Trigonometry Formula Sheet

Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 <θ < π 2 Unit circle definition For this definition θ is any angle. or 0 ° < θ < 90 ° . y ( x, y ) hypotenuse y opposite 1 θ x x θ adjacent opposite hypotenuse adjacent cosθ = hypotenuse opposite tan θ = adjacent sin θ = hypotenuse opposite hypotenuse secθ = adjacent adjacent cot θ = opposite cscθ = y =y 1 x cos θ = = x 1 y tan θ = x sin θ = 1 y 1 sec θ = x x cot θ = y csc θ = Facts and Properties Domain The domain is all the values of θ that can be plugged into the function. sinθ , cosθ , tan θ , csc θ , secθ , cotθ , θ can be any angle θ can be any angle 1⎞ ⎛ ⎝ ⎠ θ ≠ n π , n = 0, ± 1, ± 2,… 1⎞ ⎛ θ ≠ ⎜ n + ⎟ π , n = 0, ± 1, ± 2,… 2⎠ ⎝ θ ≠ n π , n = 0, ± 1, ± 2,… θ ≠ ⎜ n + ⎟ π , n = 0, ± 1, ± 2,… 2 Range The range is all possible values to get out of the function. cscθ ≥ 1 and cscθ ≤ −1 −1 ≤ sin θ ≤ 1 −1 ≤ cos θ ≤ 1 sec θ ≥ 1 and sec θ ≤ −1 −∞ ≤ tanθ ≤ ∞ −∞ ≤ cot θ ≤ ∞ Period The period of a function is the number, T, such that f (θ + T ) = f ( θ ) . So, if ω is a fixed number and θ is any angle we have the following periods. sin (ωθ ) → cos ( ωθ ) → tan ( ωθ ) → csc (ωθ ) → sec (ωθ ) → cot (ωθ ) → T = T = 2π ω 2π ω π ω 2π T = ω 2π T = ω π T = ω T= Formulas and Identities Tangent and Cotangent Identities sin θ cos θ tan θ = cot θ = cosθ sin θ Reciprocal Identities 1 1 csc θ = sin θ = sin θ csc θ 1 1 secθ = cosθ = cosθ secθ 1 1 cotθ = tanθ = tanθ cot θ Pythagorean Identities 2 2 sin θ + cos θ = 1 tan 2 θ + 1 = sec 2 θ 1 + cot 2 θ = csc 2 θ Even/Odd Formulas sin ( −θ ) = − sin θ csc( −θ ) = − cscθ cos( −θ ) = cosθ sec( −θ ) = secθ tan ( −θ ) = − tan θ cot ( −θ ) = − cot θ Periodic Formulas If n is an integer. sin ( θ + 2πn ) = sin θ csc( θ + 2π n ) = csc θ cos( θ + 2πn ) = cos θ sec( θ + 2π n ) = secθ tan (θ + π n ) = tan θ cot (θ + π n ) = cot θ Double Angle Formulas sin ( 2θ ) = 2 sinθ cosθ cos( 2θ ) = cos2 θ − sin2 θ = 2 cos 2 θ −1 = 1− 2sin2 θ 2 tan θ tan ( 2θ ) = 1 − tan 2 θ Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then π πx t 180t and x = = ⇒ t= π 180 x 180 Half Angle Formulas 1 sin 2 θ = (1 − cos (2θ )) 2 1 2 cos θ = (1 + cos (2θ ) ) 2 1 − cos ( 2θ ) tan 2 θ = 1 +cos (2θ ) Sum and Difference Formulas sin ( α ± β ) = sin α cos β ± cos α sin β cos ( α ± β ) = cos α cos β ∓ sin α sin β tan α ± tan β 1 ∓ tan α tan β Product to Sum Formulas 1 sin α sin β = ⎡⎣ cos (α − β ) − cos (α + β )⎤⎦ 2 1 cos α cos β = ⎣⎡cos (α − β ) + cos (α + β ) ⎦⎤ 2 1 sin α cos β = ⎡⎣ sin (α + β ) + sin (α − β )⎤⎦ 2 1 cos α sin β = ⎣⎡sin (α + β ) − sin (α − β )⎦⎤ 2 Sum to Product Formulas ⎛ α+ β⎞ ⎛ α− β⎞ sin α + sin β = 2 sin ⎜ ⎟ cos ⎜ 2 ⎟ ⎝ 2 ⎠ ⎝ ⎠ tan (α ± β ) = α + β ⎞ ⎛α − β ⎞ sin α − sin β = 2 cos ⎜⎛ ⎟ sin ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎛α− β⎞ ⎛α+ β⎞ cos α + cos β = 2 cos ⎜ ⎟ cos ⎜ 2 ⎟ ⎝ 2 ⎠ ⎝ ⎠ ⎛α + β ⎞ ⎛α − β ⎞ cos α − cos β = −2 sin ⎜ ⎟ sin ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ Cofunction Formulas ⎛π ⎞ sin ⎜ − θ ⎟ = cos θ ⎝2 ⎠ ⎛π ⎞ cos ⎜ − θ ⎟ = sin θ ⎝2 ⎠ ⎛π ⎞ csc ⎜ − θ ⎟ = sec θ ⎝2 ⎠ ⎛π ⎞ tan ⎜ −θ ⎟ = cot θ ⎝2 ⎠ ⎛π ⎞ sec ⎜ − θ ⎟ = csc θ ⎝2 ⎠ ⎛π ⎞ cot ⎜ −θ ⎟ = tan θ ⎝2 ⎠ Unit Circle y π ⎛ 1 3⎞ ⎜− , ⎟ ⎝ 2 2 ⎠ 2 3 ⎛ 2 2⎞ , ⎜− ⎟ ⎝ 2 2 ⎠ ⎛ 3 1⎞ ⎜− , ⎟ ⎝ 2 2⎠ 3π 4 (0,1 ) 2 π 90° 3 120° π 60° ⎛ 2 2 ⎞ , ⎜⎜ ⎟⎟ ⎝ 2 2 ⎠ 4 π 45° 135° 5π 6 ⎛ 1 3⎞ ⎜⎜ , ⎟⎟ ⎝2 2 ⎠ 30° ⎛ 3 1⎞ , ⎜⎜ 2 2 ⎟⎟ ⎝ ⎠ 6 150° ( −1,0) π 180° ⎛ 3 1⎞ ⎜− , − ⎟ 2⎠ ⎝ 2 7π 6 ⎛ 2 2⎞ ,− ⎜− ⎟ 2 ⎠ ⎝ 2 210° 4π 3 0 360° 2π 330° 11 π 6 225° 5π 4 0° 240° ⎛ 1 3⎞ ⎜ − ,− ⎟ ⎝ 2 2 ⎠ 315° 7π 300° 270° 4 5π 3π 3 2 ⎛ ⎞ (0,−1 ) 1 3 ⎜ ,− ⎟ ⎝2 2 ⎠ For any ordered pair on the unit circle ( x, y ) : cos θ = x and sin θ = y Example ⎛ 5π ⎞ = 1 cos ⎜ ⎟ ⎝ 3 ⎠ 2 ⎛ 5π ⎞ = − 3 sin ⎜ ⎟ 2 ⎝ 3 ⎠ (1,0 ) ⎛ 3 1⎞ ,− ⎟ ⎜ ⎝ 2 2⎠ ⎛ 2 2⎞ ,− ⎜ ⎟ 2 ⎠ ⎝ 2 x Inverse Trig Functions Definition y = sin −1 x is equivalent to x =sin y Inverse Properties 1 1 cos ( cos − ( x )) = x cos − ( cos ( θ ) ) = θ y = cos− 1 x is equivalent to x = cos y sin ( sin −1 ( x )) = x sin −1 ( sin (θ )) = θ tan ( tan −1 ( x ) ) = x tan −1 (tan (θ y = tan−1 x is equivalent to x = tan y Domain and Range Function Domain y = sin−1 x −1 y = cos x −1 y = tan x −1 ≤ x ≤ 1 − −1 ≤ x ≤ 1 −∞ < x < ∞ Alternate Notation sin −1 x = arcsin x Range − π π ≤ y≤ 2 2 0 ≤ y ≤π π 2 < y< cos −1 x = arccos x tan −1 x = arctan x π 2 Law of Sines, Cosines and Tangents c β a α γ b Law of Sines sin α sin β sin γ = = a b c Law of Tangents a − b tan 12 (α − β ) = a + b tan 12 (α + β ) Law of Cosines a2 = b2 + c2 − 2bc cos α b − c tan 12 ( β − γ ) = b +c tan 12 ( β + γ ) b2 = a2 + c2 − 2 ac cos β a − c tan 12 ( α − γ ) = a + c tan 12 ( α + γ ) c2 = a2 + b2 − 2 ab cos γ Mollweide’s Formula a + b cos 12 ( α − β ) = c sin 12 γ )) = θ