How can you determine theta without using trigonometric graphs?

[solve]

Homework Statement

a) 2sin3x=sqrt(2).

Find Theta.

b) Cos4x=1/2. Find Theta

Homework Equations

The Attempt at a Solution

a) sin3x=1/sqrt(2)

Theta=pi/4. This is from the 45-45-90 triangle.

Also Theta=3pi/4. How did they get this one? Can you guess it without looking at the graph of the sine function?

b) 4x=1/2=pi/3=theta. From 30-60-90 triangle.

But theta also equals 5pi/3. Again, How can we guess the other angle( 5pi/3) without looking at the graph of the cosine function? Thanks.

 

[Mentallic]

Homework Statement

a) 2sin3x=sqrt(2).

Find Theta.

a) sin3x=1/sqrt(2)

Theta=pi/4. This is from the 45-45-90 triangle.

You mean find x :tongue:

If you were given $$sin(\theta)=\frac{1}{\sqrt{2}}$$ then you'd say that $$\theta=\frac{\pi}{4}, ...$$

But if $\theta=3x$ then you solve everything in the same way, but you need to substitute in 3x for $\theta$ to obtain

$$3x=\frac{\pi}{4}, ...$$

which then means $$x=\frac{\pi}{12}, ...$$

Also Theta=3pi/4. How did they get this one? Can you guess it without looking at the graph of the sine function?

Do you know how to use the unit circle and the quadrants to find the other trig values? (it's equivalent to the sine graph but it's just another way of looking at it)
If not, you can either memorize the identities such as $\sin(x)=\sin(\pi-x)$ or just stick to looking at the graphs (I do it this way in my head, personally I find it the easiest).

Now one more thing, did the question give you a range of values x could be or are you expected to give the general solution to x?

b) 4x=1/2=pi/3=theta. From 30-60-90 triangle.

But theta also equals 5pi/3. Again, How can we guess the other angle( 5pi/3) without looking at the graph of the cosine function? Thanks.

Again, shouldn't you be solving for x?

Use the identity $\cos(x)=\cos(-x)=\cos(2\pi-x)$

or again, learn about the unit circle representation, such as from here
http://www.themathpage.com/atrig/unit-circle.htm#tan

 

[solve]

Thank You for the answer. I do have to solve for x, but that's not a problem once all the angles theta are identified.

I was just looking for a way to be able to find all the thetas without looking at the unit circle. Could you please tell me more about these specific identites and how they are related to finding all the thetas with a certain range? Thank You.

 

[Mentallic]

Thank You for the answer. I do have to solve for x, but that's not a problem once all the angles theta are identified.

As long as you remember that if x is given a certain range, say $0\leq x\leq 2\pi$ then you need to extend your range to accommodate for the multiplier, so $0\leq 4x\leq 8\pi$.

I was just looking for a way to be able to find all the thetas without looking at the unit circle. Could you please tell me more about these specific identites and how they are related to finding all the thetas with a certain range? Thank You.

I'm sorry, what do you mean about how it relates to finding all the thetas in a certain range? The trig graphs, the unit circle, and the trig identities are all related. If you know one, then you can deduce the others.

 

[solve]

As long as you remember that if x is given a certain range, say $0\leq x\leq 2\pi$ then you need to extend your range to accommodate for the multiplier, so $0\leq 4x\leq 8\pi$.
I'm sorry, what do you mean about how it relates to finding all the thetas in a certain range? The trig graphs, the unit circle, and the trig identities are all related. If you know one, then you can deduce the others.

Suppose, I haven't memorised Unit Circle and don't have one available, also have no access to trig graphs. What can I do to realize that the other angle theta for cos4x=1/2, aside from the obvious theta=pi/3 that I derive from 30-60-90 triangle, is 5pi/3 ? Thank You.

 

[Mentallic]

Suppose, I haven't memorised Unit Circle and don't have one available, also have no access to trig graphs. What can I do to realize that the other angle theta for cos4x=1/2, aside from the obvious theta=pi/3 that I derive from 30-60-90 triangle, is 5pi/3 ? Thank You.

The unit circle is explained in the link I provided earlier.

Also, you don't need access to trig graphs, you simply graph them yourself. And until you give yourself a moment to understand the trig circle, reading off the graphs would be best. This is assuming you don't remember the identities

$$\sin(x)=\sin(\pi-x)$$

$$\cos(x)=\cos(2\pi-x)$$

$$\tan(x)=\tan(\pi+x)$$

Which are all identities I don't remember off-hand, but merely thought of the unit circle and deduced it quickly and easily.

 

[solve]

The unit circle is explained in the link I provided earlier.

Also, you don't need access to trig graphs, you simply graph them yourself. And until you give yourself a moment to understand the trig circle, reading off the graphs would be best. This is assuming you don't remember the identities

$$\sin(x)=\sin(\pi-x)$$

$$\cos(x)=\cos(2\pi-x)$$

$$\tan(x)=\tan(\pi+x)$$

Which are all identities I don't remember off-hand, but merely thought of the unit circle and deduced it quickly and easily.

These are exactly the identites I needed to remember. Not even sure if I even knew them to start with. Even if I knew them I doubt I'd realize to use these exact identities to find other theta besides pi/3 for cos4x=1/2. I had to be told and shown to use these ones specifically. Thanks.