VCE Methods Question Thread!

[snowisawesome]

if cos (x) = -1/2,
can someone please explain how the base angle = 2pi/3
Very confused with this
Thanks

[VanillaRice]

if cos (x) = -1/2,
can someone please explain how the base angle = 2pi/3
Very confused with this
Thanks

To flip the question, what do you think the base angle should be?

[snowisawesome]

To flip the question, what do you think the base angle should be?

Wouldn't it be pi/3, since when you solve cos(x) = -1/2, the base angle is pi/3

[VanillaRice]

Wouldn't it be pi/3, since when you solve cos(x) = -1/2, the base angle is pi/3

Technically, you could think of the base angle as pi/3. However, you would also need to realise that cosine is negative only in the second and third quadrants. Hence, we want pi - pi/3 = 2pi/3 and pi + pi/3 = 4pi/3, which are in the second and third quadrants of the first revolution of the unit circle respectively.

I assume whatever solution you're referencing is using 2pi/3 to avoid confusion of the positive and negative values of cosine.

Hope this helps

[snowisawesome]

Technically, you could think of the base angle as pi/3. However, you would also need to realise that cosine is negative only in the second and third quadrants. Hence, we want pi - pi/3 = 2pi/3 and pi + pi/3 = 4pi/3, which are in the second and third quadrants of the first revolution of the unit circle respectively.

I assume whatever solution you're referencing is using 2pi/3 to avoid confusion of the positive and negative values of cosine.

Hope this helps

But why is it that when solving cos(x) = -1/2 for a domain solution we use base angle = pi/3
But when solving cos(x) = -1/2 for the general solution, why is the base angle 2pi/3

[VanillaRice]

But why is it that when solving cos(x) = -1/2 for a domain solution we use base angle = pi/3
But when solving cos(x) = -1/2 for the general solution, why is the base angle 2pi/3

Because cos(pi/3) = 1/2, not -1/2. What you really want is cos(pi - pi/3) = cos(2pi/3) = -1/2. We only use pi/3 as a base angle to work off with, it isn't the actual answer. After we find the base angle, we then need to consider which quadrant we want our solution to be in. In this case, it's the second/third quadrants (Pi/3 itself is in the first quadrant).

Hope that makes sense

[snowisawesome]

Because cos(pi/3) = 1/2, not -1/2. What you really want is cos(pi - pi/3) = cos(2pi/3) = -1/2. We only use pi/3 as a base angle to work off with, it isn't the actual answer. After we find the base angle, we then need to consider which quadrant we want our solution to be in. In this case, it's the second/third quadrants (Pi/3 itself is in the first quadrant).

Hope that makes sense

So whenever we solve cos(x) = -1/2 for a domain solution we use pi/3 as the base angle
and whenever we solve cos(x) = -1/2 for the general solution we use 2pi/3 as the base angle
Is this correct?

[Yertle the Turtle]

This is why symmetry of angles is SOOOO important in Circular Functions, it has got to be one of the most important things that you need to remember because it is so easy to slip up.

[VanillaRice]

So whenever we solve cos(x) = -1/2 for a domain solution we use pi/3 as the base angle
and whenever we solve cos(x) = -1/2 for the general solution we use 2pi/3 as the base angle
Is this correct?

It might help not to rigidly define a "base angle". It's not necessarily always the thing that comes after the 2pi(n) in a general solution. If you were to follow the same logic as my previous post, the base angle is Pi/3 for both cases. It's just not explicitly stated. Think of the general solution as 2pi(n) + (Pi - pi/3) instead. Once again, the base angle is Pi/3, it's just not explicitly stated when you see 2pi/3.

Regardless, as long as you know what you need to do, the definition of "base angle" doesn't matter. If you want to use your own logic, and you're able to get the right answer, then that's fine. You won't be asked specifically for the base angle in a VCAA exam. You will however, need to be aware of the properties of the unit circle.

Hope that helped

[snowisawesome]

Thanks VanillaRice
Am i right in thinking that when you solve sin(x) = -1/2 for a general solution the base angle is pi +pi/6 = 7pi/6

[TheAspiringDoc]

Thanks VanillaRice
Am i right in thinking that when you solve sin(x) = -1/2 for a general solution the base angle is pi +pi/6 = 7pi/6

Can someone please explain what a base angle is? I don't really understand any of the above^^

[VanillaRice]

Thanks VanillaRice
Am i right in thinking that when you solve sin(x) = -1/2 for a general solution the base angle is pi +pi/6 = 7pi/6

Yes, but don't forget you also need to consider -(pi)/6 (or 11pi/6).

EDIT: To be more precise, you should consider the base angle to be pi/6. This is because sin has two solutions in the unit circle: x = pi +pi/6 = 7pi/6 (like you said), and 0 - pi/6 = -pi/6 (or 2pi - pi/6 = 11pi/6).

Can someone please explain what a base angle is? I don't really understand any of the above^^

Do you know how to solve trigonometric equations, both generally and within a specified domai?. If your answer is yes, then you don't need to worry

Otherwise, the base angle is a method that is often taught to help students solve trigonometric equations. We essentially find the angle in the first quadrant that relates to the equation, and apply it to whatever quadrant is necessary.
Example: say we are solving cos(x) = -1/2 for x, and we want an answer that is in the domain [pi/2, pi]. The relevant angle here is pi/3 (since in the first quadrant, cos(pi/3) = 1/2). Since our required answer is in the second quadrant (domain [pi/2, pi]), our answer would be x = pi - pi/3 = 2pi/3.

Using this method is definitely not necessary, however many students find it helps them solve such equations.

Hope this helps

[Lear]

Quick question, Heff 2016 exam 1 question 4A
There’s two marks there and the marking guide gives one mark for two correct answers and one for the ACTS graph things with a line through it. Is that required for the second mark??

[VanillaRice]

Quick question, Heff 2016 exam 1 question 4A
There’s two marks there and the marking guide gives one mark for two correct answers and one for the ACTS graph things with a line through it. Is that required for the second mark??

No, I don't believe so. I believe that the final mark corresponds to the final line of the solution (i.e. the final answer), rather than the diagram. You definitely won't be required to draw a diagram like that to obtain the relevant marks in a VCAA exam.

[Lear]

Thank you