# Solving an Equation with Cosine

• salman213
In summary, the conversation discusses the equation x = rcos(a)cos(a) and the attempt to solve it using various methods. However, it is realized that the equation has no solution in the real numbers due to an error in squaring cos(a). The conversation ends with the person finding the correct interpretation and understanding the solution.
salman213
1.x = rcos(a)cos(a)

x is known
r is known

I can't seem to get this!

x = -2
r = 7

therefore

-2/7 = cos(a)*cos(a)

so..

-2/7 = cos^2(a)

i know cos^2(a) = 1 + cos2a / 2

-2/7 = 1 + cos2a / 2

so
-4/7 -1 = cos2a

I cannot take cos inverse of this value since it is "greater" than 1 (-1.57)

the book says the angle is 106.6 degrees!

HELP!

## The Attempt at a Solution

salman213 said:

i know cos^2(a) = 1 + cos2a / 2

Nope, it's $\cos^2(a)=(1+\cos(2a))/2$. Both terms are divided by 2.

yea sorry that's what i meant

but it still doesn't work out..

Egad, you have an even more serious problem that I missed.

salman213 said:
-2/7 = cos(a)*cos(a)

You can't square cos(a) and get a negative number. This equation has no solution in the real numbers.

NEVERMIND I figured it out, i was interpreting the solution incorrectly...

Thanks anyways!

salman213 said:

"egad" is an exclamation like "crikey" or "blimey".

## 1. What is an equation with cosine?

An equation with cosine is an equation that involves the trigonometric function cosine (cos). These types of equations typically involve finding the value of an unknown angle or side length in a right triangle.

## 2. How do I solve an equation with cosine?

To solve an equation with cosine, you can use the inverse cosine (arccos) function or the cosine ratio (adjacent/hypotenuse) to find the value of the unknown angle. You can also use the Pythagorean theorem (a^2 + b^2 = c^2) to find the value of the unknown side length.

## 3. What are the common steps to solving an equation with cosine?

The common steps to solving an equation with cosine are: 1) identify the given information and what you are trying to solve for, 2) determine which trigonometric function (cosine, sine, or tangent) is involved, 3) use the appropriate formula or ratio to set up the equation, 4) solve for the unknown angle or side length, and 5) check your answer by plugging it back into the original equation.

## 4. Can an equation with cosine have multiple solutions?

Yes, an equation with cosine can have multiple solutions. This is because cosine is a periodic function, meaning it repeats itself at regular intervals. So, the solutions to an equation with cosine can vary by multiples of 360 degrees or 2π radians.

## 5. Are there any special cases when solving an equation with cosine?

Yes, there are a few special cases to keep in mind when solving an equation with cosine. These include: 1) when the given information is an angle in the third or fourth quadrant, you may need to add 180 degrees or π radians to the solution, 2) when the given information is a ratio, you may need to use the inverse cosine function, and 3) when solving for an angle, the solution should be in the range of 0 to 360 degrees or 0 to 2π radians.

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