Trigonometry : 30-60-90 Triangles
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Example Question #1 : 30 60 90 Triangles
In a 30-60-90 triangle, the length of the side opposite the 
By definition, the length of the hypotenuse is twice the length of the side opposite the 
Recall that the hypotenuse is the side opposite the 
Thus, using the equation below, where ss represents the short side (that opposite the 
Plugging in our values for the short side we find the hypotenuse as follows:
Example Question #1 : 30 60 90 Triangles
A triangle has three angles 
A triangle with a 
Therefore, to make the above statement true 
Example Question #2 : 30 60 90 Triangles
Triangle 
What is the height of the triangle?
An equilateral triangle has internal angles of 60°, so the sin of one of those angles is equivalent to the height of the triangle divided by the side length,
so..
Example Question #2 : 30 60 90 Triangles
In a 
Based on the 30-60-90 identity, the measure of the side opposite the 30 degree angle is doubled to get the hypotenuse.
Therefore,
Example Question #1 : 30 60 90 Triangles
What is the height of an equilateral triangle with side length 8?
The altitude of an equilateral triangle splits it into two 30-60-90 triangles. The height of the triangle is the longer leg of the 30-60-90 triangle. If the hypotenuse is 8, the longer leg is 
To double check the answer use the Pythagorean Thereom:
Example Question #6 : 30 60 90 Triangles
What is the ratio of the side opposite the 
Step 1: Locate the side that is opposite the 
The shortest side is opposite the 
Step 2: Recall the ratio of the sides of a 
From the shortest side, the ratio is 
The ratio of the short side to the hypotenuse is
Example Question #4 : 30 60 90 Triangles
It is known that the smallest side of a 30-60-90 triangle is 5.
Find 
We know that in a 30-60=90 triangle, the smallest side corresponds to the side opposite the 30 degree angle.
Additionally, we know that the hypotenuse is 2 times the value of the smallest side, so in this case, that is 10.
The formula for
Example Question #5 : 30 60 90 Triangles
It is known that for a 30-60-90 triangle,
Find the area of the triangle.
Note:
First, we know that in a 30-60-90 triangle,
Also, the base is the smallest side times 
The height is just the smallest side, 
Substituting these values into the formula given for area of a triangle, we obtain the answer 
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