Solve for x in sin ⁡ ( 30 ) ⋅ sin ⁡ ( 60 ) ⋅ sin ⁡ ( x + 10 ) sin ⁡ ( 80 ) ⋅ sin ⁡ ( 40 ) ⋅ sin ⁡ ( x ) = 1

Question

Solve for x in            sin      ⁡      (      30      )      ⋅      sin      ⁡      (      60      )      ⋅      sin      ⁡      (      x      +      10      )              sin      ⁡      (      80      )      ⋅      sin      ⁡      (      40      )      ⋅      sin      ⁡      (      x      )        =  1

Madisyn Avery · Accepted Answer

Step 1Let us use the identity  sin  ⁡  θ  ⋅  sin  ⁡  (      60    ∘    −  θ  )  ⋅  sin  ⁡  (      60    ∘    +  θ  )  =            sin      ⁡      3      θ        4  to get            sin      ⁡              40        ∘            sin      ⁡              80        ∘                    sin      ⁡              60        ∘              =      1          4      sin      ⁡              20        ∘              .In your equation, leave the terms containing x in LHS and move the others to RHS, then use the derived relation.            sin      ⁡      (      x      +              10        ∘            )              sin      ⁡      x        =            sin      ⁡              40        ∘            sin      ⁡              80        ∘                    sin      ⁡              30        ∘            sin      ⁡              60        ∘              =      1          2      sin      ⁡              20        ∘              =            sin      ⁡              30        ∘                    sin      ⁡              20        ∘            From here, considering the nature of the original problem,   x  =      20    ∘   is an obvious solution. To confirm further, expand sin to get,            sin      ⁡      (      x      +              10        ∘            )              sin      ⁡      x        =            sin      ⁡      (              20        ∘            +              10        ∘            )              sin      ⁡              20        ∘              cos  ⁡      10    ∘    +  cot  ⁡  x  sin  ⁡      10    ∘    =  cos  ⁡      10    ∘    +  cot  ⁡      20    ∘    sin  ⁡      10    ∘    cot  ⁡  x  =  cot  ⁡      20    ∘    .

Solve for x in sin ⁡ ( 30 ) &

Answered question

Answer & Explanation

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