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FUNdamental Trigonometric Identities


  • Your six Trigonometric Functions are...
    • sine (sin)
    • cosine (cos)
    • tangent (tant)
    • cosecant (csc)
    • secant (sec)
    • cotangent (cot)
  • It is helpful to associate...
    • sin & csc
    • cos & sec
    • tan & cot
  • We Use Trigonometric identites to study relationships between trigonometric functions.
  • Trigonometric identities are used to solve trigonometric proofs.
  • Six of the Fundamental trigonometric identities are...
Reciprocal Identities
sine
sine
cosine
cosine
tangent
tangent
cosecant
cosecant
secant
secant
cotangent
cotangent
Quotient Identities
tangent
tangent
cotangent
cotangent

Pythagorean Identities
Pythagorean Identity
Pythagorean Identity
Pythagorean Identity
Pythagorean Identity
Pythagorean Identity
Pythagorean Identity
  • Using the triangle below, we can prove that all of these identies make sense.
    • First let's give each letter a theoretical number to help us prove that the identities are true.
      • A = 3
      • B = 4
      • C = 5
    • For example...
      • Using sohcahtoa, we can determine that the sin of theta is equal to a / c.(Sin = Opp / Hyp) When we substitute in our theoretical values for the letters, we get the sin of theta equals 3 / 5. Our trigonometric identities above say that sin theta = 1 / csc theta. First, we must determine the value of csc theta. Csc is equal to c / a. (Csc = Hyp / Opp) If we substitue in our number we get csc theta equals 5 / 3. If we put everything together, the equation should be 3 / 5 = 1 / ( 5 / 3). 1 / (5 / 3) = 3 / 5 thus demonstrating that the fundamental trigonometric functions work.

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  • In some situations, we may have to make it so that both sides of an equation look the same.
    • For example...
      • cosx*secx = 1
        • This equation is true because cosine is the reciprocal of secent. So, no matter what x is when you multiply the reciprocals you find that the answer is always one.
      • (secx + tanx)(secx - tanx) = 1
        • This equation is a little more difficult. In order to do this, first we must use the distributive property to make the equation inf the form of ax^2+bx+c. Then we need to combine like terms. When you simplify it out, you find that one equals sec^2x - tan^2x which happens to be true because it is one of the identities listed above.
  • Rules
    • Theres only one rule that must be followed when solving trigonometric proofs. It is, once you start working on one side of the equation you're only allowed to change that side.
  • Practice.
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(Need Help? Answers are at the bottom of the page.)
  • Helpful Hint for solving Trigonometric Proofs
    • Never give up. Try! Try! Try! You can do it!







Answers:
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