# High School Math : Triangles

## Example Questions

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### Example Question #83 : Trigonometry

In this figure, side , and . What is the value of angle ?

Undefined

Explanation:

Since , we know we are working with a right triangle.

That means that .

In this problem, that would be:

Plug in our given values:

### Example Question #2 : Triangles

Let ABC be a right triangle with sides  = 3 inches,  = 4 inches, and  = 5 inches. In degrees, what is the  where  is the angle opposite of side ?

Explanation:

We are looking for . Remember the definition of  in a right triangle is the length of the opposite side divided by the length of the hypotenuse.

So therefore, without figuring out  we can find

### Example Question #1 : Trigonometric Functions

In this figure, if angle , side , and side , what is the measure of angle ?

Undefined

Explanation:

Since , we know we are working with a right triangle.

That means that .

In this problem, that would be:

Plug in our given values:

### Example Question #6 : Graphing The Sine And Cosine Functions

In this figure, , and . What is the value of angle ?

Undefined

Explanation:

Notice that these sides fit the pattern of a 30:60:90 right triangle: .

In this case, .

Since angle  is opposite , it must be .

### Example Question #7 : Graphing The Sine And Cosine Functions

A triangle has angles of . If the side opposite the angle is , what is the length of the side opposite ?

Explanation:

The pattern for is that the sides will be .

If the side opposite is , then the side opposite will be .

### Example Question #3 : Triangles

In , , and . To the nearest tenth, what is  ?

A triangle with these sidelengths cannot exist.

Explanation:

By the Triangle Inequality, this triangle can exist, since .

By the Law of Cosines:

Substitute the sidelengths and solve for  :

### Example Question #11 : Graphing The Sine And Cosine Functions

A triangle has sides of length 12, 17, and 22. Of the measures of the three interior angles, which is the greatest of the three?

Explanation:

We can apply the Law of Cosines to find the measure of this angle, which we will call :

The widest angle will be opposite the side of length 22, so we will set:

### Example Question #12 : Graphing The Sine And Cosine Functions

In  , , and . To the nearest tenth, what is ?

A triangle with these characteristics cannot exist.

Explanation:

By the Law of Cosines:

or, equivalently,

Substitute:

### Example Question #4 : Triangles

In this figure, angle  and side . If angle , what is the length of side ?

Explanation:

For this problem, use the law of sines:

.

In this case, we have values that we can plug in:

Cross multiply:

Multiply both sides by :

### Example Question #5 : Triangles

In this figure  and . If , what is ?