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## Trigonometry

### Course: Trigonometry>Unit 4

Lesson 2: Sinusoidal equations

# Cosine equation algebraic solution set

Solve a cosine equation with an infinite number of solutions. Use trig identities to represent the whole solution set. Created by Sal Khan.

## Want to join the conversation?

• brain hurt
• 8cos(12x)+4=−4 at 1 point the explanation of the mastery question says
Since −1-1−1minus, 1 is a trough, it is the only solution within this interval.
What is meant by trough? Does it indicate the lowest value in the wave graph? What about the equation would tell me that? (assuming that is the correct interpretation of the word 'trough').
• It would be helpful to learn the sinusoidal expression's anatomy before doing this lesson.

I assume a "trough" would be the minimum part of the graph, which can be calculated by subtracting the absolute value of the amplitude from the midline.

For example, in the function 3 cos(x)+5, 3 is the amplitude and 5 is the midline.
Without even graphing this function, it is clear that the maximum value is 5 + 3, or 8, and the minimum value as 5 - 3, or 2.

You can graph this function and you'll see I'm correct.

I am not entirely sure what the question you had was, but hopefully this helped you (assuming you're still stuck 2 years later lol)
(1 vote)
• How does Sal get 1/8 and -1/8? Does that mean that there is an invisible 1 before the cosine?
• There is always an invisible 1, because 1 times anything = the same as before, so I can multiply anything by 1.

However, this is not the reason he used 1/8. Simply put, multiplying the rest of the equation by 1/8 is the same as dividing by 8. The answer will be the same.
Sal probably wrote it as a fraction so it looks a bit neater.

This is very basic math stuff I'm talking about, maybe 4-5th grade.
I highly recommend you revisit those basic math essentials before doing Trigonometry, or you will be confused and stuck on a lot of things.
• why negative cos theta is equal to positive cos theta?
• −cos(𝜃) ≠ cos(𝜃)

But cos(−𝜃) = cos(𝜃)
Draw 𝜃 and −𝜃 in the unit circle, and you'll see that they have the same 𝑥-coordinate.
• Where did the -(𝜋/4)n and +(𝜋/4)n come from?
• When Sal divides both sides by 8 and -8 the omitted step will be (I'll write for 8 but it's applies to -8 as well): (cos^-1(-1/6)-2pi * n)/8. So if you remember properties of ratios(sorry don't remember correct term) it'll could be also written as: (cos^-1(-1/6))/8 - (2pi * n)/8. In the second term(subtrahend) both 2(in numerator) and 8(in denominator) could be divided by 2 leaving us 1 and 4 respectively, which leads us to (1pi *n)/4 and since 1 is often omitted as a coefficient we have (pi * n)/4.
• It's incorrect to apply arccos(cos(θ))=θ to cos(-8x+2pi*n), as "-8x+2pi*n" represents angle which is outside of range of arccos function. "-8x+2pi*n" belongs to the third quarter of the unit circle, while arccos is able to spit out only angles that belong to the first and the second quarter of the unit circle.
• It's the other way around.

−8𝑥 could very well be outside the arccos range, which is why we can't say
arccos(cos(−8𝑥)) = −8𝑥

But for any given 𝑥 there will always be exactly one integer 𝑛, such that
arccos(cos(−8𝑥 + 2𝜋𝑛)) = −8𝑥 + 2𝜋𝑛
• if my answer are x=1/8cos^-1(-1/6) + pi/4n and x=-1/8cos^-1(-1/6) + pi/4n also correct ? because if you add or minus 2pi it also the same ? i normally add instead of minus
• i belive that is correct.
(1 vote)
• In one of the practices, there is a question I am confused about.

The question was 6sin(3x)+1=7 (In degrees)
I got the solution sets x = 30 * 360n & x = (180 - 30) * 360n

The correct answer out of the choices was x = 30 + 120n. However, when you take the sin(30 + 120n) and have n take on the values 2 + 3x (x being an integer =< 0) you get -1, instead of 0.5, what you should get.

Can someone please tell me what I am not understanding?
• n take on the values 2 + 3x.So this would mean

x = (n-2)/3 which does not make sense. We want x to be the subject not n.

You need let n be an integer.

So the correct answer is x = 30 + 120n.

n = 1

=> x = 150

6*sin(450)+1 = 7 since sin(450)=sin(90)=1

I have various answers on sinusoidal equations in my profile that might be useful to look at.
• Hi, I was just wondering why we had to include the x values that satisfy the equation cos(-8x)= -1/6 in our final answer. I do understand that cosine is an even function, therefore cos(x) = cos(-x), but aren't we only searching for the x values that satisfy cos(8x)=-1/6? timestamp at approximately