The problemFind side length using trig

[caters]

Homework Statement

$AB = 20 cm$, $m∠A = 30°$ , and $m∠C = 45°$ . Express the number of centimeters in the length of $BC$ in simplest radical form.

Homework Equations

$sin A = sin C$

The Attempt at a Solution

$AB = 20, BC = x$

D is the point where this obtuse triangle separates into 2 right triangles

$BD/20 = sin A$
$AD/20 = cos A$

30-60-90 triangle
$1:2:\sqrt{3}$

BD is 10 according to this ratio which means that sin A is 1/2 and AD would be $20\sqrt{3}$

sin C is the same but for a 45-45-90 triangle instead.

45-45-90 triangle
$1:1:\sqrt{2}$

But here is where I am stuck. I am trying to find the side lengths of the 45-45-90 triangle with the trigonometric ratios being the same for both triangles but the angles being different so that I know the hypotenuse BC. But I don't know what side lengths will give me the trigonometric ratios being the same and the $1:2:\sqrt{3}$ and $1:1:\sqrt{2}$ side length ratios being true.

 

[SammyS]

Homework Statement

$AB = 20 cm$, $m∠A = 30°$ , and $m∠C = 45°$ . Express the number of centimeters in the length of $BC$ in simplest radical form.

Homework Equations

$sin A = sin C$

The Attempt at a Solution

$AB = 20, BC = x$

D is the point where this obtuse triangle separates into 2 right triangles

$BD/20 = sin A$
$AD/20 = cos A$

30-60-90 triangle
$1:2:\sqrt{3}$

BD is 10 according to this ratio which means that sin A is 1/2 and AD would be $20\sqrt{3}$

sin C is the same but for a 45-45-90 triangle instead.

45-45-90 triangle
$1:1:\sqrt{2}$

But here is where I am stuck. I am trying to find the side lengths of the 45-45-90 triangle with the trigonometric ratios being the same for both triangles but the angles being different so that I know the hypotenuse BC. But I don't know what side lengths will give me the trigonometric ratios being the same and the $1:2:\sqrt{3}$ and $1:1:\sqrt{2}$ side length ratios being true.

Your result for the length of side AD is incorrect.

 

[RUber]

I second what SammyS says.
Once you know BD though, you should quickly know BC, since BC is the hypotenuse of the 45-45-90 triangle, right? You have already written the appropriate ratio for the length of a side to the hypotenuse of this triangle.

 

[HallsofIvy]

Homework Statement

$AB = 20 cm$, $m∠A = 30°$ , and $m∠C = 45°$ . Express the number of centimeters in the length of $BC$ in simplest radical form.

Homework Equations

$sin A = sin C$

No, with A= 30 degrees and B= 45 degrees sin(A) is definitely not equal to sin(C)!
Perhaps you meant the sine law:
$$\frac{sin(A)}{BC}= \frac{sin(B)}{AC}= \frac{sin(C)}{AB}$$
The cosine law might also be useful:
$$(AB)^2= (AC)^2+ (BC)^2- 2(AC)(BC) cos(C)$$
and equivalent formulas for the other two angles.

3. The Attempt at a Solution

$AB = 20, BC = x$

D is the point where this obtuse triangle separates into 2 right triangles

$BD/20 = sin A$
$AD/20 = cos A$

30-60-90 triangle
$1:2:\sqrt{3}$

BD is 10 according to this ratio which means that sin A is 1/2 and AD would be $20\sqrt{3}$

sin C is the same but for a 45-45-90 triangle instead.

45-45-90 triangle
$1:1:\sqrt{2}$

But here is where I am stuck. I am trying to find the side lengths of the 45-45-90 triangle with the trigonometric ratios being the same for both triangles but the angles being different so that I know the hypotenuse BC. But I don't know what side lengths will give me the trigonometric ratios being the same and the $1:2:\sqrt{3}$ and $1:1:\sqrt{2}$ side length ratios being true.

 

[Saracen Rue]

I'm kind of new here... can someone please tell me what the dollar signs represent?

 

[Mentallic]

I'm kind of new here... can someone please tell me what the dollar signs represent?

They should be double dollar signs, and they're just a simpler alternative to the $$tags.$$

 

[SammyS]

I'm kind of new here... can someone please tell me what the dollar signs represent?

To further explain:

The dollar signs (as well as the # sign ) are used as tags to enable using "LaTeX" for displaying mathematical expressions.

On this site, those should be doubled.

For instance, $20\sqrt{3}$ , should have been $\$ 20\text{\sqrt } 3$ ## $$ ## .

It would display $$20\sqrt{3}$$