Trigonometry has a reputation of being a pain. Ask most people what sine, cosine and tangent are for and they'll give you a blank stare - even if they've used them countless times during their school days. It's a real shame too, since the three functions are actually incredibly useful tools in mathematics, used in everything from construction to video game design, without most people realising it.
In this guide we'll be covering exactly what the three functions are, how they can be calculated, where they're used and even where you can calculate sine, cosine and tangent values at the touch of a button.
The Basics of a Right Angle Triangle
The three functions stem from the relationships between sides of a right angle triangle and the angles within it. For any right angle triangle pick an angle (apart from the right angle itself) and label it θ (theta) - this will be the angle we're focusing on.
This then defines the three sides of the triangle as:
Opposite: The side opposite angle θ
Adjacent: The side next to angle θ
Hypotenuse: The side opposite the right angle and by definition the longest side of the triangle
With these terms defined, the three main trigonometric functions are simply ratios of these sides:
Sine (sin θ): Opposite ÷ Hypotenuse
Cosine (cos θ): Adjacent ÷ Hypotenuse
Tangent (tan θ): Opposite ÷ Adjacent
That's really all there is to it. Every other rule, relationship or function within trigonometry is just a variation or evolution on this theme.
Understanding SOH-CAH-TOA
When it comes to remembering the rules for the three trigonometric functions most people are taught 'SOH-CAH-TOA' which helps them recall the relationships at a glance.
It stands for:
Sine = Opposite ÷ Hypotenuse
Cosine = Adjacent ÷ Hypotenuse
Tangent = Opposite ÷ Adjacent
Nothing fancy. It's simply a mnemonic that makes it easier to remember the basic rules.
Degrees and Radians
The primary unit of angular measurement is either degrees or radians. A full circle is 360 degrees or 2π radians (≈6.283). Most common uses of trigonometry, including many problems seen in classrooms and day-to-day use, are in degrees. While more complex mathematics makes more use of radians. If your calculator is calculating sin(x°) then it's using degrees and not radians.
Common Values of Sine, Cosine, Tangent
There are a set of angles that have particularly useful values, that it's worth remembering. Some of these are shown below:
Note that tan(90°) is undefined - this is due to properties of right angle triangles explained earlier - the adjacent side becomes 0 which makes it impossible to divide by.
Everyday Uses of Trigonometry
Trigonometry has many uses throughout construction, especially in carpentry and surveying where it allows the relationships between angles and sides to be calculated.
It's also used extensively in navigation, both on the ground and on the sea. In physics it's useful for analysing projectile motion and more specifically the forces that contribute to it - such as gravity, thrust and air resistance. Another common use is in alternating current (AC) electricity, where the voltage and current follow a sine wave pattern - meaning properties of sine curves can be used to find peak power, average power and other useful measurements. Similar uses are found in video games and animations to allow characters and objects to move in a circular path.
Working Through a Practical Example
Let's say we're building a wheelchair ramp and need to know if a 3 foot rise would be possible within a 15 degree angle limit safely. To do this we can use tangent as we know the opposite and adjacent sides. With tan(15°)=opposite ÷ adjacent we can substitute the values and rearrange to find the opposite length (rise):
opposite = tan(15°) x adjacent = tan(15°) x 3ft = 3 ÷ 0.268 ≈ 11.2ft
This means the ramp would need a run of approximately 11.2ft to acheive the 15 degree angle limit. If we didn't know this relationship we'd have to guess or build a physical model of the ramp to see how far it would extend - a time consuming process for relatively little maths gained.
Working Through the Calculations
For quick calculations a scientific calculator is ideal. Most newer calculators have sine, cosine and tangent functions built-in to them, along with their inverses. They also tend to include exponents, logs and roots which are handy if a problem requires a mix of operations. The dedicated sine calculator is good for working strictly with sine values, but for anything more involved a scientific calculator is better.
The Big Picture
Sine, cosine and tangent might seem like strange beasts to deal with at first, but they're really just shorthand for the relationships between angles and lengths in right angle triangles. At their most basic they're just ratios of triangle sides with different sides weighted more depending on the angle. Once the SOH-CAH-TOA rules are known then most problems boil down to simple substitution and rearranging of values, and from there it's simple arithmetic.