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

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

Ответ:

jrfranckowiak

02.01.2022

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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

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Ответ:

kaitlynn73

19.11.2022

Mathematics

(A) $\frac{1}{5}$

Step-by-step explanation:

The multiplicative Rate of change refers to the fixed constant which when multiplied to the output will give the increase or decrease in the next output value.

Now, We have the table of values as:

x y