# Solving Trigonometry Problems: How to Find CE Using the Cosine Rule

• terryds
In summary: AE+BE = \cos(\alpha+\beta)## and ##AE-BE = \cos(\alpha-\beta)##. Finally, solve for CE.I get this ##(\frac{1}{(sin^2(\beta))})CE^2+(\frac{2\ L\cos \alpha}{sin \ \alpha})CE-L^2=0##Then, use the quadratic formula to solve CE??But if I use quadratic formula, there will be two solutions, but there is only one CE.And I doubt if the solution obtained will be the same as in the photo I gaveCB/ sin alpha = CA / sin beta
terryds

## Homework Statement

I don't understand how to get CE.

## Homework Equations

Trigonometric identities (maybe)

## The Attempt at a Solution

[/B]
CB^2 = AC^2 + AB^2 - 2 AC AB cos α
(CE/sin(β))^2 = (CE/sin(α))^2 + L^2 - 2 (CE/sin(α)) L cos α

which is going to be a quadratic equation and it's hard to obtain what CE is (using this method)

Try the usual way you derive the cosine rule - except solving for the altitude of the triangle instead of factoring it out.

I think using sine rule to find a side first is a faster way

Simon Bridge said:
Try the usual way you derive the cosine rule - except solving for the altitude of the triangle instead of factoring it out.

I get this ##(\frac{1}{(sin^2(\beta))})CE^2+(\frac{2\ L\cos \alpha}{sin \ \alpha})CE-L^2=0##

Then, use the quadratic formula to solve CE??
But if I use quadratic formula, there will be two solutions, but there is only one CE.
And I doubt if the solution obtained will be the same as in the photo I gave

PPHT123 said:
I think using sine rule to find a side first is a faster way

CB/ sin alpha = CA / sin beta

CE = CA sin alpha
CE = CB sin beta

Then whatt??

terryds said:
CB/ sin alpha = CA / sin beta

CE = CA sin alpha
CE = CB sin beta

Then whatt??
angle ACB = 180 - a -b
you can try to use L/sin(180-a-b) = ??

PPHT123 said:
angle ACB = 180 - a -b
you can try to use L/sin(180-a-b) = ??

Angle ACB = 180 - (a + b)
sin(180-(a+b)) = sin (a+b)

L/sin(a+b) = CB/ sin (a) = CA / sin (b)

Then??
How to make CE (the altitude) show up ?

terryds said:
Angle ACB = 180 - (a + b)
sin(180-(a+b)) = sin (a+b)

L/sin(a+b) = CB/ sin (a) = CA / sin (b)

Then??
How to make CE (the altitude) show up ?
You succeeded to express CB (or CA) in terms of L , a and b. Then what is CB sin(b) ?

PPHT123 said:
You succeeded to express CB (or CA) in terms of L , a and b. Then what is CB sin(b) ?
L/sin(a+b) = CE/sin(a)Sin(b)
CE = L sin(a) sin(b)/ sin(a+b)

Thanks a lot for your help bro

Simon Bridge
terryds said:

## Homework Statement

I don't understand how to get CE.

## Homework Equations

Trigonometric identities (maybe)

## The Attempt at a Solution

[/B]
CB^2 = AC^2 + AB^2 - 2 AC AB cos α
(CE/sin(β))^2 = (CE/sin(α))^2 + L^2 - 2 (CE/sin(α)) L cos α

which is going to be a quadratic equation and it's hard to obtain what CE is (using this method)
##AE/CE = \cot(\alpha)## and ##BE/CE = \cot(\beta)##. Now add and manipulate.

## 1. What is the cosine rule?

The cosine rule, also known as the law of cosines, is a formula used in trigonometry to find the length of a side or the measure of an angle in a triangle. It states that in a triangle with sides a, b, and c and opposite angles A, B, and C, the following relationship holds: c² = a² + b² - 2ab cos C.

## 2. When should I use the cosine rule?

The cosine rule is used when you need to find the length of a side or the measure of an angle in a triangle, and you know the lengths of two sides and the measure of the angle opposite the side you want to find. It is also useful when you have a non-right triangle, as it can be used to find the missing side or angle.

## 3. How do I apply the cosine rule to find CE?

To find CE using the cosine rule, you will need to know the lengths of two sides and the measure of the angle opposite CE. Plug these values into the formula c² = a² + b² - 2ab cos C, with c representing CE, a and b representing the known sides, and C representing the known angle. Solve for c to find the length of CE.

## 4. What is the difference between the cosine rule and the Pythagorean theorem?

The Pythagorean theorem is a special case of the cosine rule, where the triangle is a right triangle and the angle opposite the hypotenuse is always 90 degrees. In this case, the formula simplifies to c² = a² + b², which is the famous Pythagorean theorem. The cosine rule, on the other hand, can be used for any triangle, whether it is a right triangle or not.

## 5. Can the cosine rule be used to find angles?

Yes, the cosine rule can be used to find angles in a triangle. You can rearrange the formula c² = a² + b² - 2ab cos C to solve for the cosine of the angle C. Once you have the cosine, you can use the inverse cosine function or a calculator to find the measure of the angle C.

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