Known in triangle ABC, B 30, AB 2 root number 3, AC 2, find triangle ABC area?

Hello, the idea of this problem is to use the sine theorem and the triangle area formula.

Triangular area = 1 2 a b sinc 1 2 a c sinb = 1 2 b c sina).

In this question, c = 2 times the root number 3 is given

b=2b=30°

Therefore, it is necessary to find the sina

The sine theorem formula (a sina = b sinb = c sinc sinc with b sinb = c sinc

1 2 = double root number three sinc

Solution. sinc = the root of the second is three.

Because c*sinb b c

Old. c = 60° or 120°

then a = 90° or 30°

Triangle area = double root number three or root number three.

This knowledge point is the compulsory five knowledge of mathematics in the high school education version.

If you don't study, you don't have to study. Hope. Thank you.

From the sinusoidal theorem: ac sinb ab sinc, i.e., sinc=absinb ac=2 3sin30° 2= 3 2, we can see that c=60° or c=120°

When c = 60° and a = 90°, then s abc = ab* ac 2 = 2 3 * 2 2 = 2 3

When c = 120°, a = 30 ° = b, bc = ac = 2, then s abc = 1 2 * ac * bcsin120 ° = 1 2 * 2 * 2 * 3 2 = 3

From the sinusoidal theorem: ac sinb ab sinc, i.e., sinc=absinb ac=2 3sin30° 2= 3 2, we can see that c=60° or c=120°

When c = 60° and a = 90°, then s abc = ab* ac 2 = 2 3 * 2 2 = 2 3

When c = 120°, a = 30 ° = b, bc = ac = 2, then s abc = 1 2 * ac * bcsin120 ° = 1 2 * 2 * 2 * 3 2 = 3

a=30°，b=135°，c=√6-√2。

Solution: Since cos15°=cos(45°-30°)=cos45cos30+sin45sin30=(6+ 2) 4, then according to the cosine theorem, c = a + b -2abcosc

So c = 6- 2

Then according to the sinusoidal theorem, a sina = b sinb = c sinc, we can get, 2 sina = ( 6- 2) [( 6- 2) 4] = 4, then sina = 1 2, because a then a = 30°, then b = 180-a-c = 135° i.e. a = 30°, b = 135°, c = 6- 2.

cos15=cos(45-30)

cos45cos30+sin45sin30=(√6+√2)/4

c²=a²+b²-2abcosc

c=√6-√2

sin15=sin(45-30)=sin45cos30-cos45sin30=(√6-√2)/4

a/sina=c/sinc

2/sina=(√6-√2)/[(√6-√2)/4]=4sina=1/2

Because a is an acute angle to a.

So a=30

b=180-a-c

So c = 6- 2

a = 30 degrees.

b = 135 degrees.

b/sinb=c/sinc

2 1 2 = 2 root number 3 sinc

4sinc=2 root number 3

sinc=root number 3 2

So, c = 120 degrees or 60 degrees.

When C = 120 degrees, the angle A = Angle B = 30 degrees, and AB is not equal to AC Therefore, the triangle ABC is not an isosceles triangle and does not fit the topic.

When c = 60 degrees.

Angle a = 90 degrees.

As can be seen from the question, C 2 root number 3, B 2.

By sinb sinc 2 root number 3. Get C 60 or 120 degrees!

1.When C 60, A root number 3

2.When C 120, A 30, then S 1 2bcsina root number 3 is actually focused on the flexible use of formulas and careful calculations. If you have any questions, you can ask them and hope to adopt them.

According to the triangular cosine theorem a sina = b sinb sinc = c sinc , ac sinb = ab sinc, and b = 30°, ab = 2 root number 3, ac = 2, then 2 sin30° = 2 root number 3 sinc , we can get sinc = root number 3 2, from which we know c = 60°, then we know a = 90° then s = 1 2 abac = 1 2x2 root number 3x2 = 2 root number 3 complete.

From the sinusoidal theorem: ac sinb ab sinc, i.e., sinc=absinb ac=2 3sin30° 2= 3 2, we can see that c=60° or c=120°

When c = 60° and a = 90°, then s abc = ab* ac 2 = 2 3 * 2 2 = 2 3

When c = 120°, a = 30 ° = b, bc = ac = 2, then s abc = 1 2 * ac * bcsin120 ° = 1 2 * 2 * 2 * 3 2 = 3

ac ab=(root number 3) 3=tan b (or from the sinusoidal theorem.)

Passing point C is CD vertical AB, COS30 root number 3 2, AD can be proved to be AB, then the area is 2 root number 3

It is a right triangle.

The a-angle is 90 degrees.

The area is 2 times the root number 3

Sine Theorem

In a triangle, the ratio of each side to the sine of the opposite angle is equal.

i.e. a sina = b sinb = c sinc = 2r (2r is a constant quantity in the same triangle, twice the radius of the circumscribed circle of this triangle).

This theorem is true for any triangle ABC.

a/sina=b/sinb=c/sinc

Therefore, the angle c is 60 degrees (or 120 degrees), so a is a right angle (or 30 degrees), so the area is 2 times the root number 3 (or root number 3).

Sine theorem:

ab/sinc=ac/sinb

sinc=absinb/ac=2sin120°/(2√3)=2*(√3/2)/(2√3)=1/2

b = 120° is the obtuse angle.

c is an acute angle. c=30°

a=180°-b-c=30°

Height on the bottom AC = absin30° = 2*1 2=1 Area = 1 2*2 3*1 = 3

Just cross the point B and make the perpendicular line perpendicular to AC. I don't want to count without a pen.

b sinb=c sinc ac sinb=ab sinc so sinc sinc=2 root number 3 c = 60° or 120° Then discuss it in different situations