Trigonometric Identities The Six Trigonometric Functions sin = = ℎ cos = = ℎ tan = = Pythagorean Identities sin + cos = 1 Reciprocal Identities 1 csc 1 cos = sec 1 tan = cot ℎ = ℎ sec = = cot = = tan = Sum or Difference of Two Angles sin 2 = 2 sin cos tan 2 = 2 tan 1 − tan Half-Angle Formulas sin tan 1 − cos = ±2 2 = csc − ./ 2 1 − cos tan = ±1 + cos 2 sin tan = 2 1 + cos 1 − cos tan = 2 sin Complex Numbers 2 30 = cos + 4 sin 2 31 = cos + 4 sin 2 $31 = 4 cos − 4 sin cos cos = sin sin = sin cos = cos sin = cos 2 = cos − sin cos 2 = 2 cos − 1 cos 2 = 1 −2 sin cos cot sin cos 1 + cos = ±2 2 = csc + ./ 2 1 + cos tan = ±2 1 − cos sin tan = 2 1 − cos 1 + cos tan = 2 sin 5 cos = 62 31 + 2 $31 7 5 sin = 362 31 − 2 $31 7 31 $31 2 −2 tan = 42 31 + 2 $31 sin ± sin = 2 sin + cos + cos = cos − cos = #±% #∓% , cos + , #&% #$% 2 cos + , cos + , #&% #$% −2 sin + , sin + , Co-Function Identities sin + $1, = cos csc + $1, = sec tan + $1, = cot cot + $1, = tan 0 0 cos + $1, = sin sec + $1, = csc 0 0 0 0 Even-Odd Identities sin− = −sin cos− = cos tan− = −tan Expansions sin = − 89 :! tan = − 89 cos = 1 − 8 ! : + 8< =! + 8 < + = © 2010 ANOVA Learning Services !"#$%& !"#&% Sum to Product Formulas Law of Cosines sin H sin J = I . cos sin !"#$%$ !"#&% "'(#&%&"'(#$% "'(#&%$"'(#$% Law of Sines sin G cot = Product to Sum Formulas sin ± = sin cos ± cos sin cos ± = cos cos ∓ sin sin tan ± tan tan ± = 1 ∓ tan tan Double Angle Formulas csc = Quotient Identities sec = 1 + tan csc = 1 + cot 1 sin 1 sec = cos 1 cot = tan sin = csc = . = 8C D! − 5= − csc− = −csc sec− = sec cot− = −cot 8> ?! 8E A! + +⋯ +⋯ 5?8 > :5= 8 ?8 9 sec = 1 + 5 8 8 + + ⋯ cot = 8 − : − + I − 2 I cos J www.anova-learning.com 5 :58 < csc = 8 + A + :AB + 5=5B + ⋯ or =8 C D 58 9 D= + A58 E ?B 8 < +⋯ − FD= − ⋯ J = cos$5 + K &L $M , KL Trigonometric Identities & Formulas NOP Q In terms of NOP Q = RSN Q UVP Q RSU Q RSN RSN Q UVP Q sin T1 * cos T1 * sin cos RNR Q NXR Q Each trigonometric function in terms of the others sin √1 * sin 1 sin 1 * sin √1 √1 * sin sin cos √1 * cos 1 cos √1 * 1 cos cos √1 * cos RNR Q tan √1 tan 1 √1 tan 1 csc tan √csc * 1 csc 1 √csc * 1 √1 tan tan T1 tan NXR Q 1 tan csc √csc sec √sec * 1 sec *1 Tcsc * 1 Tsec * 1 csc √sec * 1 sec 1 sec RSU Q 1 √1 cot cot √1 cot 1 cot T1 cot 1 √1 cot cot cot √sec * 1 Geometric Constructions of Trig Functions Unit Circle A D E C F cccc Circle with radius Jb cccc cccc bd – sine be – cosine cccc cccc JG – cosecant JH – secant G B ccccc – tangent bH cccc – cotangent bG Trigonometric Val Values for Common Angles Degrees 0° 30° 45° 60° 90° 120° 135° 150° 180° 210° 225° 240° 270° 300° 315° 330° 360° Radians 0[ [ ⁄6 [ ⁄4 [ ⁄3 [ ⁄2 2[⁄3 3[⁄4 5[⁄6 [ 7[⁄6 5[⁄4 4[⁄3 [ ⁄2 5[⁄3 7[⁄4 11[⁄6 2[ © 2010 ANOVA Learning Services sin 0 ⁄2 1⁄ √2 2⁄2 √3 3⁄2 1 √3 3⁄2 √2 2⁄2 ⁄2 1⁄ 0 *1 1⁄2 *√ √2⁄2 *√ √3⁄2 --1 *√ √3⁄2 *√ √2⁄2 *1 1⁄2 0 cos 1 √3⁄2 √2⁄2 1⁄2 0 *1⁄2 *√2⁄2 *√3⁄2 -1 *√3⁄2 *√2⁄2 *1⁄2 0 1⁄2 √2⁄2 √3⁄2 1 tan 0 √3⁄3 1 √3 undefined *√3 -1 *√3⁄3 0 √3⁄3 1 √3 undefined *√3 -1 *√3⁄3 0 www.anova-learning.com csc undefined 2 √2 2√3⁄3 0 2√3⁄3 √2 2 undefined -2 *√2 *2√3⁄3 0 *2√3⁄3 *√2 -2 undefined sec 1 2√ √3⁄3 √2 √ 2 undefined -2 *√2 * *2√ √3⁄3 -1 *2√ √3⁄3 *√2 * -2 undefined 2 √2 √ 2√ √3⁄3 1 cot undefined √3 1 √3⁄3 1 *√3⁄3 -1 *√3 undefined √3 1 √3⁄3 -1 *√3⁄3 -1 *√3 undefined Trigonometric Trigonomet Identities & Formulas