Question

If A is a quadrant 1 angle, such that sinA=2/5 and B is a quadrant 3 angle, such that tanB=1, find sin(A+B) and cos(A+B) and the quadrant in with A+B lies in. Find sin2A.

*Please Show Work*

Answer 1

Answers: sin(A+B)=
`(-2√(2)-√(42))/(10)`, cos(A+B)=
`(2√(2)-√(42))/(10)`, A+B=Quadrant 3, sin(2A)=
`(4√(21))/(25)`

NOTES:

sin A =
`(2)/(5)` ⇒ cos A =
`(√(21))/(5)`

since tan B = 1 and is in Quadrant 3, then

sin B =
`-(√(2))/(2)` and cos B =
`-(√(2))/(2)`

SIN (A + B):

sin (A + B) = (sin A * cos B) + (cos A * sin B)

= (
`(2)/(5)`)(
`-(√(2))/(2)`) + (
`(√(21))/(5)`)(
`-(√(2))/(2)`)

=
`-(2√(2))/(10)` +
`-(√(42))/(10)`

=
`(-2√(2)-√(42))/(10)`

COS (A + B):

cos (A + B) = (cos A * cos B) - (sin A * sin B)

= (
`(√(21))/(5)`)(
`-(√(2))/(2)`) - (
`(2)/(5)`)(
`-(√(2))/(2)`)

=
`-(√(42))/(10)` -
`-(2√(2))/(10)`

=
`(2√(2)-√(42))/(10)`

SIN 2A:

sin (A + A) = 2 (sin A * cos A)

= 2 (
`(2)/(5)`)(
`(√(21))/(5)`)

=
`(4√(21))/(25)`

A + B:

sin A =
`(2)/(5)`

A = sin⁻¹
`((2)/(5))`

A = 23.57°

tan B = 1

B = tan⁻¹(1)

B = 45°

= 45° + 180° in Quadrant 3

= 225°

A + B = 23.6° + 225°

= 248.6° which lies in Quadrant 3