Introduction to Trigonometry

Right Triangle Trigonometry

Introduction  Trigonometry is a branch of mathematics that uses triangles to help you solve problems.

 Trig is useful to surveyors, engineers, navigators, and machinists (and others too.)

Labeling Right Triangles

Labeling Right Triangles  The most important skill you need right now is the ability to correctly label the sides of a right triangle.

 The names of the sides are:  the hypotenuse  the opposite side  the adjacent side

Labeling Right Triangles  The hypotenuse is easy to locate because it is always found across from the right angle.

Since this side is across from the right angle, this must be the hypotenuse.

Here is the right angle ...

Labeling Right Triangles  Before you label the other two sides you must have a

reference angle

selected.

 It can be

either

of the two acute angles.

 In the triangle below, let’s pick angle B as the reference angle.

B This will be our reference angle...

C A

Labeling Right Triangles  Remember, angle B is our reference angle.

 The hypotenuse is side BC because it is across from the right angle.

B (ref. angle)

hypotenuse C A

Labeling Right Triangles  Side AC is across from our reference angle B. So it is labeled:

opposite

.

B (ref. angle)

hypotenuse A C opposite

Labeling Right Triangles Adjacent means beside or next to  The only side unnamed is side AB. This must be the

adjacent

side.

adjacent

B (ref. angle)

hypotenuse A opposite C

Labeling Right Triangles  Let’s put it all together.

 Given that angle B is the reference angle, here is how you must label the triangle:

B (ref. angle)

hypotenuse adjacent C A opposite

Labeling Right Triangles  Given the same triangle, how would the sides be labeled if angle C were the reference angle?

 Will there be any difference?

Labeling Right Triangles  Angle C is now the reference angle.

 Side BC is still the hypotenuse since it is across from the right angle.

B hypotenuse

C (ref. angle)

A

Labeling Right Triangles  However, side AB is now the side

opposite since it is across from angle C.

B opposite hypotenuse

C (ref. angle)

A

Labeling Right Triangles  That leaves side AC to be labeled as the

adjacent

side.

B opposite A hypotenuse

C (ref. angle)

adjacent

Labeling Right Triangles  Let’s put it all together.

 Given that angle C is the reference angle, here is how you must label the triangle: B hypotenuse opposite

C (ref. angle)

A adjacent

Labeling Practice  Given that angle X is the reference angle, label all three sides of triangle WXY.

 Do this on your own. Click to see the answers when you are ready.

W

X

Y

Labeling Practice  How did you do?

 Click to try another one...

W opposite Y adjacent hypotenuse

X

Labeling Practice  Given that angle R is the reference angle, label the triangle’s sides.

 Click to see the correct answers.

R

T S

Labeling Practice  The answers are shown below:

R

adjacent T hypotenuse opposite S

Which side will never be the reference angle?

The right angle What are the labels?

Hypotenuse, opposite, and adjacent