| Basic Trigonometric Functions (SOH-CAH-TOA) |
| Right Triangle Trigonometry Applications |
| More Practice |
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, for example, we can use the trig functions on angles to solve for unknown side measurements, or use inverse trig functions on sides to solve for unknown angle measurements. Thus, 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, in the Law of Sines and Cosines, and Areas of Triangles section.
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)
You may have been taught $ \boldsymbol {\text{S}\frac{\text{O}}{\text{H}}\,\,\text{C}\frac{\text{A}}{\text{H}}\,\,\text{T}\frac{\text{O}}{\text{A}}}$ (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.
Note that the second set of three trig functions are just the reciprocals of the first three; this makes it a little easier! 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. (Note: We do have to be careful when using $ \displaystyle \frac{1}{{\tan \left( x \right)}}$ for $ \cot \left( x \right)$ in the calculator. For angles $ \displaystyle \frac{\pi }{2},\frac{{3\pi }}{2}$, the results won’t be correct; it shows an error, instead of 0 (try it!). It would be better to use $ \displaystyle \frac{{\cos \left( x \right)}}{{\sin \left( x \right)}}$ in this case.)
Here are the 6 trigonometric functions, shown with both the SOHCAHTOA and Coordinate System Methods. Remember that the sin (cos, and so on) of an angle is just a number; it’s unitless, since it’s basically a ratio.
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. For 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°.
Here are some problems where we need to think about ratios of sides of right triangles:
Right Triangle Trigonometry Applications
Here are some types of word problems (applications) 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 (from Geometry!) to put that angle in the triangle on the ground.
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 or fraction.
And, as always, always draw pictures!
Angle of Elevation Problems:
Angle of Depression Problem:
Right Triangle Systems Problem:
Here’s a problem where it’s easiest to solve it using a System of Equations:
Trig Shadow Problem:
Trig Grade Problem:
For Practice: Use the Mathway widget below to try a problem. Click on Submit (the blue arrow to the right of the problem) and click on Solve the Triangle to see the answer.
You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.
If you click on Tap to view steps, or Click Here, you can register at Mathway for a free trial, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).
On to Angles and the Unit Circle – you’re ready!