Right Triangle Trigonometry

This section covers:

You may have been introduced to Trigonometry in Geometry, when you had to find either a side length or angle measurement of a triangle.  Trigonometry is basically the study of triangles, and was first used to help in the computations of astronomy.  Today it is used in engineering, architecture, medicine, physics, among other disciplines.

The 6 basic trigonometric functions that you’ll be working with are sine (rhymes with “sign”), cosine, tangent, cosecant, secant, and cotangent.  (Don’t let the fancy names scare you; they really aren’t that bad).

With Right Triangle Trigonometry, we use the trig functions on angles, and get a number back that we can use to get a side measurement, as an example.    Sometimes we have to work backwards to get the angle measurement back so we have to use what a call an inverse trig function But basically remember that we need the trig functions so we can determine the sides and angles of a triangle that we don’t otherwise know.

Later, we’ll see how to use trig to find areas of triangles, too, among other things.

You may  have been taught SOH – CAH – TOA  (SOHCAHTOA)  (pronounced “so – kuh – toe – uh”) to remember these.  Back in the old days when I was in high school, we didn’t have SOHCAHTOA, nor did we have fancy calculators to get the values; we  had to look up trigonometric values in tables.

Remember that the definitions below assume that the triangles are right triangles, meaning that they all have one right angle (90°).   Also note that in the following examples, our angle measurements are in degrees; later we’ll learn about another angle measurement unit, radians, which we’ll discuss here in the Angles and Unit Circle section.

Basic Trigonometric Functions (SOH – CAH – TOA)

Here are the 6 trigonometric functions, shown with both the SOHCAHTOA and Coordinate System Methods.   Note that the second set of three trig functions are just the reciprocals of the first three; this makes it a little easier!

Note that the cosecant (csc), secant (sec), and cotangent (cot) functions are called reciprocal functions, or reciprocal trig functions, since they are the reciprocals of sin, cos, and tan, respectively.

For the coordinate system method, assume that the vertex of the angle in the triangle is at the origin (0, 0):

Here are some example problems.  Note that we commonly use capital letters to represent angle measurements, and the same letters in lower case to represent the side measurements opposite those angles.  We also use the theta symbol θ  to represent angle measurements, as we’ll see later.

Note also in these problems, we need to put our calculator in the DEGREE mode.

And don’t forget the Pythagorean Theorem ($${{a}^{2}}+{{b}^{2}}={{c}^{2}}$$,  where a and b are the “legs” of the triangle, and c is the hypotenuse), and the fact that the sum of all angles in a triangle is 180°.

Trigonometry Word Problems

Here are some types of word problems that you might see when studying right angle trigonometry.

Note that the angle of elevation is the angle up from the ground; for example, if you look up at something, this angle is the angle between the ground and your line of site.

The angle of depression is the angle that comes down from a straight horizontal line in the sky.  (For example, if you look down on something, this angle is the angle between your looking straight and your looking down to the ground).   For the angle of depression, you can typically use the fact that alternate interior angles of parallel lines are congruent (sorry, too much Geometry!) to put that angle in the triangle on the ground (we’ll see examples).

Note that shadows in these types of problems are typically on the ground.   When the sun casts the shadow, the angle of depression is the same as the angle of elevation from the ground up to the top of the object whose shadow is on the ground.

Also, the grade of something (like a road) is the tangent (rise over run) of that angle coming from the ground.  Usually the grade is expressed as a percentage, and you’ll have to convert the percentage to a decimal to use in the problem.

And, as always, always draw pictures!

Angle of Elevation Problem:

Devon is standing 100 feet from the Eiffel Tower and sees a bird land on the top of the tower (she has really good eyes!).   If the angle of elevation from Devon to the top of the Eiffel Tower is close to 84.6°, how tall is the tower?

Solution:

This is a good example how we might use trig to get distances that are typically difficult to measure.   Note that the angle of elevation comes up off of the ground.

Angle of Depression Problem:

From the top of a building that is 200 feet tall, Meryl sees a car coming towards the building.  (Somehow she knows that) the angle of depression when she first saw the car was 20° and when she stopped looking at it was 40° degrees.  How far did the car travel?

Solution:

The first step is to draw a picture, and note that we can sort of “reflect” the angles of depression down to angles of elevation, since the horizon and ground are parallel.  Then we get to use trig!

Right Triangle Systems Problem:

Here’s a problem where it’s easiest to solve it using a System of Equations:

Two girls are standing 100 feet apart.  They both see a beautiful seagull in the air between them.  The angles of elevation from the girls to the bird are 20° and 45°, respectively.   How high up is the seagull?

Solution:

The length of a tree’s shadow is 20 feet when the angle of elevation to the sun is 40°.  How tall is the tree?

Solution:

Again, note that shadows in these types of problems are on the ground.   When the sun casts the shadow, the angle of depression is the same as the angle of elevation from the ground up to the top of the tree.

So let’s solve using trig:

Chelsea walked up a road that has a 20% grade (she could feel it!) to get to her favorite store.  At what angle does the road come up from the ground (at what angle is the road inclined from the ground)?

Solution:

Remember that the grade of a road can be thought of as  $$\frac{\text{rise}}{\text{run}}$$  and you usually see it as a percentage.   So a 20% grade is the same as a grade of  $$\frac{\text{20}}{\text{100}}$$; for every 20 feet the road goes up vertically, it goes 100 feet horizontally.

Understand these problems, and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Angles and the Unit Circle – you’re ready!