Chapter 5 Trigonometric Ratios of Compound Angles Exercise Ex. 7.1

Question 34

I f space tan space left parenthesis A plus B right parenthesis equals p comma space tan space left parenthesis A minus B right parenthesis equals q comma space t h e n space s h o w space t h a t space tan space 2 A equals fraction numerator p plus q over denominator 1 minus p q end fraction

Solution 34

R H S comma fraction numerator p plus q over denominator 1 minus p q end fraction equals fraction numerator tan left parenthesis A plus B right parenthesis plus tan thin space left parenthesis A minus B right parenthesis over denominator 1 minus tan left parenthesis A plus B right parenthesis. tan left parenthesis A minus B right parenthesis end fraction equals fraction numerator begin display style fraction numerator tan space A plus tan thin space B over denominator 1 minus tan space A. tan space B end fraction end style plus begin display style fraction numerator tan space A minus tan space B over denominator 1 plus tan space A. tan space B end fraction end style over denominator 1 minus fraction numerator tan space A plus tan thin space B over denominator 1 minus tan space A. tan space B end fraction. fraction numerator tan space A minus tan space B over denominator 1 plus tan space A. tan space B end fraction end fraction equals fraction numerator begin display style fraction numerator left parenthesis tan thin space A plus tan space B right parenthesis left parenthesis 1 plus tan space A. tan space B right parenthesis plus left parenthesis tan space A minus tan space B right parenthesis left parenthesis 1 minus tan space A. space tan thin space thin space B right parenthesis over denominator left parenthesis 1 minus tan space A. tan space B right parenthesis left parenthesis 1 plus tan space A. tan space B right parenthesis end fraction end style over denominator begin display style fraction numerator left parenthesis 1 minus tan space A. tan space B right parenthesis left parenthesis 1 plus tan space A. tan space B right parenthesis minus left parenthesis tan space A plus tan space B right parenthesis. left parenthesis tan space A minus tan space B right parenthesis over denominator left parenthesis 1 minus tan space A. space tan space B right parenthesis left parenthesis 1 plus tan space A. tan space B right parenthesis end fraction end style end fraction equals fraction numerator tan space A plus tan space B plus tan squared A. tan space B plus tan space A. tan squared B plus tan space A minus tan thin space B minus tan thin space squared A. tan space B plus tan space A. tan squared B over denominator 1 minus tan squared space A. tan squared space B minus tan squared space A plus tan squared space B end fraction equals fraction numerator 2 space tan space A plus 2 space tan space A. tan squared B over denominator left parenthesis 1 minus tan squared A right parenthesis left parenthesis 1 plus tan squared B right parenthesis end fraction equals fraction numerator 2 space tan space A left parenthesis 1 plus tan squared space B right parenthesis over denominator left parenthesis 1 minus tan squared space A right parenthesis left parenthesis 1 plus tan squared space B right parenthesis end fraction equals fraction numerator 2 space tan space A over denominator 1 minus tan squared space A end fraction equals tan space 2 A equals L H S H e n c e space P r o v e d

Question 19

I f space fraction numerator sin space left parenthesis x plus y right parenthesis over denominator sin space left parenthesis x minus y right parenthesis end fraction equals fraction numerator a plus b over denominator a minus b end fraction comma space s h o w space t h a t space fraction numerator tan space x over denominator tan space y end fraction equals a over b

Solution 19

fraction numerator sin space x. cos space y plus sin space y. cos space x over denominator sin space x. cos space y minus sin space y. cos space x end fraction equals fraction numerator a plus b over denominator a minus b end fraction rightwards double arrow fraction numerator sin space x. cos space y plus sin space y. cos space x plus sin space x. cos space y minus sin space y. cos space x over denominator sin space x. cos space y plus sin space y. cos space x minus sin space x. cos space y plus sin space y. cos space x end fraction equals fraction numerator a plus b plus a minus b over denominator a plus b minus a plus b end fraction left square bracket U sin g space C o m p o n e n d o space a n d space D i v i d e n d o right square bracket rightwards double arrow fraction numerator 2 sin space x. cos space y over denominator 2 sin space y. cos space x end fraction equals fraction numerator 2 a over denominator 2 b end fraction rightwards double arrow fraction numerator tan space x over denominator tan space y end fraction equals a over b H e n c e space P r o v e d

Question 32

I f space a n g l e space theta space i s space d i v i d e d space i n t o space t w o space p a r t s space s u c h space t h a t space t h e space tan g e n t s space o f space o n e space p a r t space i s space lambda space p a r t t i m e s space t h e space tan g e n t space o f space t h e space o t h e r comma space a n d space Ï• space i s space t h e i r space d i f f e r e n c e space t h e n space s h o w space t h a t space space sin space theta equals fraction numerator lambda plus 1 over denominator lambda minus 1 end fraction sin space Ï•.

Solution 32

Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Question 33

I f space tan space theta equals fraction numerator sin space alpha minus cos space alpha over denominator sin space alpha plus cos space alpha end fraction comma space t h e n space s h o w space t h a t space sin space alpha plus cos space alpha equals square root of 2 cos space theta

Solution 33

tan space theta equals fraction numerator sin space alpha minus cos space alpha over denominator sin space alpha plus cos space alpha end fraction rightwards double arrow tan space theta equals space fraction numerator tan space alpha minus 1 over denominator tan space alpha plus 1 end fraction left square bracket D i v i d i n g space b o t h space N u m e r a t o r space a n d space D e n o m i n a t o r space b y space cos space alpha right square bracket rightwards double arrow tan space theta equals fraction numerator tan space alpha minus tan space begin display style pi over 4 end style over denominator 1 plus tan space begin display style pi over 4 end style. tan space alpha end fraction rightwards double arrow tan space theta equals tan space open parentheses alpha minus pi over 4 close parentheses rightwards double arrow theta equals alpha minus pi over 4 space space space left square bracket R e m o v i n g space tan space f r o m space b o t h space s i d e s right square bracket rightwards double arrow cos space theta equals cos open parentheses alpha minus pi over 4 close parentheses space left square bracket T a k i n g space cos space o n space b o t h space s i d e s right square bracket rightwards double arrow cos space theta equals cos space alpha. cos space pi over 4 plus sin space alpha. sin space pi over 4 rightwards double arrow cos space theta equals cos space alpha. fraction numerator 1 over denominator square root of 2 end fraction plus sin space alpha. fraction numerator 1 over denominator square root of 2 end fraction rightwards double arrow cos space theta equals fraction numerator cos space alpha plus sin space alpha over denominator square root of 2 end fraction rightwards double arrow square root of 2 space cos space theta equals sin space alpha plus cos space alpha H e n c e space P r o v e d

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 2(a)

Solution 2(a)

Question 2(b)

Solution 2(b)

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 7

Solution 7

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 8(iii)

Solution 8(iii)

Question 9

Solution 9

Question 10

Solution 10

Question 11(i)

Solution 11(i)

Question 11(ii)

Solution 11(ii)

Question 11(iii)

Solution 11(iii)

Question 12(i)

Prove that:

Sin (60o – q) cos (30o + q) + cos (60o – q) sin (30o + q) = 1Solution 12(i)

Question 12(iii)

Prove that:

Solution 12(iii)

Question 12(ii)

Prove that:

Solution 12(ii)

Question 13

Solution 13

Question 14(i)

Solution 14(i)

Question 14(ii)

Solution 14(ii)

Question 15(i)

Solution 15(i)

Question 15(ii)

Solution 15(ii)

Question 16(i)

Solution 16(i)

Question 16(ii)

Solution 16(ii)

Question 16(iii)

Solution 16(iii)

Question 16(iv)

Solution 16(iv)

Question 16(v)

Solution 16(v)

Question 16(vi)

Solution 16(vi)

Question 17(i)

Solution 17(i)

Question 17(ii)

Solution 17(ii)

Question 17(iii)

Solution 17(iii)

Question 17(iv)

Solution 17(iv)

Question 18

Solution 18

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28(i)

Solution 28(i)

Question 28(ii)

Solution 28(ii)

Question 29(i)

Solution 29(i)

Question 29(ii)

Solution 29(ii)

Question 29(iii)

Solution 29(iii)

Question 30

Solution 30

Question 31

Solution 31

Chapter 17 Trigonometric Ratios of Compound Angles Exercise Ex. 7.2

Question 4

Solution 4

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 3

Solution 3

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