Principles of Mathematics 12: Explained!

Transcription

1 Principles of Mathematics : Eplained!

2 PART I MULTIPLICATION & DIVISION IDENTITLES Algebraic proofs of trigonometric identities In this lesson, we will look at various strategies for proving identities. Try to memorize all the different types, as it will make things much simpler for you when they are mied together. Type I: Identities with multiplication & division: In these proofs, you will need to convert everything to sine and cosine, then use fraction multiplication & division to simplify. Eample : Prove: sinsec tan Fraction Review Eample : Prove: tancos sin Multiplying Fractions: To multiply fractions, simply multiply the numerators together, and the denominators together. Canceling: When multiplying fractions you will frequently find factors that can be cancelled. You can cancel something on top with something identical on the bottom. Eample 3: Prove: csc sec cot Dividing Fractions: When dividing fractions, rewrite the top fraction, then multiply by the reciprocal of the bottom fraction. Principles of Mathematics : Eplained! 3

3 PART I MULTIPLICATION & DIVISION IDENTITLES For each of the following, write an algebraic proof. ) Prove: cot tan Identities will always have the following two properties: ) If you graph the left and right sides, you will obtain eactly the same graph. ) Prove: csc cos cot ) If you plug in the same angle for on both sides, you will obtain eactly the same number. 3) Prove: sin cos tan 4) Prove: sec cot cos tan Principles of Mathematics : Eplained! 4

4 PART I MULTIPLICATION & DIVISION IDENTITLES 5) Prove: tan sin csc cos 6) Prove: tan sin sec 7) Prove: cos sin cos cot sec csc 8) Prove: sec cot 9) Prove: sec csc tan 0) Prove: csc tan cos sin sec Principles of Mathematics : Eplained! 5

5 Answers ) TRIGONOMETRY LESSON PART I MULTIPLICATION & DIVISION IDENTITLES 5) 8) ) 6) 9) 3) 7) 4) 0) Principles of Mathematics : Eplained! 6

6 PART I I ADDITION & SUBTRACTION IDENTITLES We will now look at identities where adding & subtracting is involved. You will first convert everything to sine & cosine, then use a common denominator to simplify the fractions. Eample : + sincos Prove: sec + sin cos sec + sin sin + cos sin cos + cos cos sincos + cos cos + sincos cos Eample : cos + sin Prove cot + sec sincos cot + sec cot + cos cos + sin cos cos cos sin + sin cos cos sin cos sin + sincos sincos cos + sin sincos In multiplying & dividing fractions, you don t need a common denominator. In adding & subtracting fractions, you always need a common denominator. Eample : + cos sin - Multiply the first fraction by the denominator of the second fraction. Multiply the second fraction by the denominator of the first fraction. sin- cos + cos sin - sin - cos Now multiply the fractions together and simplify sin - cos + cos(sin -) cos(sin -) sin +cos - cos(sin -) Eample : + cos Multiply the second fraction by the denominator of the first. We don't need to do anything with the first fraction since we will now have the same denominator. cos + cos cos cos + cos cos + cos cos Fraction Review Principles of Mathematics : Eplained! 7

7 PART I I ADDITION & SUBTRACTION IDENTITLES Questions: For each of the following, write an algebraic proof. ) sin cos sec sin ) cos sin cos + sin sin + tan sin cos 3) sec 3 sin + cos + cot 4) cos sin csc 3 cos sin tan sin cos 5) cos sin sin csc sec 6) sec tan sin cos cos Principles of Mathematics : Eplained! 8

8 PART I I ADDITION & SUBTRACTION IDENTITLES 7) cos + sin cos + tan 8) cos cos + sin cot + sin sin 9) cos + sin + tan 0) cos + sin csc + sin Principles of Mathematics : Eplained! 9

9 PART I I ADDITION & SUBTRACTION IDENTITLES ) 5) ) 6) 9) 7) 3) 0) 8) 4) Principles of Mathematics : Eplained! 30

10 PART I I I THREE SPECIAL IDENTITLES The three special identities (below) are critical when simplifying trigonometric epressions. The basic idea is that when you come across one of these special identities during simplification, you should immediately replace it with whatever that identity is equal to. Watch out for manipulations of these identities, as you will be epected to recognize them as well. Eample : Prove: - cos tan cos - cos tan sin + cos sin - cos -sin cos - cos - sin -cos sin - Eample : Use Special Identity Here. Prove: sin - csc -cotcos tan + sec tan sec - -tan - sec cot + csc cot csc - -cot - csc Use Special Identity Here. Spread out the two cosines so you can form cot. Principles of Mathematics : Eplained! 3

11 PART I I I THREE SPECIAL IDENTITLES Questions: Use the special identities to do each of the following proofs. ) sec tan sin cos ) cos + tan sin sec 3) tan + cot sec csc 4) + tan sec 5) sec cos tan sin 6) sin + cot cos csc Principles of Mathematics : Eplained! 3

12 PART I I I THREE SPECIAL IDENTITLES 7) sec sin sec 8) csc cot 9) csc sin cos cot 0) sec tan Principles of Mathematics : Eplained! 33

13 ) TRIGONOMETRY LESSON PART I I I THREE SPECIAL IDENTITLES 5) 8) ) 6) 9) 3) 7) 0) 4) Principles of Mathematics : Eplained! 34

14 PART I V COMPOUND FRACTIONS & SPECIAL IDENTITLES Net we ll look at compound fractions. Everything here is basically the same as in the previous section, just be sure to follow your rules for dividing fractions & watch for special identities. Eample : Prove: tan + sec tan csc - Without using the special identities csc tan + sin + cos sin With the special identities tan + csc sec cot sec tan sin cos + cos cos sin sin sin sin cos + cos cos sin sin sin sin + cos cos sin sin cos cos sin sin cos cos sec tan Principles of Mathematics : Eplained! 35

15 PART I V COMPOUND FRACTIONS & SPECIAL IDENTITLES Questions: Prove each of the following: ) sec sin sin + tan ) tan cot + tan cos + 3) cos csc sin + cos cot 4) sin cos sin sec sec + csc 5) tan sin cos + cos 6) cot tan sin sin tan + sin Principles of Mathematics : Eplained! 36

16 PART I V COMPOUND FRACTIONS & SPECIAL IDENTITLES + tan + 7) c ot tan 8) sec + csc 9) + tan tan + cot cos cos 0) + cot sec sec + ) tan sin + tan sin+ cos sin sin ) + tan sin + sin Principles of Mathematics : Eplained! 37

17 ) TRIGONOMETRY LESSON PART I V COMPOUND FRACTIONS & SPECIAL IDENTITLES ) 3) 5) 6) 4) Principles of Mathematics : Eplained! 38

18 PART I V COMPOUND FRACTIONS & SPECIAL IDENTITLES 7) 8) + tan 9) + cot sec csc cos sin sin cos sin cos tan + sec csc cos + sin 0) ) ) Principles of Mathematics : Eplained! 39

19 PART V OTHER PROOFS Difference of Squares: See the review on the side of the page, then study the eample. Eample : Prove: 4 sec (sec )(sec + ) tan (sec + ) sin + cos cos sin cos + cos cos cos sin + cos cos cos sin + sin cos 4 cos 4 sec - sin + sin cos 4 cos Difference of Squares You can recognize a difference of squares by the following: It is a binomial, with a minus in the middle. (Watch out for binomials with a plus, this will not be a difference of squares.) The first & last terms are perfect squares. Rearranging an epression to make an identity: Eample : Prove that ( cot -) csc - cot (cot ) (cot )(cot ) cot cot + cot + cot *Rearranging to allow use of special identity csc cot Factoring out a negative to make an identity: Eample 3: Prove: 4- sec +4-tan 4 sec + 4 (sec ) 4 tan Eample : Factor 4sin - 9cos 8 8 Eample : Factor -y (Watch for multiple difference of squares) 8 8 y ( y )( + y ) 4 4 ( y )( + y )( + y ) y + y + y + y 4 4 ( )( )( )( ) Principles of Mathematics : Eplained! 40

20 PART V OTHER PROOFS ) 3tan 3sin cos + tan ) + cot sin 3) sec cos sin tan 4) (sin + cos ) + (sin cos ) 5) (+ sin ) + cos (+ sin ) 6) 4 4 sin cos sin Principles of Mathematics : Eplained! 4

21 PART V OTHER PROOFS 7) (tan ) sec tan 8) ( sec )( sin ) sin 9) 3 cos csc csc 3 sin 0) 4 csc + 4 sin cos ( sin ) ) tan sin cos cot sin cos Principles of Mathematics : Eplained! 4

22 PART V OTHER PROOFS ) ) 3) 4) 5) 6) 7) 8) 0) 9) ) Principles of Mathematics : Eplained! 43

23 PART V I CONJUGATES Sometimes you will get identities that can t be broken down any further. In these cases, you can multiply numerator & denominator by the conjugate. This will convert the fraction into something that will give you identities to work with. The conjugate is obtained by taking a binomial from the original epression and changing the sign in the middle. + cos Eample : Prove sin sin - cos + cos has the conjugate: - cos sin + cos sin + cos cos sin cos cos sin ( cos ) sin sin ( cos ) sin cos Prove each of the following identities: ) cos + sin + sin ) 3) sin cos sin cos cos sin sin cos 4) sin + cos cos + sin Principles of Mathematics : Eplained! 44

24 PART V I CONJUGATES ) ) 3) 4) Principles of Mathematics : Eplained! 45