Easier Trigonometric Equations
By the nature of the trig functions,
trig equations will yield an infinite number of solutions. Watch
closely to the domain of x as we do each problem. Sometimes we
ask for all possible solutions, called the general
solutions of a trig equation. These are generally
by finding particular solutions
adding in a factor for the period length of the particular
Many times we will restrict the domain of x to the first period of the
given function or one time around the unit circle, 2p.
1) Find the values of x
0 and 2pfor which cos x = .4
Solution: To find x, we are looking
for the angle measured in rads, one time around the unit circle.
Since the cos is positive in quads 1 and 4, we should find two answers
in the given domain.
x = cos-1
Use your calculator to get the answer in
first quadrant. Round to 4 decimal places.
x = 1.1593
This is the solution in quadrant 1.
other solution is in quad 4. Remembering our reference angles,
other answer is found by taking 2p-1.1593 =
Therefore, the answers are
= 1.1593 and 5.1239
2) Solve 4 sin
q + 5 = 2 for 0o < q
Isolate the variable.
4 sin q
The answers this
are in quadrants III and IV because the sin is negative in those
Find the reference angle first then apply the correct reference angle
This is rounded to
place. Now find the answers in the correct quadrants.
In three, it's 180
48.6 = 228.6o
In four, it's 360 -
Therefore, the answers
q= 228.6o and 311.4o
3) Solve the equation
sec x = 4.2 for x between 0 and 2p
Solution: Solve the same way
as the first example
x = sec-1
x = 1.3304
This is the answer in the first
Sec is also positive in quadrant IV. Using reference angles, we
2p- 1.3304 = 4.9528
Therefore the solutions are
= 1.3304 and 4.9528
( Note: If we wanted
all solutions, we would simply add multiples of 2pto
each of the above answers. x = 1.3304 + 2np and
4.9528 + 2np)
4) Solve the
equation for all solutions. Round answers to the nearest
of a radian.
Solve for x:
12 cot x = 35
cot x = 35/12
x = cot-1
x = .33
Since the cot is positive in Quads I and
we want the above answer plus this one: 3.14 + .33 = 3.47
Therefore, all answers
x = .33 + 2np and x = 3.47 + 2np
of a line is the angle a, where 0o
< a< 180o, that is
from the positive x-axis to the line. Lines going uphill will
a slope smaller than 90o and lines going downhill will have
slopes greater than 90o. Study the two graphs below.
For any line with
m and inclination a,
m = tan a
if adoesn't = 0
then the line has undefined slope.
1) To the nearest degree, find the
of the line 3x + 4y = 8
Find the slope of the equation.
3x + 4y = 8
4y = -3x + 8
y = (-3/4)x + 2
m = -3/4
The line is going downhill, so the
will be greater than 90o.
So our inclination
a= 180 - 37 = 143o
2) To the nearest
find the inclination of the line -2x + 5y = 15
Same as above.
-2x + 5y = 15
5y = 2x + 15
y = (2/5)x + 3
m = 2/5
The line is going
because the slope is positive, so the inclination will be less than 90o.
So our inclination
it for this section. On to the next section!!