If sin(6m-8) = cos (2m), find the value of m.

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Solve the trigonometric equation, assuming that arcs are in degrees:

sin(6m – 8°) = cos(2m)

The cosine of an angle is equal to the sine of its complementary:

•  cos(2m) = sin(90° – 2m)

so the equation becomes

sin(6m – 8°) = sin(90° – 2m)

sin(6m – 8°) – sin(90° – 2m) = 0

Convert that difference into a product by applying this trigonometric identity:



for  p = 6m – 8°  and  q = 90° – 2m, and that gives you



Combine like terms together in brackets:





If a product is zero, then one of the factors must be zero:



Solving them separately:

•  



Divide both sides by  4, and you have



          ✔

where  k₁  is an integer.

•  



Divide both sides by  2, and you have also



          ✔

where  k₂  is an integer.

Solution set:

S = {m ∈ R:  m = 12.25° + k₁ · 45°   or   m = 24.5° + k₂ · 90°,   k₁, k₂ ∈ Z}.

I hope this helps. =)

Tags:  trigonometric trig equation sum difference product transformation formula relation trigonometry