Sum and difference identities

Sum and difference identities

Trig identities are formulas developed based on Pythagorean Theorem. These identities show us how and where to find the sine, cosine, and tangent of the sum and difference of two given angles.

Lessons

Download the Trigonometry identities chart here

Formulas:
sin(A+B) \sin (A + B)
sin(AB) \sin (A - B)
cos(A+B) \cos (A + B)
cos(AB) \cos (A - B)
tan(A+B) \tan (A + B)
tan(AB) \tan (A - B)
  • 1.
    Simplify expressions

    a)
    sin 24°cos 36° + cos 24°sin 36°

    b)
    tan2π5tan3π201+tan2π5tan3π20\frac{tan {2 \pi \over 5 } - tan {3 \pi \over 20}}{1 + \tan {2 \pi \over 5} \cdot \tan {3 \pi \over 20}}


  • 2.
    Prove Identities
    a)
    sin(AB)sinB+cos(AB)cosB=sinAsinBcosB \frac{\sin (A - B)}{\sin B} + \frac{\cos (A - B)}{\cos B} = \frac{\sin A}{\sin B \cos B}

    b)
    1+tanAtan(A+π4)=1tanA\frac{1 + \tan A}{\tan (A + {\pi \over 4})} = 1 - \tan A


  • 3.
    Without using a calculator, evaluate:
    a)
    sin 15°

    b)
    sec (-105°)

    c)
    tan19π12 tan {19\pi \over 12}


  • 4.
    Given sinA=45\sin A = -{4 \over5} and cosB=1213, \cos B = {12 \over 13} ,
    where πA3π2 \pi \leq A \leq {3 \pi \over 2} and 3π2B2π, {3 \pi \over 2} \leq B \leq 2\pi ,
    find the exact value of cos(A+B) \cos (A + B)

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