Section 6.1 Verifying Trigonometric Identities

Guidelines for Verifying Trigonometric Identities

Strategy: Try rewriting the more complicating side (left side). Verify the identity:   Strategy: Try rewriting the more complicating side (left side).   Apply a reciprocal identity     Cancel out numerator with denominator   Reciprocal identity

Strategy: Try rewriting the more complicating side (left side).   Verify the identity: Strategy: Try rewriting the more complicating side (left side). Apply a reciprocal identity     Cancel out numerator with denominator     Reciprocal identity

Apply a quotient identity Verify the identity:   Strategy: Try rewriting the more complicating side, the left side, using identities that contain sine and cosine. Apply a quotient identity   Multiply  

Continued from previous slide… Verify the identity:   Continued from previous slide…   Get a common denominator, which is     Multiply  

Continued from previous slide… Verify the identity:   Continued from previous slide…   Add the numerators   Do you recognize an identity? Pythagorean Identity  

Continued from previous slide… Verify the identity:   Continued from previous slide…   Do you recognize a identity?  

Apply a quotient identity Verify the identity:   Strategy: Try rewriting the more complicating side, the left side, using identities that contain sine and cosine. Apply a quotient identity   Multiply  

Continued from previous slide… Verify the identity:   Continued from previous slide…   Get a common denominator, which is   Multiply  

Continued from previous slide… Verify the identity:   Continued from previous slide…   Add the numerators   Do you recognize an identity?  

Continued from previous slide… Verify the identity:   Continued from previous slide…   Do you recognize a identity?  

Strategy: start with the more complicated side, the left side.   Strategy: start with the more complicated side, the left side. Is there a greatest common factor?     Is there a variation of an identity here?   Multiply  

Strategy: start with the more complicated side, the left side.   Strategy: start with the more complicated side, the left side. Is there a greatest common factor?     Is there a variation of an identity here?   Multiply  

Strategy: separate the term on the left into two terms   Strategy: separate the term on the left into two terms   Are there any identities here? Reciprocal identity Quotient Identity  

Strategy: separate the term on the left into two terms   Strategy: separate the term on the left into two terms   Are there any identities here? Reciprocal identity Quotient Identity  

Strategy: start with the more complicated side, the left side.   Strategy: start with the more complicated side, the left side. What must we do to add the two fractions on the left? Find the least common denominator, which is…    

       

      Do you see an identity? Pythagorean Identity  

Can we find a greatest common factor in the numerator?       Can we find a greatest common factor in the numerator?  

    Can we cancel anything?        

  Strategy: Both sides are equally complicated. Everything is expressed in sines and cosines. Let’s work on the left side to make it look like the right.     Do you see an identity?    

What do we know about sin(-x)?     What do we know about sin(-x)?    

    Do you see an identity? Quotient identity  

We need to find the least common denominator in the numerator.       We need to find the least common denominator in the numerator.  

         

How about a common factor in the numerator?   How about a common factor in the numerator?    

           

Verifying Trig Functions Review Learn the fundamental identities. Try to rewrite the more complicated side of the equation so that it is identical to the simpler side. It is often helpful to express all functions in terms of sine and cosine and then simplify the result. Usually, any factoring or indicated algebraic operations should be performed. For example, As you select substitutions, keep in mind the side you are not changing, because it represents your goal. If an expression contains 1 + sin x, multiplying both numerator and denominator by 1 – sin x would give 1 – sin² x, which could be replaced with cos² x.