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### Course: Trigonometry > Unit 4

Lesson 6: Challenging trigonometry problems- Trig challenge problem: area of a triangle
- Trig challenge problem: area of a hexagon
- Trig challenge problem: cosine of angle-sum
- Trig challenge problem: arithmetic progression
- Trig challenge problem: maximum value
- Trig challenge problem: multiple constraints
- Trig challenge problem: system of equations

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# Trig challenge problem: cosine of angle-sum

Sal is given cos(θ) and cos(φ) and he finds cos(θ+φ). To do that, he must first find sin(θ) and sin(φ) using the Pythagorean identity. Created by Sal Khan.

## Want to join the conversation?

- At3:33Sal says that Theta is between pi and 2pi, so the angle is going to be in the third or fourth quadrant.

I'm confused how Sal came to this conclusion. We're told the cosine of theta is negative, so surely this means the angle is in the second or third quadrant?(7 votes)- Theta between pi and 2pi is given as part of the problem. Without that information we wouldn't be able to tell if we were in the 2nd or 3rd quadrant.(14 votes)

- can you simplify that once more to:

7sqrt(3)/25?(3 votes)- If you mean, can you simplify (24 - 7√3)/50 to (7√3)/25, the answer is "no"

You can rewrite (24 - 7√3)/50 as 24/50 - (7√3)/50 and then you can simplify that first part as 12/25, but you are stuck with the second term of the required exact answer languishing over a denominator of 50. In other words, you are still stuck with

12/25 - (7√3)/50

There is no common factor of 2 buried in the 7 that you can use to simplify that second fraction.(6 votes)

- I think we could still solve this exercise by not making the explicit assumption that phi is a positive angle, as Sal did.

We know phi has a positive cosine. So this angle must lie either on the 1st or 4th quadrant. We are also told it is an acute angle. So, by my reasoning, that must be on the 1st quadrant. On the 4th quadrant phi would be neither acute or obtuse.

To be acute we would have to change phi sign to negative. But the question doesn't say "negative phi is an acute angle", it says "phi is an acute angle".

Would you agree with this? It is true that we could look at negative phi as the inverse of any phi angle (including an originally negative angle). But I believe it is accepted practice to not name positive angles unless explicitly required to do so. So, we just know that an acute angle is a positive angle.(3 votes)- Well you're wrong, cos(phi) is positive, not phi. So phi could still lie on the 1st or 4th quadrant.

For example if we knew phi was negative then it would lie in the 4th quadrant but we only know cos of phi, which is positive and therefore phi is in the 1st or 4th quadrant.(4 votes)

- how come you can't just add the cos(phi) (7/25) and cos(theta) (-sqrt3/2)(1 vote)
- The reason you can't just add cos(phi) and cos(theta) is because what the question is asking you to find cos(phi + theta) not cos(phi) + cos(theta), the second being what you described. If you just add the two different cosine values, that's all fine and dandy, but you're not answering the question. What you're supposed to do is find the radian amount of theta and phi, and then take the cosine of the value of them added. That's why you use the cosine angle addition formula and find the sine values for theta and phi.(6 votes)

- At0:10, Sal says "we can assume [phi] is a
**positive**acute angle..." What is he basing that statement on?(3 votes)- He realised that the question would not have a unique solution unless this assumption is made. However, somebody mentions below that the word "acute" may carry with it an implication of positivity, therefore this may just have been a clarification.(2 votes)

- why costheta(costheta) not equal to cosquarethetasqaure.?(2 votes)
- "cos" is not a number, and "cos(theta)" is not multiplication.(4 votes)

- Why can't you simply apply the arccos function to find theta and phi, and add them and apply the cosine function? It seems long and unnecessary to use the cos addition identity, compared to finding the angles and adding them.(2 votes)
- Yes, that method would have been way more easier, but only if one has access to a calculator so that we can use the inverse cosine function. If this question comes in a test which does not allow calculators, this is the method which is best.(4 votes)

- Bit out-the-book question, but, trig identities seems to be a chapter which is an incorporation of both concept and formulas. All I want to ask is, 'What are the real-life applications of trig identities?' Trig is an important chap for sure, but identities seems to be a bit off-grid, doesn't it?(3 votes)
- Sometimes we use sin or cos to model some scenario, due to its Periodic property.

But this function might be too complicated, thus we use the identities to convert the function into something we actually can calculate.(1 vote)

- at0:17why is fie or whatever that symbol is , can you assume that it is positive?(1 vote)
- The formula works for any angles, positive or negative, however you need to know both the cosine and sine of the angles. If phi were in the fourth quadrant, then the sine of the angle would be -24/25 instead of 24/25, and the answer would be different.(4 votes)

- Completely unrelated question, but maybe some teachers that might be present here will be able to answer :) Why Americans don't use the first letters of the Greek alphabet to name angles? Alpha, beta, gamma... Why is theta so popular? (I am from a European country, and in our schools, angles are almost never named "theta" - we usually start with alpha, so for an angle to be named theta, there must be 6 angles in a problem or example). Just curious :)(2 votes)
- It is simply convention. However we do use α and β a lot of times for identities involving more than one angle. The other Greek letters are used in different areas of mathematics and physics.(3 votes)

## Video transcript

Voiceover:We're told theta
is between pi and 2pi, and cosine of theta is equal to negative square root of 3 over 2, and phi is an acute angle, and we can assume it's
a positive acute angle. So we could say an acute positive angle or as a positive acute angle. And cosine of phi is equal to 7/25. Find cosine of phi plus theta exactly. So essentially, can we figure
it out without a calculator? I encourage you to pause this video and think about it on your own. Let's see if we can work through it. When we see, "Find cosine
of phi plus theta," we're finding the cosine of
the addition of 2 angles, so to me at least, that screams out that maybe
the angle addition formula can help us evaluate this, especially because we know
what cosine of theta is, cosine of phi is, and then maybe we can also use those to figure out what sine of
theta and sine of phi are. So let's just write out
the angle addition formula. It tells us that cosine of phi plus theta is equal to cosine of
both of those angles, the product of the cosines
of both of those angles. So cosine phi times cosine theta minus, so if this was a positive,
this is going to be a negative, if this was a negative,
this would be a positive, minus the product of the
sines of both of these angles, so sine of phi times sine of theta. And we already know some
of this information. We know what cosine of phi is. Cosine of phi is 7/25. So that is 7/25. We know what cosine of theta is. Cosine of theta is negative
square root of 3 over 2. Negative square root of 3 over 2, so we're going to take a
product here for this term. Now we need to figure out what sine of phi and sine of theta are. Lucky for us, we have
the Pythagorean identity. The Pythagorean identity tells
us that sine squared theta plus cosine squared theta is equal to 1. Or we could say that sine squared theta is equal to 1 minus cosine squared theta, or that sine of theta is
equal to the plus or minus square root of 1 minus
cosine squared theta. For example, we could use
this now to figure out what sine of theta is. We could say sine of
theta is going to be equal to the plus or minus square root of 1 minus cosine squared theta. Cosine squared theta is negative
square root of 3 over 2. If you square it, that's
going to be positive, and if you square 3, if you square the square root of 3, you're going to get 3, and if you square 2,
you're going to get 4. The plus or minus square
root of 1 minus 3/4, which is equal to the plus
or minus square root of 1/4, which is equal to plus or minus 1/2. Now which one is it going to be? Is sine of theta going to be
positive 1/2 or negative 1/2? To think about that, we could draw ourselves a
little unit circle here. That's my Y-axis. That is my X-axis. Let me draw a little unit circle here, as neatly as I can. A little unit circle right over there. Now what do they tell us about theta? They tell us that theta
is between pi and 2 pi, so it's between pi and 2 pi. So our angle, our terminal, I guess the terminal ray of the angle is going to sit, is going to be in the
third or fourth quadrants. We're saying sine of theta is equal either positive 1/2 or negative 1/2, so it's either positive 1/2, which could mean it's one of
these angles right over here, or it's negative 1/2, which means it's one of
these angles right over here. This tells us that we're in
the third or fourth quadrant, so sine of theta, we don't know if theta is this angle or if theta is this angle right over here, but we know if it's in the
third or fourth quadrant, the sine of it is going
to be non-positive. We know that for this theta, sine of theta is going to be negative 1/2, negative 1/2. So this right over here is negative 1/2. Now let's think about sine of phi. Sine of phi is going to be equal to plus or minus square root of 1 minus cosine of phi squared. Cosine of phi is 7/25, so that's 49 over 625. Let's see. What is that going to be? Let me do it over here. 625 over 625 minus 49 over 625, I just rewrote 1 as 625 over 625. That's going to be 625
minus 50 would be 575. That's going to be one more. That's going to be 576 over 625, so it's equal to the
plus or minus square root of 576 over 625. And let's see. I know what the square root of 625 is. It's 25. 576 , is it 24? 24 times 24, yup, it is 576. So this is equal to the
plus or minus 24/25. So sine of phi is 24/25. Remember, the sine of an
angle is the Y-coordinate of where the terminal ray
intersects the unit circle. So we're either looking
at one of these angles. We're either looking
at one of those angles, if the sine is a positive, so we're either looking at
this angle or that angle, or we're looking at a
terminal ray down here again. Now they tell us that phi
is a positive acute angle. So we know that we're dealing
actually with this scenario, this scenario right over here. Sine of phi is going to
be the positive 24/25, so that's 24/25. Now we just have to multiply the numbers and then do the subtraction. This is going to be equal to 725 ... let me just write it down. This is going to be equal to
negative 7 square roots of 3 over 25 times 2 is 50, over 50, minus, but then we're going to
have a negative out here, so we could say plus. Negative times negative is positive. And then 24 over 25
times 1/2 is 12 over 25, so plus 12 over 25. But actually, let me just write it over 50 since we have a 50 right over here. This is going to be plus 24 over 50. And so this is going to be equal to 24 minus 7 times the
square root of 3 over 50. And we are done.