30-60-90 Triangles - Trigonometry

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Question

In a triangle, the side opposite the degree angle is . How long is the side opposite the degree angle?

Answer

Based on the 30-60-90 identity, the measure of the side opposite the 30 degree angle is doubled to get the hypotenuse.

Therefore,

$h=15\cdot2$

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Question

In a 30-60-90 triangle, the length of the side opposite the $30^\circ$ angle is . What is the length of the hypotenuse?

Answer

By definition, the length of the hypotenuse is twice the length of the side opposite the $30^\circ$ angle.

Recall that the hypotenuse is the side opposite the $90^\circ$ angle.

Thus, using the equation below, where ss represents the short side (that opposite the $30^\circ$ angle) we get:

$H=2\cdot SS$

Plugging in our values for the short side we find the hypotenuse as follows:

$5\cdot2 = 10$

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Question

A triangle has three angles , and such that and . The side opposite to measures units in length. How long is the side opposite of ?

Answer

A triangle with a angle relation is a , , degree triangle. The side opposite the smallest angle of a triangle is the shortest side, of length . The side opposite the largest angle is the longest side, measuring twice the length of the shortest side for this triangle, units.

$\frac{1}{2}=\sin{30}$

$=\frac{opp}{hyp}=\frac{3}{h}$

Therefore, to make the above statement true .

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Question

Triangle is equilateral with a side length of .

What is the height of the triangle?

Answer

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,

$\sin(60)=\frac{height}{side}$

so..

$h=s*\sin(60)=s\frac{\sqrt3}{2}$

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Question

What is the height of an equilateral triangle with side length 8?

Answer

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 $4\sqrt{3}$ .

To double check the answer use the Pythagorean Thereom:

$\(4\sqrt{3})^2+4^2= \16(3)+16= \48+16=64 \ \sqrt{64}=8$

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Question

What is the ratio of the side opposite the $30^{\circ}$ angle to the hypotenuse?

Answer

Step 1: Locate the side that is opposite the $30^{\circ}$ side..

The shortest side is opposite the $30^\circ$ angle. Let's say that this side has length .

Step 2: Recall the ratio of the sides of a $30^\circ-60^\circ-90^\circ$ triangle:

From the shortest side, the ratio is $1:\sqrt {3}:2$ .

is the hypotenuse, which is twice as big as the shortest side..

The ratio of the short side to the hypotenuse is

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Question

It is known that the smallest side of a 30-60-90 triangle is 5.

Find $\sin(30\degree)$ .

Answer

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

$\sin=\frac{opposite}{hypotenuse}$ , so $\frac{5}{10}$ or $\frac{1}{2}$ .

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Question

It is known that for a 30-60-90 triangle,

$\sin(30\degree)=\frac{6}{12}$ .

Find the area of the triangle.

Note:

$\textup{Area}=\frac{1}{2}bh$

Answer

First, we know that in a 30-60-90 triangle,

$\sin(30)=\frac{height}{hypotenuse}$ .

Also, the base is the smallest side times $\sqrt3$ , so in our case it is $6\sqrt3$ .

The height is just the smallest side, .

Substituting these values into the formula given for area of a triangle, we obtain the answer $18\sqrt3$ .

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