Visualizing the Area of a Triangle with Varying Angles

[zeion]

Homework Statement

Show that for all $$\theta \epsilon (0, \pi)$$, the area of a triangle with side lengths a and b with included angle $$\theta is A = \frac{1}{2} a b sin \theta$$. (Hint: You need to consider two cases)

Homework Equations

The Attempt at a Solution

I have just begun working on this problem.. not really sure where to start.

Does $$\theta \epsilon (0, \pi)$$ mean that the angle is > than 0 and < than pi?
Am I supposed to show that when the angle is less than or greater than the condition then the equation to find area is not valid?

 

[Dick]

Yes, that's what (0,pi) means. The only cases where the area is not ab*sin(theta) is where sin(theta) might be negative. They aren't in (0,pi). What's the area in that case?

 

[zeion]

The area is bh/2

 

[Dick]

The area is bh/2

They want you to give an answer in terms of the sides a and b. Not the base and the height.

 

[zeion]

Can you give me a little more hint -_-;

What are the two cases that I need to consider?

 

[Dick]

Can you give me a little more hint -_-;

What are the two cases that I need to consider?

Use trig and A=bh/2. What's h in terms of a and the included angle? Draw a right triangle. And I'm really not sure what the 'two cases' they are talking about are.

 

[zeion]

h = b(sin theta)
or
h = b(sin 180 - theta)

 

[Dick]

sin(theta) and sin(180-theta) are the same number. Aren't they?

 

[zeion]

So can I show this by drawing a picture?

 

[Dick]

There's a variety of ways to draw a picture to show sin(pi-x)=sin(x). Which sort did you have in mind? How do you picture sin(x)?