- #1

- 2,370

- 303

- Homework Statement
- Solve the equation##sin (∅+45^0)=2 cos (∅-30^0)##

giving all solutions in the interval ##0^0< ∅<180^0##

- Relevant Equations
- Trigonometric identities

Find the Mark scheme solution here;

Now find my approach;

Using the trig. identities It follows that,

##\frac {1}{\sqrt 2}##⋅ ##sin ∅##+##\frac {1}{\sqrt 2}##⋅ ##cos ∅##=##{\sqrt 3}##⋅ ##cos ∅##+##sin ∅##

→##sin ∅##[##\frac {1}{\sqrt 2}##-##1]##=##cos ∅##[##\frac {-1}{\sqrt 2}##+##{\sqrt 3}##]

→##sin ∅##⋅[##\frac {1-{\sqrt 2}}{\sqrt 2}]##=##cos ∅##⋅[##\frac {\sqrt 6 -1}{\sqrt 2}]##

→##tan ∅##=##[\frac {\sqrt 6 -1}{1-{\sqrt 2}}]##

## ∅##=##-74.051^0##, but we want our solutions to be in the domain, ##0^0< ∅<180^0##,

therefore, ## ∅##=##-74.051^0 + 180^0##=##105.9^0##

I would definitely be interested in another approach...

Now find my approach;

Using the trig. identities It follows that,

##\frac {1}{\sqrt 2}##⋅ ##sin ∅##+##\frac {1}{\sqrt 2}##⋅ ##cos ∅##=##{\sqrt 3}##⋅ ##cos ∅##+##sin ∅##

→##sin ∅##[##\frac {1}{\sqrt 2}##-##1]##=##cos ∅##[##\frac {-1}{\sqrt 2}##+##{\sqrt 3}##]

→##sin ∅##⋅[##\frac {1-{\sqrt 2}}{\sqrt 2}]##=##cos ∅##⋅[##\frac {\sqrt 6 -1}{\sqrt 2}]##

→##tan ∅##=##[\frac {\sqrt 6 -1}{1-{\sqrt 2}}]##

## ∅##=##-74.051^0##, but we want our solutions to be in the domain, ##0^0< ∅<180^0##,

therefore, ## ∅##=##-74.051^0 + 180^0##=##105.9^0##

I would definitely be interested in another approach...

Last edited: