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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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,

$h=15\cdot2$

Compare your answer with the correct one above

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:

$5\cdot2 = 10$

Compare your answer with the correct one above

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..

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

Compare your answer with the correct one above

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$

Compare your answer with the correct one above

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

Compare your answer with the correct one above

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 $18\sqrt3$ .

Compare your answer with the correct one above

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

Compare your answer with the correct one above

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$

Compare your answer with the correct one above

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,

$h=15\cdot2$

Compare your answer with the correct one above

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:

$5\cdot2 = 10$

Compare your answer with the correct one above

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