# How to solve this question

#### MethMath11

##### Junior Member

Here's what I got so far
16(2cos^2 A -1)

#### MarkFL

##### Super Moderator
Staff member
I would recommend trying to see if you can spot two angles A and B that make the given equations true.

#### Jomo

##### Elite Member
This does not seem to be all that easy to do.

#### Dr.Peterson

##### Elite Member
I'd start with the second equation, which tells you something simple about A and B. (It may be helpful to also use some implications from the first equation about the possible quadrants of the angles.) Then you can put the resulting expression for B into the first equation.

You can then solve to find all solutions for A and B. (All this can be practically "by inspection" if you think about it the right way, but it can be more formal. There are some subtleties you might miss, but they turn out not to cause trouble.)

I don't see a way to solve this without finding A and B.

#### Harry_the_cat

##### Senior Member
Assume A is acute (for now). Let cos A = x, then get cos B, sin A and sin B in terms of x.
Give that a go.

#### JeffM

##### Elite Member
I think this may easier if we start with some substitutions and algebra.

$$\displaystyle w = sin(a),\ x = sin(b),\ y = cos(a), \text { and } z =cos(b).$$

Then from the given equations, we can work with:

$$\displaystyle w + x = 1,\ y + z = 0, \ w^2 + y^2 = 1, \text { and } x^2 + z^2 = 1.$$

$$\displaystyle w + x = 1 \implies w = 1 - x \implies w^2 = 1 - 2x + x^2.$$

$$\displaystyle y + z = 0 \implies z = -\ y \implies z^2 = y^2.$$

$$\displaystyle w^2 + y^2 = 1 \implies 1 - 2x + x^2 + y^2 = 1 \implies x = \dfrac{2 \pm \sqrt{4 - 4y^2}}{2} = 1 \pm \sqrt{1 - y^2}.$$

$$\displaystyle x^2 + z^2 = 1 \implies x^2 + y^2 = 1 \implies x = \pm \sqrt{1 - y^2}.$$

$$\displaystyle \therefore \pm \sqrt{1 - y^2} = 1 \pm \sqrt{1 - y^2}.$$

If the signs are the same, this is impossible. So the signs must be different.

$$\displaystyle \pm 2 \sqrt{1 - y^2} = 1 \implies 4(1 - y^2) = 1 \implies y^2 = \dfrac{3}{4} \implies y = \pm \dfrac{\sqrt{3}}{2}.$$

#### Otis

##### Senior Member
If one is familiar with the unit circle and the special angles (I think Dr. Peterson hinted at this), it's not hard to find supplimentary angles A and B.

MarkFL suggested trying to spot A and B, so naturally I took the given equations (evaluated) to be 1/2+1/2=1 and -1+1=0, respectively, because I'm familiar with those special-angle trig values. The unit-circle concept memorized helps reveal A and B readily.

#### MarkFL

##### Super Moderator
Staff member
If one is familiar with the unit circle and the special angles (I think Dr. Peterson hinted at this), it's not hard to find supplimentary angles A and B.

MarkFL suggested trying to spot A and B, so naturally I took the given equations (evaluated) to be 1/2+1/2=1 and -1+1=0, respectively, because I'm familiar with those special-angle trig values. The unit-circle concept memorized helps reveal A and B readily.

Yes, I looked for two angles symmetric about $$\displaystyle \frac{\pi}{2}$$, and readily found 2 angles satisfying both equations.

#### Harry_the_cat

##### Senior Member
Yes, I looked for two angles symmetric about $$\displaystyle \frac{\pi}{2}$$, and readily found 2 angles satisfying both equations.
Mark FL yes I did that too and you get the answer quite easily. I just wasn't 100% sure if that was a complete solution. What if there were other values of A and B?

I suppose the nature of the question "If this … then this" implies that even if you only find one solution to the original 2 simultaneous equations (and ignore any others) then the next part is true. Of course that is relying on the last statement being true for all possible solutions of A and B, and that is relying on author of the question.

I hope that makes sense?? Here' an example of what I'm getting at:

If A2 = 4 and B2 = 9 then A + B = ?.

Isn't your logic used above the same as "Oh I can see two answers that work namely A=2 and B=3 so then A+B=5" missing something? Obviously the "then" statement is not correct for all A and B.

I think I agree with myself!!! Thoughts?

#### Jomo

##### Elite Member
Mark FL yes I did that too and you get the answer quite easily. I just wasn't 100% sure if that was a complete solution. What if there were other values of A and B?

I suppose the nature of the question "If this … then this" implies that even if you only find one solution to the original 2 simultaneous equations (and ignore any others) then the next part is true. Of course that is relying on the last statement being true for all possible solutions of A and B, and that is relying on author of the question.

I hope that makes sense?? Here' an example of what I'm getting at:

If A2 = 4 and B2 = 9 then A + B = ?.

Isn't your logic used above the same as "Oh I can see two answers that work namely A=2 and B=3 so then A+B=5" missing something? Obviously the "then" statement is not correct for all A and B.

I think I agree with myself!!! Thoughts?
I too saw the obvious answers but were there others? I still think that this problem was a difficult one although JeffM made it look easy.

#### Harry_the_cat

##### Senior Member
I too saw the obvious answers but were there others? I still think that this problem was a difficult one although JeffM made it look easy.
Yes there will be others because of the periodic nature of the functions involved.

#### Otis

##### Senior Member
… saw the obvious answers but were there others? …
It's multiple choice, so we only need one.

#### Harry_the_cat

##### Senior Member
Yeah I know that but "maybe" another value is also valid. (It's not in this case but could be.)

#### Dr.Peterson

##### Elite Member
It's multiple choice, so we only need one.
Yes, as long as you only want to get the "right" answer, and trust that they got it correct.

If you care about truth, instead, you have to consider that maybe there is no answer, because the value is not determined by the given facts. And if there were a choice like "insufficient information", you would have to dig deeper.

This is one of the reasons I hate multiple choice! It distorts your perspective on math, removing incentive to look for certainty.

#### MarkFL

##### Super Moderator
Staff member
Mark FL yes I did that too and you get the answer quite easily. I just wasn't 100% sure if that was a complete solution. What if there were other values of A and B?

I suppose the nature of the question "If this … then this" implies that even if you only find one solution to the original 2 simultaneous equations (and ignore any others) then the next part is true. Of course that is relying on the last statement being true for all possible solutions of A and B, and that is relying on author of the question.

I hope that makes sense?? Here' an example of what I'm getting at:

If A2 = 4 and B2 = 9 then A + B = ?.

Isn't your logic used above the same as "Oh I can see two answers that work namely A=2 and B=3 so then A+B=5" missing something? Obviously the "then" statement is not correct for all A and B.

I think I agree with myself!!! Thoughts?
Initially, I looked at squaring the given equations and trying to derive an answer based solely on that. But, in the end, I found I agree with this:

I don't see a way to solve this without finding A and B.
I would have much preferred to be able to algebraically derive the double-angle expression from the two given equations, but I was unable to, given the amount of time I could afford to spend on it.

The problem says "suppose A and B are two angles such that..." and imply that only one of the given choices can be correct. Having found two such angles, and that one of the answers is the result, I was satisfied.

#### Otis

##### Senior Member
… Yes, as long as you only want to get the "right" answer …

I do understand tutors' interest in the side discussion; however, my desire is for MethMath to understand what's expected here (in a fairly simple exercise).

#### JeffM

##### Elite Member
Initially, I looked at squaring the given equations and trying to derive an answer based solely on that. But, in the end, I found I agree with this:

I would have much preferred to be able to algebraically derive the double-angle expression from the two given equations, but I was unable to, given the amount of time I could afford to spend on it.

The problem says "suppose A and B are two angles such that..." and imply that only one of the given choices can be correct. Having found two such angles, and that one of the answers is the result, I was satisfied.
I started off as Mark did, but when that did not work, I did not next look for A and B.

I fully agree with otis that experimenting with special right triangles as a speedy first approach to apparently difficult trigonometric problems is undoubtedly what the person writing the exercise intended to teach. I further agree with otis that someone who is studying for an exam in trigonometry will often profit from that approach. It did not occur to me to start there because I have not had to take an exam in trigonometry for well over 50 years.

So yes, the lesson for the student is: if a problem in a text or examination on trigonometry seems weird, see whether a special right triangle gives the answer. That is a great lesson for taking exams. I do not, however, see it teaching much in the way of mathematical reasoning.

What I saw was four unknowns and two equations. That lead me to ask myself whether I could find two additional equations. That in turn led me to one of the basic trigonometric identities, namely

$$\displaystyle sin^2( \theta ) + cos^2( \theta ) = 1.$$

That approach, can we find apparently missing equations and can we take advantage of basic trig identities, strikes me as being a more useful general approach for learning mathematics. I see now that I skipped over both points in my answer so the student could not see those ideas in my initial response.

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