Trig Cheat Sheet PDF

Trig Cheat Sheet Definition of the Trig Functions Right triangle definition Unit Circle Definition For this definition we assume that For this definition θ is any angle. π 0 < θ < or 0◦ < θ < 90◦. 2...

Trig Cheat Sheet Definition of the Trig Functions Right triangle definition Unit Circle Definition For this definition we assume that For this definition θ is any angle. π 0 < θ < or 0◦ < θ < 90◦. 2 opposite hypotenuse sin(θ) = csc(θ) = y 1 hypotenuse opposite sin(θ) = =y csc(θ) = adjacent hypotenuse 1 y cos(θ) = sec(θ) = x 1 hypotenuse adjacent cos(θ) = = x sec(θ) = opposite adjacent 1 x tan(θ) = cot(θ) = y x adjacent opposite tan(θ) = cot(θ) = x y Facts and Properties Domain Period The domain is all the values of θ that can be The period of a function is the number, T , such plugged into the function. that f (θ + T ) = f (θ). So, if ω is a fixed number sin(θ), θ can be any angle and θ is any angle we have the following periods. cos(θ), θ can be any angle 2π   sin (ω θ) → T = 1 ω tan(θ), θ 6= n + π, n = 0, ±1, ±2,... 2 2π cos (ω θ) → T = ω csc(θ), θ 6= nπ, n = 0, ±1, ±2,... π   tan (ω θ) → T = 1 ω sec(θ), θ 6= n + π, n = 0, ±1, ±2,... 2 2π csc (ω θ) → T = ω cot(θ), θ 6= nπ, n = 0, ±1, ±2,... 2π sec (ω θ) → T = ω π cot (ω θ) → T = ω Range The range is all possible values to get out of the function. −1 ≤ sin(θ) ≤ 1 −1 ≤ cos(θ) ≤ 1 −∞ < tan(θ) < ∞ −∞ < cot(θ) < ∞ sec(θ) ≥ 1 and sec(θ) ≤ −1 csc(θ) ≥ 1 and csc(θ) ≤ −1 © Paul Dawkins - https://tutorial.math.lamar.edu  Trig Cheat Sheet Formulas and Identities Tangent and Cotangent Identities Half Angle Formulas sin(θ) cos(θ)   θ r 1 − cos(θ) tan(θ) = cot(θ) = sin =± cos(θ) sin(θ) 2 2 Reciprocal Identities   r θ 1 + cos(θ) 1 1 cos =± csc(θ) = sin(θ) = 2 2 sin(θ) csc(θ) s   1 1 θ 1 − cos(θ) sec(θ) = cos(θ) = tan =± cos(θ) sec(θ) 2 1 + cos(θ) 1 1 cot(θ) = tan(θ) = Half Angle Formulas (alternate form) tan(θ) cot(θ) Pythagorean Identities sin2 (θ) = 1 2 (1 − cos(2θ)) 1 − cos(2θ) tan2 (θ) = cos2 (θ) = 12 (1 + cos(2θ)) 1 + cos(2θ) sin2 (θ) + cos2 (θ) = 1 tan2 (θ) + 1 = sec2 (θ) Sum and Difference Formulas 2 1 + cot (θ) = csc2 (θ) sin(α ± β) = sin(α) cos(β) ± cos(α) sin(β) Even/Odd Formulas cos(α ± β) = cos(α) cos(β) ∓ sin(α) sin(β) sin(−θ) = − sin(θ) csc(−θ) = − csc(θ) tan(α) ± tan(β) tan(α ± β) = cos(−θ) = cos(θ) sec(−θ) = sec(θ) 1 ∓ tan(α) tan(β) tan(−θ) = − tan(θ) cot(−θ) = − cot(θ) Product to Sum Formulas 1 Periodic Formulas sin(α) sin(β) = 2 [cos(α − β) − cos(α + β)] 1 If n is an integer then, cos(α) cos(β) = 2 [cos(α − β) + cos(α + β)] 1 sin(θ + 2πn) = sin(θ) csc(θ + 2πn) = csc(θ) sin(α) cos(β) = 2 [sin(α + β) + sin(α − β)] 1 cos(θ + 2πn) = cos(θ) sec(θ + 2πn) = sec(θ) cos(α) sin(β) = 2 [sin(α + β) − sin(α − β)] tan(θ + πn) = tan(θ) cot(θ + πn) = cot(θ) Sum to Product Formulas     α+β α−β Degrees to Radians Formulas sin(α) + sin(β) = 2 sin cos 2 2 If x is an angle in degrees and t is an angle in  α+β   α−β  radians then sin(α) − sin(β) = 2 cos sin 2 2 π t πx 180t = ⇒ t= and x=     180 x 180 π α+β α−β cos(α) + cos(β) = 2 cos cos 2 2 Double Angle Formulas  α+β   α−β  cos(α)−cos(β) = −2 sin sin sin(2θ) = 2 sin(θ) cos(θ) 2 2 cos(2θ) = cos2 (θ) − sin2 (θ) Cofunction Formulas π  π  2 = 2 cos (θ) − 1 sin − θ = cos(θ) cos − θ = sin(θ) 2π  π 2  = 1 − 2 sin2 (θ) csc − θ = sec(θ) sec − θ = csc(θ) 2 tan(θ)  π2   π2  tan(2θ) = tan − θ = cot(θ) cot − θ = tan(θ) 1 − tan2 (θ) 2 2 © Paul Dawkins - https://tutorial.math.lamar.edu  Trig Cheat Sheet For any ordered pair on the unit circle (x, y) : cos(θ) = x and sin(θ) = y Example     √ 5π 1 5π 3 cos = sin =− 3 2 3 2 © Paul Dawkins - https://tutorial.math.lamar.edu  Trig Cheat Sheet Inverse Trig Functions Definition Inverse Properties −1 cos cos−1 (x) = x cos−1 (cos(θ)) = θ
y = sin (x) is equivalent to x = sin(y) sin sin−1 (x) = x sin−1 (sin(θ)) = θ
y = cos−1 (x) is equivalent to x = cos(y) tan tan−1 (x) = x tan−1 (tan(θ)) = θ
y = tan−1 (x) is equivalent to x = tan(y) Domain and Range Alternate Notation Function Domain Range sin−1 (x) = arcsin(x) π π y = sin−1 (x) −1 ≤ x ≤ 1 − ≤y≤ cos−1 (x) = arccos(x) 2 2 y = cos−1 (x) −1 ≤ x ≤ 1 0≤y≤π tan−1 (x) = arctan(x) π π y = tan−1 (x) −∞ < x < ∞ −