Question

X 8 17 Find the exact value of x. X=

Answer 1

Answer:

621

629

Step-by-step explanation:

We know that

sin

(

x

+

y

)

=

sin

x

cos

y

+

sin

y

cos

x

If

cos

x

=

8

17

and

sin

y

=

12

37

We can use,

cos

2

x

+

sin

2

x

=

1

and

cos

2

y

+

sin

2

y

=

1

To calculate

sin

x

and

cos

y

sin

2

x

=

1

−

cos

2

x

=

1

−

(

8

17

)

2

=

225

17

2

sin

x

=

15

17

cos

2

y

=

1

−

sin

2

y

=

1

−

(

12

37

)

2

=

1225

37

2

cos

y

=

35

37

so,

sin

(

x

+

y

)

=

15

17

⋅

35

37

+

12

37

⋅

8

17

=

621

629

Answer link

Shwetank Mauria

Nov 22, 2016

sin

(

x

+

y

)

=

621

629

or

−

429

629

depending on the quadrant in which sine and cosine lie.

Explanation:

Before we commence further, it may be mentioned that as

cos

x

=

8

17

,

x

is in

Q

1

or

Q

4

i.e.

sin

x

could be positive or negative and as

sin

y

=

12

37

,

y

is in

Q

1

or

Q

2

i.e.

cos

y

could be positive or negative.

Hence four combinations for

(

x

+

y

)

are there and for

sin

(

x

+

y

)

=

sin

x

cos

y

+

cos

x

sin

y

, there are four possibilities.

Now as

cos

x

=

8

17

,

sin

x

=

√

1

−

(

8

17

)

2

=

√

1

−

64

289

=

√

225

289

=

±

15

17

and

as

sin

y

=

12

37

,

cos

y

=

√

1

−

(

12

37

)

2

=

√

1

−

144

1369

=

√

1225

1369

=

±

35

37

Hence,

(1) when

x

and

y

are in

Q

1

sin

(

x

+

y

)

=

15

17

×

35

37

+

8

17

×

12

37

=

525

+

96

629

=

621

629

(2) when

x

is in

Q

1

and

y

is in

Q

2

sin

(

x

+

y

)

=

15

17

×

−

35

37

+

8

17

×

12

37

=

−

525

+

96

629

=

−

429

629

(3) when

x

is in

Q

4

and

y

is in

Q

2

sin

(

x

+

y

)

=

−

15

17

×

−

35

37

+

8

17

×

12

37

=

525

+

96

629

=

621

629

(4) when

x

is in

Q

4

and

y

is in

Q

1

sin

(

x

+

y

)

=

−

15

17

×

35

37

+

8

17

×

12

37

=

−

525

+

96

629

=

−

429

629

Hence,

sin

(

x

+

y

)

=

621

629

or

−

429

629