Question
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Tutor-Verified Answer
Answer
The solutions are:
-
with or for any integer . -
or .
Solution
-
Begin with the equation
-
Isolate the cosine term by subtracting 3 from both sides:
-
Divide both sides by 5:
-
The general solutions for an equation of the formare given bywhere
is any integer. Thus, forthe solutions are -
It appears that the subsequent steps involve the equation of a circle,If we interpret the value
as the -coordinate on a circle with radius , then substitute and into the circle’s equation: -
Compute:
-
Solve for
: -
Take the square root of both sides. Remember that both positive and negative roots are possible:
Thus, the solutions are:
-
For the trigonometric equation:
-
For the circle equation with
and :
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Beyond the Answer
Oops! Seems like there’s a little mix-up here. While your manipulation of the cosine equation is on track, you took a wrong turn when you ventured into the sine territory. Remember,
tells you about the x-coordinate on a unit circle, while
would relate to the y-coordinate. So instead of
, let’s stick with the original cosine pathway!
Now, about the circle equation—you’ve got that part right! So after finding that
, you can determine the angle
that corresponds to that cosine value. Dig into your trigonometric identities and fear not, for angles in the second and third quadrants will fit the bill!
Watch out for common mistakes! Misplacing your sine and cosine can lead to a real head-scratcher. Make sure you stay on the right function path. When dealing with circular coordinates, remember which trigonometric function corresponds to which axis—
for x and
for y. It’ll save you from going on a wild goose chase!