonometry · Unit Circle
cos θ = x-coordinate. sin θ = y-coordinate.
Unit Circle — every point is (cosθ, sinθ) at angle θ from the positive x-axis
Read any trig value directly from the unit circle
Every point on the unit circle is (cosθ, sinθ). At θ=0°: (1,0). At θ=90°: (0,1). At θ=180°: (−1,0). At θ=270°: (0,−1). Memorize the 16 common angles and their exact coordinates.
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🃏 Trigonometry · Unit Circle
Unit circle — which coordinate is sin, which is cos?
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🃏 Answer
cos θ = x-coordinate. sin θ = y-coordinate.
Unit Circle — every point is (cosθ, sinθ) at angle θ from the positive x-axis — Every point on the unit circle is (cosθ, sinθ). At θ=0°: (1,0). At θ=90°: (0,1). At θ=180°: (−1,0). At θ=270°: (0,−1). Memorize the 16 common angles and their exact coordinates.
cscθ=1/sinθ=Hyp/Opp. secθ=1/cosθ=Hyp/Adj. cotθ=1/tanθ=Adj/Opp. Key: Cosecant pairs with Sine, Secant pairs with Cosine — the "co" prefix always goes with the OTHER function.
C (CHO)
Csc = Hypotenuse / Opposite = 1/sin — pairs with Sine
Radian Conversion — to radians: multiply by π/180 · to degrees: multiply by 180/π — Key values to memorize: 180°=π, 90°=π/2, 60°=π/3, 45°=π/4, 30°=π/6. Arc length formula: s=rθ (θ must be in radians). Area of sector: A=½r²θ.
Even-Odd Identities — Cosine is EVEN (symmetric about y-axis) · Sine and Tangent are ODD (symmetric about origin)
Simplify negative angle expressions in seconds
Cosine is EVEN: cos(−θ)=cos(θ). Sine is ODD: sin(−θ)=−sin(θ). Tangent is ODD: tan(−θ)=−tan(θ). Even = graph symmetric about y-axis. Odd = symmetric about the origin.
Even-Odd Identities — Cosine is EVEN (symmetric about y-axis) · Sine and Tangent are ODD (symmetric about origin) — Cosine is EVEN: cos(−θ)=cos(θ). Sine is ODD: sin(−θ)=−sin(θ). Tangent is ODD: tan(−θ)=−tan(θ). Even = graph symmetric about y-axis. Odd = symmetric about the origin.
Expanding trig functions of sums and differences of two angles
The cosine formula flips the sign — where sin keeps the same sign, cos uses the opposite
sin(A+B) = sinA cosB + cosA sinB. sin(A-B) = sinA cosB - cosA sinB. cos(A+B) = cosA cosB - sinA sinB. cos(A-B) = cosA cosB + sinA sinB. Use to find exact values like sin75 = sin(45+30). Double angle formulas are special cases where A equals B.
sin(A+B)
sinAcosB plus cosAsinB — sign stays the same as original
cos(A+B)
cosAcosB minus sinAsinB — sign FLIPS
Exact values
Split angle into 30, 45, or 60 degree combinations
sin(A+B)sinAcosB plus cosAsinB — sign stays the same as original
cos(A+B)cosAcosB minus sinAsinB — sign FLIPS
Exact valuesSplit angle into 30, 45, or 60 degree combinations
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Inverse Trig Functions
arcsin range -90 to 90. arccos range 0 to 180. arctan range -90 to 90 (open). Output is always an ANGLE.
The restricted domains and ranges of inverse trig functions
arcsin(1/2) = 30 degrees only, not 150, even though sin(150) = 1/2. Restricted range applies.
arcsin: domain [-1,1], range [-90,90] covering quadrants 4 and 1. arccos: domain [-1,1], range [0,180] covering quadrants 1 and 2. arctan: domain all reals, range (-90,90) covering quadrants 4 and 1 but never reaching the endpoints. Composition: sin(arcsin x) always equals x. arcsin(sin x) equals x only if x is inside [-90,90].
arcsin range
-90 to 90 degrees — quadrants 4 and 1 only
arccos range
0 to 180 degrees — quadrants 1 and 2 only
arctan range
Open interval -90 to 90 — never reaches plus or minus 90
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🃏 Inverse Trig Functions
Inverse trig functions — the ranges?
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🃏 Answer
arcsin range -90 to 90. arccos range 0 to 180. arctan range -90 to 90 (open). Output is always an ANGLE.
arcsin range-90 to 90 degrees — quadrants 4 and 1 only
arccos range0 to 180 degrees — quadrants 1 and 2 only
arctan rangeOpen interval -90 to 90 — never reaches plus or minus 90
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Solving Trig Equations
Isolate the trig function. Find reference angle. Use CAST for quadrants. Add the period for the general solution.
Step-by-step method for finding all solutions to a trigonometric equation
There are infinitely many solutions — add n times 360 degrees (or n times 2pi) for the general solution
Steps: Isolate the trig function. Find the reference angle using inverse trig. Use CAST to determine quadrants (All positive in Q1, Sin in Q2, Tan in Q3, Cos in Q4). Find all angles in 0 to 360 degrees. Add n times 360 for the general solution. Sin and cos have period 2pi. Tan has period pi. For multiple angle equations, solve first then divide.
Reference angle
Acute angle with the x-axis — always positive and less than 90
CAST quadrants
Which quadrants give positive values — two solutions per period
Add n times period
Add multiples of 360 or 2pi for all infinitely many solutions
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🃏 Solving Trig Equations
Solving trig equations — the steps?
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🃏 Answer
Isolate the trig function. Find reference angle. Use CAST for quadrants. Add the period for the general solution.
Reference angleAcute angle with the x-axis — always positive and less than 90
CAST quadrantsWhich quadrants give positive values — two solutions per period
Add n times periodAdd multiples of 360 or 2pi for all infinitely many solutions
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Half-Angle Formulas
sin(A/2) = sqrt of (1-cosA)/2. cos(A/2) = sqrt of (1+cosA)/2. Sign depends on quadrant of A/2 not A.
Half-angle formulas for finding exact values and simplifying trig integrals
The sign is determined by the quadrant of the HALF angle A/2, not the original angle A
sin(A/2) equals plus or minus square root of (1 minus cosA) over 2. cos(A/2) equals plus or minus square root of (1 plus cosA) over 2. Note that cosine has plus inside while sine has minus inside. Power-reducing forms: sin squared x equals (1 minus cos2x) over 2, cos squared x equals (1 plus cos2x) over 2. Essential for integrating powers of trig functions.
sin(A/2)
Square root of (1 minus cosA) over 2 — minus inside for sine
cos(A/2)
Square root of (1 plus cosA) over 2 — plus inside for cosine
Integration use
sin squared x = (1-cos2x)/2 — essential for trig integrals
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🃏 Half-Angle Formulas
The half-angle formulas?
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🃏 Answer
sin(A/2) = sqrt of (1-cosA)/2. cos(A/2) = sqrt of (1+cosA)/2. Sign depends on quadrant of A/2 not A.
sin(A/2)Square root of (1 minus cosA) over 2 — minus inside for sine
cos(A/2)Square root of (1 plus cosA) over 2 — plus inside for cosine
Integration usesin squared x = (1-cos2x)/2 — essential for trig integrals
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Polar Coordinates
To rectangular: x = r cosine theta, y = r sine theta. To polar: r = sqrt(x squared + y squared), theta = arctan(y/x) then check quadrant.
Converting between rectangular (x,y) and polar (r,theta) coordinate systems
A point has infinitely many polar representations — (r,theta) and (-r, theta+pi) describe the same point
Polar coordinates use r for distance from origin and theta for angle from positive x-axis. Convert to rectangular: x equals r times cosine theta, y equals r times sine theta. Convert to polar: r squared equals x squared plus y squared, tangent theta equals y over x — always check the quadrant. Negative r means go in the opposite direction of theta. Area in polar: one half times integral of r squared d-theta.
To polar
r = sqrt(x squared + y squared), theta = arctan(y/x) — check quadrant
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🃏 Polar Coordinates
Polar ↔ rectangular conversion?
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🃏 Answer
To rectangular: x = r cosine theta, y = r sine theta. To polar: r = sqrt(x squared + y squared), theta = arctan(y/x) then check quadrant.
To polarr = sqrt(x squared + y squared), theta = arctan(y/x) — check quadrant
To rectangularx = r cosine theta, y = r sine theta
Negative rGo in the opposite direction of theta
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Proving Trig Identities
Work ONE side only. Convert to sin and cos. Apply Pythagorean identities. Factor or combine fractions.
Strategy for proving trig identities without crossing the equals sign
Never move terms across the equal sign — transform one side until it matches the other side exactly
Rules: Work only ONE side, usually the more complex one. Never cross the equals sign. Convert everything to sine and cosine first. Apply Pythagorean identities: sin squared plus cos squared equals 1. Combine fractions over LCD. Factor when possible. Multiply by conjugate for expressions like 1 minus cosine theta. Replace sec with 1/cos, csc with 1/sin, tan with sin/cos.
One side only
Never cross equals — transform one side into the other
Convert to sin/cos
Replace sec, csc, tan, cot with sin and cos first
Pythagorean subs
sin squared + cos squared = 1 — look for substitution opportunities