7 3: Trigonometric Identities Fundamental Identities! Sum and Difference Identities

Transcription

1 7 3: Trigonometric Identities Fundamental Identities! Sum and Difference Identities sin 2 θ + cos 2 θ! sin( θ) sinθ! cos(a + B) cos A cosb sin A sinb tan 2 θ + sec 2 θ! cos( θ) cosθ! cos(a B) cos A cosb + sin A sinb + cot 2 θ csc 2 θ! tan( θ) tan θ! sin(a + B) sin A cosb + sinb cos A!! sin(a B) sin A cosb sinb cos A Double Angle Identities! Half Angle Identities sin(2a) 2sin A cos A! cos A 2 cos(2a) cos 2 A sin 2 A cos(2a) 2 sin 2 A! sin A 2 cos(2a) 2cos 2 A! tan(2a) 2tan A tan 2 A! tan A 2 + cos A 2 cos A 2 cos A + cos A tan 2 θ + sec 2 θ really means tan 2 θ + sec 2 θ When you see the equal sign in the identity tan 2 θ + sec 2 θ you may state that it means that if you plug any radian value into the variable θ the equation will be true. That is true if you are considering the identity as an equation. Equations allow the movement of the terms from one side of he equal sign to the other if you consider the identity as an equation you could also subtract from both sides to get a new identity tan 2 θ sec 2 θ. Neither of the these two interpretations are what we mean when we state a Trigonometric identity. You could say that there are two expressions, one on the left hand side of the equals sign (LHS) and the other on the right hand side (RHS) and there expressions are equivalent expressions. In other words you could use either expression in a problem if you wanted to. We say that you can substitute one expression for the other. If you see the expression sec 2 θ you could substitute tan 2 θ in it place. In the case of identities the equal sign really means the two statements are separate but equivalent. A better way to state the identity would be to use a in place of the equal sign,tan 2 θ + sec 2 θ but textbooks have chosen to use the equal sign and expect you to understand what the identity really meas. if In future courses, especially calculus, you will have a complex statements with several expressions that are terms or factors in the expression. You will simplify that expression by taking complex terms or factors and replacing them with simpler equivalent expressions that are found in the list of trig identities. Math Trig Identity Lecture! Page of! 205 Eitel

2 Example Simplify tan x tan x is an idenity so we can replace tan x with to get cancel the comon factor to get sec x is an idenity so we can replace sec x with sec x to get so we know that tan x sec x You may ask why didn't we stop when we got the expression. How did we know to continue to get simplifying to get sec x. That's a good question. To make it easier to know when to stop we will make it a lot easier in this course. We will tell you what the answer will be when you do all the correct steps. We do this by writing the expression to be simplified and then setting it equal to what the reduced answer will be. The process of showing the steps you take to get the answer is called proving the identity. If I know the answer in advance what does proving an identity ask me to do You will be given a complex expression on one side of the equal sign and told what the simplified expression will be on the other side. The process of showing all the steps you take to get the answer in a clear and detailed manner is called proving the identity. In calculus it will be a harder task to reduce an expression. You will not be given the answer as a help o guide you in the simplification process. You will have to know what the good expressions we use in the course are and reduce your expression to one of them. For this reason we put the expression to be reduced on the left hand side and the reduced expression on the right hand side. Math Trig Identity Lecture! Page 2 of! 205 Eitel

3 What are the steps I am allowed to use to simplify the left side so that it is the same expression as the right side of the equation.. Use the identity list. FInd a complex expression and substitute an alternate expression using a trigonometric Identity found on he identity lest. 2. Convert everything to sine and cosine. This reduces the number of identities you will need to consider in the next steps to ones with sine or cosine in them. 3. Factor an expression. Look for a common factor. Look for the difference of squares or a trinomial that can be factored. The factored expression may have common factors that will cancel out or a factor may be an expression that can be replaced with a simpler expression found by using the identity list. 4. Distribute or FOIL. This may create a second degree expression that is on the identities list. All or party of the new expression may be an expression that can be replaced with a simpler expression found by using the identity list. 4. Use the identity list. FInd a complex expression and substitute a alternate expression using a trigonometric Identity the identity lest. 5. Multiply the numerator and denominator of a fraction by a well chosen term to create an identity. Use the identity to simplify the expression. 6. Add two fractions to get one fraction. If you have an expression with two separate fractions, Add the separate fractions by getting a common numerator and denominator. Do this by multiplying each term by a well chosen term. Add the numerators to create a useful identity or an expression that factors. Use the identity to simplify the expression or factor and cancel to reduce the expression. 7. Make one fraction into two fractions. If you have a fraction that has a polynomial in the numerator and a monomial in the denominator break the fraction into two separate fractions each with a common denominator and reduce each separate fraction. 8. Break up a polynomial into more terms so that a part of of the terms in the sum form an identity. Use the identity to simplify the expression. Math Trig Identity Lecture! Page 3 of! 205 Eitel

4 Use the identities and steps listed above to prove each Identity. Example Convert everything to sine and cosine. You must show all steps and work so that it is clear what was done on each step. I did it in my head is not an approved step. cot x csc x convert to sine and cosine invert and multiply cancel Example 2 Break up a polynomial into more terms so that a part of of the terms in the sum form an identity. Use the identity to simplify the expression. 2sin 2 x + 3cos 2 x 2 + cos 2 x break up the polynominal into more terms 2sin 2 x + 2cos 2 x +cos 2 x 2 + cos 2 x factor out a 2 in the first two terms 2(sin 2 x + cos 2 x) +cos 2 x 2 + cos 2 x use sin 2 x + cos 2 x 2() + cos 2 x 2 + cos 2 x 2 + cos 2 x 2 + cos 2 x Math Trig Identity Lecture! Page 4 of! 205 Eitel

5 Example 3 Add separate fractions by getting a common numerator and denominator. Do this by multiplying each term by a well chosen term. Add the numerators to create a useful identity or an expression that factors. Use the identity to simplify the expression or factor and cancel to reduce the expression. The most common way to lose points on the test is to fail to show all steps in a clear manner. + + cot x 2 (state cot x in terms of and ) (multiply each fractions by a "well chosen " to get a CD ) distriubte (combine the numerators) (cancel) 2 2 Math Trig Identity Lecture! Page 5 of! 205 Eitel

6 Example 4 Factor expressions to get common factors that cancel out. sin 2 x + tan x tan x tan x ( factor each expression ) ( +) ( ) tan x ( ) + tan x (cancel) + tan x + tan x Math Trig Identity Lecture! Page 6 of! 205 Eitel

7 Example 5 Multiply the numerator and denominator by a well chosen term to create an identity. Use the identity to simplify the expression. The most common way to lose points on the test is to fail to show all steps in a clear manner. + multiply the left side of the equation by distrubite and FOIL + ( ) sin 2 x use sin 2 x cos 2 x ( ) cos 2 x cancel Math Trig Identity Lecture! Page 7 of! 205 Eitel

8 Example 6 Add separate fractions by getting a common numerator and denominator. Do this by multiplying each term by a well chosen term. Add the numerators to create a useful identity or an expression that factors. Use the identity to simplify the expression or factor and cancel to reduce the expression. The most common way to lose points on the test is to fail to show all steps in a clear manner csc x (multiply each fraction by a "well chosen " to get a CD ) csc x muiltiply or FOIL the numerators sin 2 x ( + ) cos2 x ( + ) 2csc x (combine the numerators) sin 2 x cos 2 x ( + ) 2csc x use sin 2 + cos 2 x ++ 2 ( + ) 2csc x ( + ) 2csc x (factor and cancel) 2( + ) ( + ) 2csc x 2 2csc x use csc x 2csc x 2csc x Math Trig Identity Lecture! Page 8 of! 205 Eitel

9 Example 7 Add separate fractions by getting a common numerator and denominator. Do this by multiplying each term by a well chosen term. Add the numerators to create a useful identity or an expression that factors. Use the identity to simplify the expression or factor and cancel to reduce the expression. The most common way to lose points on the test is to fail to show all steps in a clear manner. tan x + cot x sec x csc x (change to and ) + + (invert and mutiply) + multiply each term in the right bracket by a "well chosen " to get a CD + muiltiply the numerators sin 2 x + cos 2 x (combine the numerators) sin 2 x + cos 2 x use sin 2 x + cos 2 x (cancel) Math Trig Identity Lecture! Page 9 of! 205 Eitel

10 Example 8 Use an identity to simplify the expression. tan 2 x ( + cot 2 x) sin 2 x (rewrite cot 2 x in terms of tan 2 x ) tan 2 x + tan 2 x sin 2 x distribute tan 2 x + sin 2 x use tan 2 x + sec 2 x sec 2 x sin 2 x use sec 2 x cos 2 x use sec 2 x cos 2 x sec 2 x sec 2 x Math Trig Identity Lecture! Page 0 of! 205 Eitel

11 Example 9 Write as sin (x) and cos (x) and break a fraction into 2 separate fractions with a CD tan x cot x sec2 x csc 2 x (rewrite cot x and tan x in terms of sinx and ) sec 2 x csc 2 x Multiply by a "well chosen ") sec 2 x csc 2 x sin 2 x cos 2 x sin 2 x cos 2 x sec 2 x csc 2 x the right side has 2 seperate terms so we break the fraction on the left into 2 seperate fractions each of which has the common denominator sin 2 x sin 2 x cos 2 x cos 2 x sin 2 x cos 2 x sec2 x csc 2 x reduce each fraction on the left side cos 2 x sin 2 x sec2 x csc 2 x use cos 2 x sec2 x and sin 2 x csc2 x sec 2 x csc 2 x sec 2 x csc 2 x Math Trig Identity Lecture! Page of! 205 Eitel