Higher Engineering Mathematics, Sixth Edition

164 Higher Engineering Mathematics

tan

(

x+

π

4

)

=

tanx+tanπ 4

1 −tanxtanπ 4

from the formula fortan(A+B)

=

tanx+ 1

1 −(tanx)( 1 )

=

(

1 +tanx

1 −tanx

)

since tan

π

4

= 1

tan

(

x−

π

4

)

=

tanx−tanπ 4

1 +tanxtanπ 4

=

(

tanx− 1

1 +tanx

)

Hence tan

(

x+

π

4

)

tan

(

x−

π

4

)

=

(

1 +tanx

1 −tanx

)(

tanx− 1

1 +tanx

)

=

tanx− 1

1 −tanx

=

−( 1 −tanx)

1 −tanx

=− 1

Problem 4. If sinP= 0 .8142 and cosQ= 0. 4432

evaluate, correct to 3 decimal places:

(a) sin(P−Q),(b)cos(P+Q)and

(c) tan(P+Q), using the compound-angle

formulae.

Since sinP= 0 .8142 then

P=sin−^10. 8142 = 54. 51 ◦.

Thus cosP=cos54. 51 ◦= 0 .5806 and

tanP=tan54. 51 ◦= 1. 4025

Since cosQ= 0 .4432,Q=cos−^10. 4432 = 63. 69 ◦.

Thus sinQ=sin63. 69 ◦= 0 .8964 and

tanQ=tan63. 69 ◦= 2. 0225

(a) sin(P−Q)

=sinPcosQ−cosPsinQ

=( 0. 8142 )( 0. 4432 )−( 0. 5806 )( 0. 8964 )

= 0. 3609 − 0. 5204 =− 0. 160

(b) cos(P+Q)

=cosPcosQ−sinPsinQ

=( 0. 5806 )( 0. 4432 )−( 0. 8142 )( 0. 8964 )

= 0. 2573 − 0. 7298 =− 0. 473

(c) tan(P+Q)

=

tanP+tanQ

1 −tanPtanQ

=

( 1. 4025 )+( 2. 0225 )

1 −( 1. 4025 )( 2. 0225 )

=

3. 4250

− 1. 8366

=− 1. 865

Problem 5. Solve the equation

4sin(x− 20 ◦)=5cosx

for values ofxbetween 0◦and 90◦.

4sin(x− 20 ◦)=4[sinxcos20◦−cosxsin20◦],

from the formula forsin(A−B)

=4[sinx( 0. 9397 )−cosx( 0. 3420 )]

= 3 .7588sinx− 1 .3680cosx

Since 4sin(x− 20 ◦)=5cosxthen

3 .7588sinx− 1 .3680cosx=5cosx

Rearranging gives:

3 .7588sinx=5cosx+ 1 .3680cosx

= 6 .3680cosx

and

sinx

cosx

=

6. 3680

3. 7588

= 1. 6942

i.e. tanx= 1 .6942, andx=tan−^11. 6942 = 59. 449 ◦or

59 ◦ 27 ′

[Check: LHS=4sin( 59. 449 ◦− 20 ◦)

=4sin39. 449 ◦= 2. 542

RHS=5cosx=5cos59. 449 ◦= 2 .542]

Now try the following exercise

Exercise 72 Further problems on

compound angle formulae

1. Reduce the following to the sine of one
   angle:
   (a) sin37◦cos21◦+cos37◦sin21◦
   (b) sin7tcos3t−cos 7tsin3t
   [(a) sin58◦(b) sin4t]


2. Reduce the following to the cosine of one
   angle:
   (a) cos71◦cos33◦−sin71◦sin33◦

(b) cos

π

3

cos

π

4

+sin

π

3

sin

π

4

⎡

⎣

(a)cos104◦≡−cos76◦

(b)cos

π

12

⎤

⎦