How Do I Deal with Angles?
Trigonometry in the Geosciences

Introduction to Trigonometry

How steep is this hillside, and will it fail? How high is that mountain? These sorts of questions pop up all over in geosciences, from plate tectonics to maps to ocean waves, and they require you to find either an angle or a distance. To do this, we often use trigonometry, which is much easier when a right triangle is involved.

A right triangle (like the one in the figure to the right) has one angle that is 90°. The other two angles are always less than 90° and together add up to 90°. Note that the triangle on the right has three sides (A, B, and H) and three angles (a, b, and c). The side "opposite" an angle (in this case) is labeled with a capital letter corresponding to the label on the angle. The side opposite the right angle, H, is always the longest side and is called the hypotenuse.

Some Rules/Guidelines for Trigonometry of Right Triangles

We can use ratios (or the quotient) of the lengths of a right triangle's sides to figure out the angles in a right triangle. Three trigonometric ratios that we use in the geosciences are called the sine, the cosine, and the tangent, although they are often abbreviated sin, cos, and tan, respectively.

Because the trigonometric functions come up over and over, it is a good idea to figure out a way to remember their definitions. For most students (and, to be honest, your professors!) the mnemonic SOH-CAH-TOA (said "so-kah-toe-ahh") is helpful.

The mnemonic device uses the first letters in the following equations:

`text{Sin}=\frac{text{Opposite}}{text{Hypotenuse}` (this is SOH, where S = sin, O = opposite, and H = hypotenuse)

`text{Cos}=\frac{text{Adjacent}}{text{Hypotenuse}}` (this is CAH, where C = cos, A = adjacent, and H = hypotenuse)

`text{Tan}=\frac{text{Opposite}}{text{Adjacent}}` (this is TOA, where T = tan, O = opposite, and A = adjacent)

For more information about what is meant by "adjacent" and "opposite," see the problems below

You can use the equations above to determine the angle (or the sides) in any right triangle (like the figure to the right). An example of this is how to determine the size of the angle a:

`text{sin a}=\frac{text{length A}}{text{length H}}`, `text{cos a}=\frac{text{length B}}{text{length H}}`, `text{tan a}=\frac{text{length A}}{text{length B}}`

Because these are equations, they can be rearranged so that you can also figure out the length of a side (if you know the angle a in the example above)!

Calculating Distances and Angles Using Trigonometry

In order to do trigonometry calculations, you should have a calculator with buttons for tan, cos, and sin. Before you begin, make sure that your calculator is in degree mode, not radian mode. If you need help with this, skip to the How Do I Do This on My Calculator? section.

Show the difference between a radian and a degree

How do I calculate distance?

Often when figuring out the distance to a faraway object, you know an angle and one length but need to know a second leg of a triangle.

Measuring bank slope

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To figure out the missing side, you will substitute values into one of the trig formulas above, and solve for the missing value. For example, if you know the side adjacent to the angle and need to know the hypotenuse, you can use the formula for cosine and solve for the length you need.

Show a specific example

Often in the geosciences, we want to know the distance to a hill or other object. We can do this by triangulation. So if you measure the angle to a hill to be 135 degrees from north, and then walk 200 meters perpendicular to that and measure the angle to be 120 degrees from north, how far from the second position is the hill?

You will probably need to make a sketch of this problem to solve it, and to figure out which of the equations—sine, cosine, or tangent—you will need.
Show me how to do it
Choose the correct formula, given the angle and side you know and which side you want.
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Rearrange the equation so what you know is on one side and what you don't know is on the other, then substitute in the known values.
Show me how to do this
Use your calculator to determine the value of the trigonometric function (i.e. sine, cosine, or tangent) and solve for the unknown value.
Show me how to do this

How do I calculate the angle?

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Often geologists need to know a gradient of a road up a hill or the angle that rocks are tilted; to calculate these angles you will need to know the lengths of two sides of triangle. From that you can determine the sine, cosine, or tangent of the angle you need.

If you have calculated the sine, cosine, or tangent of the angle, and you need to know how big it is (in degrees), you will have to use the inverse functions on your calculator. (If you don't know how to use these on your calculator, jump to the How Do I Do This on My Calculator? section.) For example, if you want to find the size of an angle with a sine of 0.6, you need to find inverse sine of 0.6 (written sin^-1(0.6)). Try doing this on your calculator. You should get 36.87, so the angle in this case is a bit less than 37 degrees.

Sin^-1 is called "inverse sine" because it does the opposite of the sine function. In the same way, the inverse cosine (cos^-1) will give the value of an angle if you know its cosine, and tan^-1 will give you the angle if you know the tangent. In this way, you can find the size of any unknown angle of a right triangle if you just know 2 sides of that triangle.

Show a specific example

You are in charge of building a road to the top of a hill. From topographic maps, you know that the hill is 150 feet high, and the horizontal distance on the map is 1300 feet. How steep will the road be?

Figure out what is known, and which angle you want, and then pick the correct equation, for sine, cosine or tangent.
Show me how to do this
Substitute the known sides into the equation you picked.
Show me how to do this
Use the inverse function to solve this equation for the angle that you need.
Show me how to do this

Are you ready to try some sample problems? Go on to the Trig Practice Problems page

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How Do I Do This on My Calculator?

Often not knowing how to use your calculator is the hardest thing about trig problems. To figure out if your calculator is in degrees or radians mode, try finding the sin(30). If your calculator gives a result of 0.5, it is in degrees mode, and if you get -.988031, it is in radian mode, and you need to switch it!

To use the inverse buttons, typically you will need to press a button labeled 2^nd on your calculator and then the sin, cos, or tan button. This produces sin^-1 on the calculator screen.

Many of the calculator companies have online tutorials if you need more help, including TI, Sharp, and HP.

Why Does It Work?

Three similar triangles. Angle c is the same in each case and so is the tangent of c.

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While you may want to just "plug and chug" and use the trigonometry equations without understanding them, some folks find it helpful to understand why they work. The secret is in something called similar triangles. Similar triangles are triangles with three identical angles but different length sides, such as the ones in the diagram to the right.

Similar triangles are just "blown-up" or "shrunk" (the correct term is scaled) versions of each other and so the ratio of the length of the sides remains the same. In other words, the shortest side divided by the hypotenuse will always give the same result if the triangles are similar. You may already recognize that this ratio is the cosine or sine. Since similar triangles always have the same angles, then the ratio is the same for any given angle. So each ratio (whether the sin, cos, or tan) has a particular angle that goes with it, and each angle has a particular ratio of sides.

Where Is Trigonometry Used in the Geosciences?

You will find that you use trigonometry in the following areas typically taught in introductory geoscience classes, plus many others:

Mass wasting: Calculating slope steepness, gradients and forces on a hillslope, determining slope stability
Topographic maps: determining compass direction and orienteering, calculating slopes, locating objects and landmarks, calculating the distance to objects far away
Structural geology: measuring the tilting of rocks, determining the location of rocks in the subsurface, making geologic maps.
Oceanography: Describing ocean waves, calculating wind and current direction
Plate tectonics: Measuring the steepness of subduction zones
Seismology: measuring seismic waves, determining the nature of the interior of Earth through refraction

Next Steps

I am ready to PRACTICE!

If you think you have a handle on the steps above, click on this bar to try practice problems with worked answers.
Or, if you would like more help, see the resources in the links below.

Resources for Trigonometry

Dave's Short Trig Course has a section on angle measurement, trigonometry in right triangles, and the trigonometric functions, as well as many other trigonometric topics which should be helpful.

Mathworld's Trigonometry page has a huge number or resources, that go far beyond the material typically covered in an introductory course.

S.O.S. Mathematics has a trigonometry index (more info) featuring a table of trigonometric identities, lessons on functions and formulae, and a section of exercises and solutions.

Math Help's trigonometry index (more info) provides a good refresher on basic trig identities, functions, and angles. Also includes diagrams for unit circle and triangle trigonometry.

This page was written and compiled by Dr. Eric M. Baer, Geology Program, Highline Community College, and Dr. Jennifer M. Wenner, Geology Department, University of Wisconsin, Oshkosh

The Math You Need, When You Need It

How Do I Deal with Angles?
Trigonometry in the Geosciences

Introduction to Trigonometry

Some Rules/Guidelines for Trigonometry of Right Triangles