Question

Find all solutions if 0° ≤ θ < 360°. Verify your answer graphically. (Enter your answers from smallest to largest.)
cos(3theta)=1

Answer 1

\[
\cos(3\theta)=1 \\
\text{From unit circle, cosine is 1 when theta = 0} \\
\text{Also cosine has a period of 2pi. Therefore, 0, 0+2pi, 0+4pi, ... are solutions.} \\
3\theta = 2n\pi ~~ \text{where }n = 0, 1, 2, 3, ....
\]

Answer 2

In degrees, 0, 0 +360, 0 + 720, .... are all solutions.
\[
3\theta = 0, ~~360, ~~720, ~~1080,~~ 1440, ....\\
\theta =~~0, ~~120, ~~240, ~~~~360
\]

Answer 3

wait zero is an option?

Answer 4

are u in the red cross? or are u swiss.. ur image is distracting

Answer 5

Neither. Just an image off the net.

Answer 6

hahah ok...

Answer 7

So 0, 120, 240 and 360 are the solutions.

Answer 8

wouldnt 240 be an option?

Answer 9

oh yeah

Answer 10

yay thank you! got it right... i had the right answers but i forgot zero

Answer 11

can you help me on another one?

Answer 12

yw.

Answer 13

go ahead.

Answer 14

cos(2theta)= - 1/2 so i got... -60... so 300 and 60 because its positive... right? so i thought it either has to be in q1 or q4 ... so i got 150, 30 etc... well i got it wrong...

Answer 15

Do they specify the domain of theta like they did in the previous problem?

Answer 16

360

Answer 17

The cosine function is negative in the second and third quadrant.
From unit circle, cosine is -1/2 when angle = 120 and 240.
cosine had a periodicity of 360 degrees.
So 120, 240, 120+360=480, 240+360=600 all give -1/2 for cosine.
2(theta) = 120, 240, 480, 600, ...
theta = 60, 120, 240, 300.

Answer 18

so subtract 60 from 180 and add to 180?
?

Answer 19

An acronym to remember is ASTC or All Students Take Calculus.
A: All trig functions are positive in the first quadrant.
S: Sine & Csc are positive in the second quadrant.
T: Tan & Cot are positive in the third quadrant.
C: Cosine and Sec are positive in the fourth quadrant.

Answer 20

ohh thanks thats pretty cool

Answer 21

Here cosine is negative half. So it must be in the second and third quadrant.
Cosine is positive half when angle = 60 degrees. So the reference angle is 60 degrees to get half.
To get negative half just use the reference angle of 60 degrees to arrive at angles in q@ and Q3. (Remember reference angle is the angle with the x-axis).
So 180 - 60 = 120
180 + 60 = 240
So cos(120) = cos(240) = -1/2.

After that make use of the periodicity of cosine which is 360 degrees.
Keep adding as many multiples of 360 degrees to 120 and 240 to arrive at other solutions.

So cos(2*theta) = -1/2
2*theta = 120, 240, 120+360=480, 240+360=600, ....
theta = 60, 120, 240, 300.

Answer 22

To get negative half just use the reference angle of 60 degrees to arrive at angles in Q2 and Q3.

Answer 23

All trig functions have this property: Measure the angle made with the x-axis. It can be with the positive x-axis if that gives the acute angle. Or it can be with the negative x-axis if that gives the acute angle. |sin(x)| = |sin(180-x)| = |sin(180+x)| = |sin(360-x)|.
This is true for all trig functions. The magnitude is the same. They differ only in sign. By remembering ASTC we can figure out the sign.

Answer 24

|tan(x)| = |tan(180-x)| = |tan(180+x)| = |tan(360-x)|
For example, in the first quadrant if you just remembered that tan(45) = 1 and that tan is positive in the first and third quadrant, you can say tan(180+45) = tan(45) = 1
and tan(180-45) = tan(360-45) = -1

Answer 25

So knowing the value of the trig function for angle x in the first quadrant, you will know the value of the trig functions for the angles: x (Q1), 180-x (Q2), 180+x (Q3), and 360-x (Q4) right away.

Answer 26

Example sin(30) = 1/2
Sine is positive in Q1 and Q2 (from ASTC).
30 is Q1, 180-30=150 is Q2, 180+30=210 is Q3, 360-30 = 330 is Q4.
sin(30) = 1/2. Therefore,
sin(150) = 1/2
sin(210) = -1/2
sin(330) = -1/2

Answer 27

Another example,
cos(60) = 1/2. Reference angle is 60 degrees. The other angles in Q2, Q3, Q4 from the reference angle of 60 degrees are:
Q1: 60
Q2: 180 - 60 =120
Q3: 180 + 60 = 240
Q4: 360 - 60 = 300
Cosine is positive in Q1 and Q4.
Therefore, from cos(60) = 1/2 we can conclude:
cos(120) = -1/2
cos(240) = -1/2
cos(300) = 1/2

Answer 28

I think I may have confused you enough for the day. :)

Answer 29

i totally get it.. i just got lost with other problems in my hw