the graph of 6a is shown below:
the graph of 6b is shown below:
the graph of 6c is shown below:
to make graphing easier, i used x instead of theta.
basically i created separate equations from the expression on the left side of the equal sign and the expression on the right side of the equal sign.
the intersection of those 2 equations is the common solution to both.
i only looked for a solution between 0 and 180 degrees.
that's the unshaded region of the graphs.
to solve algebraically, i did the following/
6a equation is sin^2(x) = 1.
remember, i'm using x in place of theta.
this makes it easier to type and also easier to graph.
take the square root of both sides of the equation to get:
sin(x) = plus or minus sqrt(1) = plus or minus 1.
my calculator tells me:
arcsin(1) = 90 degrees.
arcsin(-1) = -90 degrees.
to convert to all positive angles between 0 and 360 degrees, i do the following:
90 degrees = 90 degrees
-90 degrees = 360 - 90 = 270 degrees.
i therefore get:
arcsin(1) = 90 degrees.
arcsin(-1) = 270 degrees.
the only solution between 0 and 180 degrees is arcsin1) = 90 degrees.
therefore 90 degrees is my only solution.
6b equation is 3sin(x) = 2cos(x).
i'm using x instead of theta for purposes already described above.
divide both sides of this equaation by 3 and divide both sides of this equation by cos(x) to get:
sin(x)/cos(x) = 2/3.
since sin(x)/cos(x) is equivalent to tan(x), the equation becomes:
tan(x) = 2/3.
arctan(2/3) = 33.69006753 degrees which can be rounded to 33.69 degrees as shown in the graph for 6b.
that would be in the first quadrant.
tangent is positive in the first and third quadrant.
third quadrant is not between 0 and 180 degrees, therefore the only solution is:
x = 33.69 degrees, as shown in the graph for 6b.
6c equation is sin(x) - 4cos(x) = 0
again, i use x in place of theta.
add 4cos(x) to both sides of the equation to get:
sin(x) = 4cos(x)
divide both sides of the equation by cos(x) to get:
sin(x) / cos(x) = 4
this is equivalent to tan(x) = 4
arctan(4) = 75.96375653 degrees.
this can be rounded to 75.964 degrees as shown in the graph for 6c.
tan is positive in the first and third quadrants only.
third quadrant is not between 0 and 180 degrees.
therefore the only solution is 75.964 degrees.