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### Course: Trigonometry>Unit 4

Lesson 6: Challenging trigonometry problems

# 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?

• At Sal 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?
• 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.
• can you simplify that once more to:

7sqrt(3)/25?
• 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.
• 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.
• 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.
• 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.
• At , Sal says "we can assume [phi] is a positive acute angle..." What is he basing that statement on?
• 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.
• why costheta(costheta) not equal to cosquarethetasqaure.?
• "cos" is not a number, and "cos(theta)" is not multiplication.
• 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.
• 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.
• 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?
• 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)
• at why 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.