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

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. A B C This will be our reference angle...

Labeling Right Triangles Remember, angle B is our reference angle. The hypotenuse is side BC because it is across from the right angle. A B (ref. angle) C hypotenuse

Labeling Right Triangles Side AC is across from our reference angle B. So it is labeled: opposite. A B (ref. angle) C opposite hypotenuse

Labeling Right Triangles The only side unnamed is side AB. This must be the adjacent side. ( ) A B (ref. angle) C adjacent hypotenuse opposite

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: A B (ref. angle) C hypotenuse opposite adjacent

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. A B C (ref. angle) hypotenuse

Labeling Right Triangles However, side AB is now the side opposite since it is across from angle C. A B C (ref. angle) opposite hypotenuse

Labeling Right Triangles That leaves side AC to be labeled as the adjacent side. A B C (ref. angle) adjacent hypotenuse opposite

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: A B C (ref. angle) hypotenuse opposite adjacent

You Try 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

You Try - Answer How did you do? Click to try another one... W X Y hypotenuse opposite adjacent

You Try Given that angle R is the reference angle, label the triangle’s sides. Click to see the correct answers. R S T

The answers are shown below: R S T hypotenuse opposite adjacent You Try - Answer

Which angle will never be the reference angle? What are the labels? The right angle Hypotenuse, opposite, and adjacent

Trigonometric Ratios There are 3 basic trigonometric ratios called the sine, cosine, and tangent We will be defining these 3 ratios and then finding the ratios given a triangle

Sine Rato The sine is the ratio of the opposite over the hypotenuse We will be using Θ (theta) to represent our reference angle We abbreviate sine of Θ as

Example Given the triangle, find sin x Given this triangle the side opposite x has a length of 5 and the hypotenuse has a length of 13 Remember, sine is the opp/hyp so W X Y

Cosine Ratio The cosine is the ratio of the adjacent side over the hypotenuse We abbreviate cosine of Θ as

Example

Tangent Ratio The tangent is the ratio of the opposite side over the adjacent side We abbreviate tangent as tan

Example

Summary There is a shortcut to memorizing your 3 trigonometric ratios. It is called SOHCAHTOA SOH : Sine = opp/hyp CAH : Cosine = adj/hyp TOA : Tangent = opp/adj