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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:
\(\mathsf{sin\,p-sin\,q=2\cdot sin\!\left(\dfrac{p-q}{2}\right)\cdot cos\!\left(\dfrac{p+q}{2}\right)}\)
for p = 6m – 8° and q = 90° – 2m, and that gives you
\(\mathsf{2\cdot sin\!\left[\dfrac{(6m-8\°)-(90\°-2m)}{2}\right]\cdot cos\!\left[\dfrac{(6m-8\°)+(90\°-2m)}{2}\right]=0}\\\\\\ \mathsf{2\cdot sin\!\left[\dfrac{6m-8\°-90\°+2m}{2}\right]\cdot cos\!\left[\dfrac{6m-8\°+90\°-2m}{2}\right]=0}\)
Combine like terms together in brackets:
\(\mathsf{2\cdot sin\!\left[\dfrac{6m+2m-8\°-90\°}{2}\right]\cdot cos\!\left[\dfrac{6m-2m-8\°+90}{2}\right]=0}\\\\\\ \mathsf{2\cdot sin\!\left[\dfrac{8m-98\°}{2}\right]\cdot cos\!\left[\dfrac{4m+82\°}{2}\right]=0}\\\\\\ \mathsf{2\cdot sin\!\left[\dfrac{\diagup\hspace{-7}2\cdot (4m-49\°)}{\diagup\hspace{-7}2}\right]\cdot cos\!\left[\dfrac{\diagup\hspace{-7}2\cdot (2m+41\°)}{\diagup\hspace{-7}2}\right]=0}\\\\\\ \mathsf{\diagup\hspace{-8}2\cdot sin(4m-49\°)\cdot cos(2m+41\°)=0}\)
\(\mathsf{sin(4m-49\°)\cdot cos(2m+41\°)=0}\)
If a product is zero, then one of the factors must be zero:
\(\mathsf{sin(4m-49\°)=0}~~\textsf{ or }~~\mathsf{cos(2m+41\°)}\)
Solving them separately:
• \(\mathsf{sin(4m-49\°)=0}\)
\(\mathsf{4m-49\°=k_1\cdot 180\°}\\\\ \mathsf{4m=49\°+k_1\cdot 180\°}\)
Divide both sides by 4, and you have
\(\mathsf{m=\dfrac{49\°}{4}+k_1\cdot \dfrac{180\°}{4}}\)
\(\mathsf{m=12.25\°+k_1\cdot 45^\circ}\) ✔
where k₁ is an integer.
• \(\mathsf{cos(2m+41\°)=0}\)
\(\mathsf{2m+41\°=90\°+k_2\cdot 180\°}\\\\ \mathsf{2m=90\°-41\°+k_2\cdot 180\°}\\\\ \mathsf{2m=49\°+k_2\cdot 180\°}\)
Divide both sides by 2, and you have also
\(\mathsf{m=\dfrac{49\°}{2}+k_2\cdot \dfrac{180\°}{2}}\)
\(\mathsf{m=24.5\°+k_2\cdot 90\°}\) ✔
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
