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°)
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.
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
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).
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.
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.
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
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.
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
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
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
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
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
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
Q: What does SOH-CAH-TOA stand for and how is it used to find missing sides in a right triangle?
A: SOH-CAH-TOA: Sin=Opposite/Hypotenuse · Cos=Adjacent/Hypotenuse · Tan=Opposite/Adjacent. To find a missing side: (1) Label the triangle — identify hypotenuse (opposite the right angle), side opposite the known angle, side adjacent to the known angle. (2) Choose the trig ratio involving your known side. (3) Set up and solve. Example: find side x opposite 35° in a triangle with hypotenuse 12 → sin(35°)=x/12 → x=12·sin(35°)≈6.88.
Q: Explain the ASTC rule and give the sign of every function in every quadrant.
A: "All Students Take Calculus" — ASTC tells which trig functions are POSITIVE. Q1 (0°–90°): All positive. Q2 (90°–180°): Sine and csc positive; cos, tan, sec, cot negative. Q3 (180°–270°): Tangent and cot positive; sin, cos, csc, sec negative. Q4 (270°–360°): Cosine and sec positive; sin, tan, csc, cot negative. Reciprocal functions always match the sign of their base function.
Q: State the three Pythagorean identities and show how the second and third are derived from the first.
A: (1) sin²θ+cos²θ=1. (2) tan²θ+1=sec²θ. (3) 1+cot²θ=csc²θ. Derivation: Start with sin²θ+cos²θ=1. Divide every term by cos²θ: sin²/cos²+1=1/cos² → tan²θ+1=sec²θ. Divide every term by sin²θ: 1+cos²/sin²=1/sin² → 1+cot²θ=csc²θ. All three come from x²+y²=1 on the unit circle where x=cosθ, y=sinθ.
Q: What are the 30-60-90 and 45-45-90 triangle ratios and what exact values do they give?
A: 30-60-90 (sides 1:√3:2): sin30°=½, cos30°=√3/2, tan30°=√3/3. sin60°=√3/2, cos60°=½, tan60°=√3. 45-45-90 (sides 1:1:√2): sin45°=cos45°=√2/2, tan45°=1. These let you find exact values (no calculator) at 30°, 45°, 60° and their equivalents in other quadrants using ASTC.
Q: When do you use Law of Sines vs. Law of Cosines, and what is the ambiguous case?
A: Law of Sines (a/sinA=b/sinB=c/sinC): use for AAS or ASA. Also SSA — the AMBIGUOUS CASE: depending on the side length given, there may be 0, 1, or 2 valid triangles. Always check for a second solution by testing 180°−A. Law of Cosines (c²=a²+b²−2ab·cosC): use for SAS or SSS. When C=90°, cos90°=0 and it reduces to the Pythagorean theorem.