Trigonometry Cheat Sheet: Formulas & Identities

TRIGONOMETRY Angle measurements 1 rev = 360° = 𝟒𝟎𝟎𝒈 = 𝟐𝝅 𝒓𝒂𝒅 1 NATO = 0.05625 deg CENTESIMAL SYSTEM Deg  grad CIRCULAR OR NATURAL SYSTEM Deg  rad 𝟒𝟎𝟎𝒈 ቇ 𝒅𝒆𝒈 ቆ 𝟑𝟔𝟎° 𝒅𝒆𝒈 ൬ 𝝅 ൰ 𝟏𝟖𝟎° rad  Deg SEXAGESIMAL SYSTEM 𝒓𝒂𝒅 ൬ 𝟏 𝒓𝒆𝒗 = 𝟑𝟔𝟎° 𝟏𝟖𝟎° ൰ 𝝅 ANGLES 180 – x = Supplement  sum is 180° 90 – x = Complement  sum is 90° 360 – x = Conjugate/Explementary  sum is 360° TRIGONOMETRIC FUNCTIONS 𝒐 − 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝜽 𝒂 − 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 SOHCAHTOA 𝒔𝒊𝒏𝜽 = 𝒐 𝒉 RECIPROCAL 𝒄𝒐𝒔𝜽 = 𝒂 𝒉 𝒕𝒂𝒏𝜽 = 𝒐 𝒂 𝑪𝒔𝒄𝜽 = 𝒉 𝒐 𝒔𝒆𝒄𝜽 = 𝒉 𝒂 𝒄𝒐𝒕𝜽 = 𝒂 𝒐 RECIPROCAL RELATIONS QUOTIENT RELATIONS 𝑨) 𝒔𝒊𝒏𝜽 = 𝟏 𝒄𝒔𝒄𝜽 𝑫) 𝒄𝒔𝒄𝜽 = 𝟏 𝒔𝒊𝒏𝜽 𝑩) 𝒄𝒐𝒔𝜽 = 𝟏 𝒔𝒆𝒄𝜽 𝑬) 𝒔𝒆𝒄𝜽 = 𝟏 𝒄𝒐𝒔𝜽 𝑪) 𝒕𝒂𝒏𝜽 = 𝟏 𝒄𝒐𝒕𝜽 𝑭) 𝒄𝒐𝒕𝜽 = 𝟏 𝒕𝒂𝒏𝜽 𝑨) 𝒕𝒂𝒏𝜽 = 𝒔𝒊𝒏𝜽 𝒄𝒐𝒔𝜽 𝑩) 𝒄𝒐𝒕𝜽 = 𝒄𝒐𝒔𝜽 𝒔𝒊𝒏𝜽 OTHER TRIGONOMETRIC FUNCTIONS y UNIT CIRCLE 𝑵𝑶𝑻𝑬: 𝑺𝒊𝒏𝒄𝒆 𝑼𝒏𝒊𝒕 𝑪𝒊𝒓𝒄𝒍𝒆, 𝒓𝒂𝒅𝒊𝒖𝒔 = 𝟏 𝟏 𝒚 = 𝒔𝒊𝒏𝜽 𝜽 𝒙 = 𝒄𝒐𝒔𝜽 𝒗𝒆𝒓𝒔𝒊𝒏𝜽 𝟏 x 𝒗𝒆𝒓𝒄𝒐𝒔𝜽 JUST REMEMBER THESE TWO EQUATIONS: IF THERE IS A: “Co” as prefix  Just change the “𝒄𝒐𝒔𝜽” to “𝒔𝒊𝒏𝜽” “Ha” as prefix  Just divide the equation by 2 “HA”  half of the function 𝟏 − 𝒄𝒐𝒔𝜽 𝑯𝒂𝒗𝒆𝒓𝒔𝒊𝒏𝜽 = 𝟐 𝑯𝒂𝒗𝒆𝒓𝒄𝒐𝒔𝜽 = 𝟏 + 𝒄𝒐𝒔𝜽 𝟐 𝒗𝒆𝒓𝒔𝒊𝒏𝜽 = 𝟏 − 𝒄𝒐𝒔𝜽 𝒗𝒆𝒓𝒄𝒐𝒔𝜽 = 𝟏 + 𝒄𝒐𝒔𝜽 “Co”  Equivalent complement angle 𝑪𝒐𝒗𝒆𝒓𝒔𝒊𝒏𝜽 = 𝟏 − 𝒔𝒊𝒏𝜽 𝑪𝒐𝒗𝒆𝒓𝒄𝒐𝒔𝜽 = 𝟏 + 𝒔𝒊𝒏𝜽 COMBINATION OF “HA” AND “CO” 𝑯𝒂𝑪𝒐𝒗𝒆𝒓𝒔𝒊𝒏𝜽 = 𝟏 − 𝒔𝒊𝒏𝜽 𝟐 𝑯𝒂𝑪𝒐𝒗𝒆𝒓𝒄𝒐𝒔𝜽 = 𝟏 + 𝒔𝒊𝒏𝜽 𝟐 ADDITIONAL TRIGO FUNCTIONS: 𝑬𝒙𝒔𝒆𝒄𝜽 = 𝒔𝒆𝒄𝜽 − 𝟏 𝑬𝒙𝒄𝒔𝒄𝜽 = 𝒄𝒔𝒄𝜽 − 𝟏 𝑪𝒉𝒐𝒓𝒅 = 𝟐𝒔𝒊𝒏 𝜽 𝟐 Sign per quadrant QII QI QIIi QIv POSITIVE NEGATIVE 𝒔𝒊𝒏𝜽 QI and QII QIII and QIV 𝒄𝒐𝒔𝜽 QI and QIV QII and QIII 𝒕𝒂𝒏𝜽 QI and QIII QII and QIV TRIGONOMETRIC IDENTITIES PYTHAGOREAN IDENTITIES Double angle IDENTITIES 𝟏 = 𝒔𝒊𝒏𝟐 𝒙 + 𝒄𝒐𝒔𝟐 𝒙 𝒔𝒊𝒏𝟐𝜽 = 𝟐𝒔𝒊𝒏𝜽𝒄𝒐𝒔𝜽 𝟏 = 𝒔𝒆𝒄𝟐 𝒙 − 𝒕𝒂𝒏𝟐 𝒙 𝒄𝒐𝒔𝟐𝜽 = 𝒄𝒐𝒔𝟐 𝜽 − 𝒔𝒊𝒏𝟐 𝜽 𝟏 = 𝒄𝒔𝒄𝟐 𝒙 − 𝒄𝒐𝒕𝟐 𝒙 𝒕𝒂𝒏𝟐𝜽 = 𝟐𝒕𝒂𝒏𝜽 𝟏 − 𝒕𝒂𝒏𝟐 𝜽 SUM AND DIFFERENCE OF TWO ANGLES OR IDENTITIES 𝒔𝒊𝒏(𝑨 + 𝑩) = 𝒔𝒊𝒏𝑨𝒄𝒐𝒔𝑩 + 𝒄𝒐𝒔𝑨𝒔𝒊𝒏𝑩 𝒄𝒐𝒔(𝑨 + 𝑩) = 𝒄𝒐𝒔𝑨𝒄𝒐𝒔𝑩 + 𝒔𝒊𝒏𝑨𝒔𝒊𝒏𝑩 𝒕𝒂𝒏(𝑨 + 𝑩) = Half angle formulas 𝒔𝒊𝒏 𝜽 𝟏 − 𝒄𝒐𝒔𝜽 = ±ඨ 𝟐 𝟐 𝜽 𝟏 + 𝒄𝒐𝒔𝜽 𝒄𝒐𝒔 = ±ඨ 𝟐 𝟐 𝒕𝒂𝒏 𝜽 𝟏 − 𝒄𝒐𝒔𝜽 = 𝟐 𝒔𝒊𝒏𝜽 𝒕𝒂𝒏𝑨 + 𝒕𝒂𝒏𝑩 𝟏 − 𝒕𝒂𝒏𝑨𝒕𝒂𝒏𝑩 PRODUCT OF FUNCTIONS 𝑺𝒊𝒏𝒙𝑪𝒐𝒔𝒚 = 𝟏 ൣ𝒔𝒊𝒏൫𝒙 + 𝒚൯ + 𝒔𝒊𝒏൫𝒙 + 𝒚൯൧ 𝟐 𝑺𝒊𝒏𝒙𝑺𝒊𝒏𝒚 = 𝟏 ൣ𝒄𝒐𝒔൫𝒙 − 𝒚൯ − 𝒄𝒐𝒔൫𝒙 + 𝒚൯൧ 𝟐 𝑪𝒐𝒔𝒙𝑪𝒐𝒔𝒚 = 𝟏 ൣ𝒄𝒐𝒔൫𝒙 + 𝒚൯ + 𝒄𝒐𝒔൫𝒙 − 𝒚൯൧ 𝟐 PARITY 𝒔𝒊𝒏(−𝜽) = −𝒔𝒊𝒏𝜽 𝒄𝒐𝒔(−𝜽) = 𝒄𝒐𝒔𝜽 𝒕𝒂𝒏(−𝜽) = −𝒕𝒂𝒏𝜽 𝒄𝒔𝒄(−𝜽) = −𝒄𝒔𝒄𝜽 𝒔𝒆𝒄(−𝜽) = 𝒔𝒆𝒄𝜽 𝒄𝒐𝒕(−𝜽) = −𝒄𝒐𝒕𝜽 WAVE CHARACTERISTICS GIVEN SIN AND COS GIVEN TANGENT 𝒚 = 𝑨𝒔𝒊𝒏(𝑩𝒙 + 𝒄) + 𝑫 𝒚 = 𝑨𝒄𝒐𝒔(𝑩𝒙 + 𝒄) + 𝑫 𝒚 = 𝑨𝒕𝒂𝒏(𝑩𝒙 + 𝒄) + 𝑫 𝝅 90° = 𝟐 180° = 𝝅 𝒚 = 𝑨𝒔𝒊𝒏(𝑩𝒙 + 𝒄) + 𝑫 𝒚 = 𝑨𝒄𝒐𝒔(𝑩𝒙 + 𝒄) + 𝑫 𝟑𝝅 270° = 𝟐 AMPLITUDE = A 360° = 𝟐𝝅 PERIOD, T = 𝟐𝝅 AMPLITUDE = ∞ 𝒐𝒓 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅 𝑩 FREQUENCY, f = 𝑩 PERIOD, T = 𝟐𝝅 −𝒄 𝝅 𝑩 𝑩 PHASE SHIFT = 𝑩 FREQUENCY, f = 𝝅 VERTICAL SHIFT = 𝑫 PHASE SHIFT = 𝑩 −𝒄 VERTICAL SHIFT = 𝑫 TRIANGLES Types of triangle with respect to side EQUILATERAL TRIANGLE ISOSCELES TRIANGLE SCALENE TRIANGLE 𝟔𝟎° 𝟔𝟎° 𝟔𝟎° 𝜽 𝜽 Types of triangle with respect to ANGLE 𝑹𝒊𝒈𝒉𝒕 𝑻𝒓𝒊𝒂𝒏𝒍𝒆 = 𝟗𝟎° 𝑨𝒄𝒖𝒕𝒆 𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆 = 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 𝟗𝟎° 𝑶𝒃𝒕𝒖𝒔𝒆 𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆 = 𝒈𝒓𝒆𝒂𝒕𝒆𝒓 𝒕𝒉𝒂𝒏 𝟗𝟎° PYTHAGOREAN THEOREM 𝒓𝟐 = 𝒙𝟐 + 𝒚𝟐 𝒚 𝒙 = 𝒓𝒄𝒐𝒔𝜽 𝒚 = 𝒓𝒔𝒊𝒏𝜽 𝜽 𝒙 SINE LAW AND COSINE LAW 𝑩 SINE LAW 𝑨 𝒂 𝒃 𝒄 = = 𝒔𝒊𝒏𝑨 𝒔𝒊𝒏𝑩 𝒔𝒊𝒏𝑪 𝒂 𝒄 𝒃 𝑪 Cosine law 𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐 − 𝟐𝒂𝒃𝒄𝒐𝒔𝑪 𝒂𝟐 = 𝒃𝟐 + 𝒄𝟐 − 𝟐𝒃𝒄𝒄𝒐𝒔𝑨 𝒃𝟐 = 𝒂𝟐 + 𝒄𝟐 − 𝟐𝒂𝒄𝒄𝒐𝒔𝑩 COMPLEMENTARY TRIGO FUNCTIONS 𝒔𝒊𝒏𝜽 = 𝒄𝒐𝒔(𝟗𝟎 − 𝜽) 𝒕𝒂𝒏𝜽 = 𝒄𝒐𝒕(𝟗𝟎 − 𝜽) 𝒔𝒆𝒄𝜽 = 𝒄𝒔𝒄(𝟗𝟎 − 𝜽) 𝒄𝒐𝒔𝜽 = 𝒔𝒊𝒏(𝟗𝟎 − 𝜽) 𝒄𝒐𝒕𝜽 = 𝒕𝒂𝒏(𝟗𝟎 − 𝜽) 𝒄𝒔𝒄𝜽 = 𝒔𝒆𝒄(𝟗𝟎 − 𝜽)

Trigonometry Cheat Sheet: Formulas & Identities

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Trigonometry Cheat Sheet: Formulas & Identities

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