Trigonometry
Trigonometry Complete Cheatsheet
Sin, cos, tan, identities, graphs, sine/cosine rule, and solving equations — complete trig reference.
📖 6 sections
⏰ 18 min read
✅ Quizzes included
🌙 Dark mode
01Right Triangle Trig▼
SOH CAH TOA
sin=Opp/Hyp, cos=Adj/Hyp, tan=Opp/Adj
Fundamental ratios.
Reciprocals
csc=1/sin, sec=1/cos, cot=1/tan
Less common.
Pythagorean
sin^2(t)+cos^2(t)=1
Fundamental identity.
TRIGExact values
theta     sin       cos       tan
0 deg     0         1         0
30 deg    1/2       sqrt3/2   1/sqrt3
45 deg    1/sqrt2   1/sqrt2   1
60 deg    sqrt3/2   1/2       sqrt3
90 deg    1         0         undefined
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Memory: sin values for 0,30,45,60,90 degrees = sqrt(0/4), sqrt(1/4), sqrt(2/4), sqrt(3/4), sqrt(4/4)
02CAST Diagram▼
Q1 (0-90)
ALL positive: sin,cos,tan
Q2 (90-180)
SIN positive only
Q3 (180-270)
TAN positive only
Q4 (270-360)
COS positive only
TRIGRelated angles
sin(180-t)=sin(t),  cos(180-t)=-cos(t)
sin(180+t)=-sin(t), cos(180+t)=-cos(t)
sin(360-t)=-sin(t), cos(360-t)=cos(t)
sin(-t)=-sin(t),    cos(-t)=cos(t)
03Identities▼
Pythagorean
sin^2+cos^2=1
Divide by cos^2: tan^2+1=sec^2
Double angle sin
sin2A=2sinA*cosA
Double angle cos
cos2A=cos^2A-sin^2A = 1-2sin^2A = 2cos^2A-1
Three forms!
Sum formula sin
sin(A+B)=sinA*cosB+cosA*sinB
Sum formula cos
cos(A+B)=cosA*cosB-sinA*sinB
❓ Quiz
Dividing sin^2+cos^2=1 by sin^2 gives?
Divide by sin^2: 1 + cos^2/sin^2 = 1/sin^2 -> 1+cot^2=csc^2.
04Trig Graphs▼
FunctionPeriodAmplitudeKey feature
y=sin x2*pi1Starts at 0
y=cos x2*pi1Starts at 1
y=tan xpiinfiniteAsymptotes at pi/2+n*pi
y=a*sin(bx+c)+d2*pi/|b||a|d=vert shift
TRIGGraph transformations
y=3*sin(2x-pi/4)+1
Amplitude=3, Period=pi
Phase shift=pi/8 right, Vertical shift=1 up
05Sine and Cosine Rules▼
Sine rule
a/sinA=b/sinB=c/sinC
Two angles+side, or two sides+non-included angle.
Cosine rule (side)
a^2=b^2+c^2-2bc*cosA
Two sides+included angle.
Cosine (angle)
cosA=(b^2+c^2-a^2)/2bc
Three sides known.
Area
Area=(1/2)*ab*sinC
Two sides+included angle.
TRIGExamples
# Sine rule: a=8,b=11,A=40 deg
sinB/11=sin40/8
sinB=11*sin40/8=0.884 -> B=62 deg

# Cosine rule: a=5,b=7,C=50 deg
c^2=25+49-2(5)(7)cos50=29.01 -> c=5.39
06Solving Trig Equations▼
TRIGSolving equations
# sin(x)=0.5 for 0 to 360 degrees
Principal: x=30 deg
Sin positive in Q1 and Q2
Solutions: x=30 and x=150 deg

# cos(2x)=-0.5 for 0 to 360 degrees
2x: cos-1(-0.5)=120 deg
2x = 120, 240, 480, 600 deg
x = 60, 120, 240, 300 deg

# tan(x)=1 for 0 to 360 degrees
Principal=45. Tan repeats every 180.
x = 45 and 225 deg
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Always find the principal angle first, then use CAST or symmetry to find all solutions in the given range.