Answer:
x = {π/6, π/2, 5π/6, 3π/2}
Step-by-step explanation:
The equation can be solved using a double-angle trig identity and factoring.
Simplify
Dividing the equation by 7 and substituting for sin(2x), we have ...
7sin(2x) = 7cos(x)
sin(2x) = cos(x)
2sin(x)cos(x) = cos(x)
2sin(x)cos(x) -cos(x) = 0
cos(x)(2sin(x) -1) = 0
Zero product rule
The product of factors is zero when one or more of the factors is zero.
cos(x) = 0 ⇒ x = {π/2, 3π/2}
2sin(x) -1 = 0 ⇒ x = arcsin(1/2) = {π/6, 5π/6}
Solutions in the given interval are ...
x = {π/6, π/2, 5π/6, 3π/2}
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Additional comment
When the equation is of the form f(x) = 0, then the x-intercepts of f(x) are its solutions. We can rearrange this one to ...
sin(2x) -cos(x) = 0
The solutions identified above match those shown in the graph.
