During high school and the early parts of college, students usually have to take a trigonometry class.
The purpose of trigonometry is to determine unknown lengths and angles.
Unfortunately, this branch of math is also complex to learn.
It’s not uncommon for students to struggle a lot with it when they’re first introduced.
Here are 10 reasons trigonometry can be so hard to learn.
Is Trigonometry Hard? (10 Reasons It Can Be)
One of the reasons trigonometry is hard is that it uses functions.
Almost any type of math uses functions, but the functions involved in trigonometry are particularly complex.
That’s because there are six main functions, which then branch off to form a series of other functions.
The student has to learn and memorize all these various functions to then calculate various angles and lengths.
If they don’t remember a particular function or variation of a function, then they’re never going to be able to solve the problem.
It isn’t as straightforward as basic algebra.
With algebra, there are only a few rules that you must remember.
With trigonometry, you have to remember what sine and cosine mean, for example.
You have to remember what they represent and the various ways they impact angles and lengths.
Trigonometry is difficult because it involves a lot of memorization of different functions which can then deviate into other functions.
Perhaps the biggest reason why trigonometry is hard is that it’s non-linear.
With basic algebra, the math is pretty straightforward.
You have to solve for X.
While this sometimes means you have to work on both sides of the equation, it’s still a pretty linear process.
Trigonometry is not linear.
That’s because it all relates to a circle.
A circle isn’t linear.
This makes trigonometry an elevated form of math that requires the individual to think past a basic linear form.
This is complex.
The student has to understand how a circle splits into various angles.
Then they need to understand how those angles work together to generate distances and other angles.
The math can become very involved in a non-linear way.
This makes trigonometry incredibly difficult to understand.
3. Unit Circle
Trigonometry all relates back to the unit circle.
If students don’t understand how a unit circle works or what it looks like, then they won’t be able to understand further concepts of trigonometry.
This makes the math difficult since it’s reliant on a student’s ability to remember how a unit circle functions.
A unit circle is, essentially, a circle that has a radius of 1.
Trigonometry becomes extremely challenging and advanced if the radius becomes larger than 1.
When constructing a unit circle, the center of the circle goes on an area where the X-axis and Y-axis cross on a graph.
The unit circle then defines the values of sine and cosine.
When performing trigonometry functions, you rely on the unit circle to help you establish various values of sine and cosine.
Sine is not the same for one angle as it is for another.
Trigonometry is hard because it uses a unit circle which only adds further complexity to this branch of mathematics.
4. Radians And Degrees
It isn’t enough that students have to solve for degrees with trigonometry.
As the math becomes more complex, they also then have to convert their answers into radians.
The radian is a pure measure that gets its basis from the radius of the unit circle.
A single radian is an angle that’s made when the radius wraps around the unit circle.
One radian is equal to about 57.2958 degrees.
That’s because, when using radians, you replace degrees with the pi symbol (π).
You also do most of the calculations with pi.
Since pi isn’t a clean number, it generates unclean solutions.
Converting between degrees and radians is both simple and complex.
The formula itself is simple, but if something else goes wrong in your calculations, then it’s easy to get the entire thing wrong.
To convert from radians to degrees, you take your answer and multiply it by 180, then divide it by pi.
If you want to convert your degrees into radians, then you take your answer and multiply it by π, then divide it by 180.
Radians are essential for getting a more accurate answer.
When planning out architecture or performing surveys, the radian is the preferred unit to use since it’s more accurate than degrees.
However, the fact that you can’t just take the answer in degrees as it is makes trigonometry more complex.
Trigonometry is hard because the final answer you receive when doing your calculations may not be the final answer after all.
You may have to convert it into a different unit which adds another step of complexity and an increased possibility of getting the answer wrong.
5. Memorization Of Known Values
There are a few known values that you have to memorize to perform trigonometric calculations.
For a unit circle, when calculating in degrees, you need to know the exact values of sine, cosine, and tangent.
These aren’t actual numbers.
Instead, they’re functions.
They’re also different based on the degree that you’re working with.
As an example, the value of sine at 30 degrees is 1/2.
However, the value of sine at 45 degrees is the square root of two divided by two.
There’s one more you need to know for sine, too.
The value of sine at 60 degrees is the square root of three divided by two.
These values are also shared among the other functions.
The cosine of 30 degrees is the same thing as the sine at 60 degrees.
It’s the square root of three divided by two.
Because you have several values to remember, and those values are the same for the other functions at certain degrees, it’s easy to get them mixed up.
If you do mix them up, then your calculation is entirely thrown off.
Degrees aren’t the only thing you need to memorize either.
You also have to remember their radian counterparts.
For example, the radian value of 30 degrees is π divided by six.
The radian value of 60 degrees is π divided by three.
This goes all the way down the list until you reach 360 degrees, which has a radian value of 2 multiplied by π.
Trigonometry is hard because you have to memorize various values of different functions in both degrees and radians.
If you don’t remember them, or if you mix them up, then your calculations will be incorrect.
6. Cartesian Coordinate Rules
Another reason trigonometry is hard is that it relies on the Cartesian Coordinate rules for the basis of its math.
If students don’t understand how these rules work, then they may only have a limited understanding of how to generate trigonometric functions.
Cartesian coordinates basically tell someone where a point is on a map or graph.
They’re calculated by determining how far along and how far the point is on a graph.
An example is the coordinates of (12, 2).
This tells us that the point on the graph is 12 units along the X-axis and 2 points up the Y-axis.
However, when first learning about Cartesian coordinates, it can be easy to get them wrong.
In particular, the students must learn and understand the concept of an ordered pair.
An ordered pair means there is a specific order that numbers have to follow.
For Cartesian coordinates, the ordered pair relies on the first number representing the X-axis and the second number representing the Y-axis.
It’s commonly written as (X, Y).
Thus, a coordinate like (12, 2) tells us that the point on the graph has an X-coordinate of 12 and a Y-coordinate of 2.
If students mix those up—for example, if they forget that the 2 is with the Y and instead write it with the X—then the very foundations of all trigonometric calculations in the future are going to be incorrect.
Cartesian coordinates can also become complex.
They can have negative values.
For example, students might see the following coordinate: (−5, 12).
This means that the point on the graph is on the X-coordinate at five units to the left rather than the right.
Since the Y-coordinate is still positive, it means it’s still at the top of the graph rather than below it.
A negative value symbolizes the crossing of that particular coordinate’s line.
If an X-coordinate is negative, then it’s going to cross the X-axis to the left side.
If a Y-coordinate is negative, then it’s going to cross the Y-axis to the bottom of the graph.
Including negatives can make reading or calculating values from the graph even more difficult.
Trigonometry is difficult because it uses Cartesian coordinates, which can be easy to get wrong initially.
7. Polar Coordinates
While Cartesian coordinates are necessary for the initial understanding of trigonometry, most of the math involved comes from polar coordinates.
Polar coordinates are a two-dimensional coordinate system.
Instead of using X and Y as its units, it uses r and theta (θ).
These represent the distance from a reference point, r, and the angle from a reference direction, θ.
While Cartesian writes their coordinates as (X, Y), polar coordinates write theirs as (r, θ).
The graph for a polar grid is also different from the one used for Cartesian coordinates.
A polar grid consists of concentric circles that radiate out from the pole.
The pole is the origin of the coordinate plane.
When plotting or defining a polar grid, most people use either degrees or radians.
They then mark the various angles in that particular unit.
When plotting on a polar grid, the coordinates look a little different.
For example, a polar coordinate of (2, 30°) means that the coordinate is 2 units to the right of the origin on the r-axis with a 30-degree angle.
When using radians, an example might be (2, (π/4)).
A lot of trigonometry problems will ask students to convert from a Cartesian coordinate to polar coordinates.
Students will then need to understand how to convert between the two.
Like any conversion, if something goes wrong, then their answers are also going to be wrong.
An example of converting between the two different coordinates runs as follows.
To convert cos θ from Cartesian to polar, you’d need to first list X and divide it by r.
This comes out to the polar coordinate which is X = r cos θ.
It’s also worth it to know that r2 = x2 + y2.
Before you can even start to get into the nitty-gritty of trigonometry, you already have to do a lot of math just to get the right functions that you’re working with.
Because of polar coordinates, trigonometry is hard.
Trigonometry requires several different types of thinking in order to solve its equations.
It requires critical thinking and visual thinking.
Because trigonometry relates to graphs, circles, and triangles, it can help an individual learn this branch of math more if they’re already a visual learner.
A visual learner is a type of individual who finds concepts easier to learn after seeing them.
They usually need their teacher to go through a few different examples visually to learn the concepts.
These particular individuals may have an easier time learning trigonometry since they have an easier time visualizing coordinates and angles.
However, this also means that those who are not visual learners may have a more difficult time learning trigonometry.
Since they’re not as effective at visualizing, they may take a bit longer to understand the concepts.
Trigonometry involves math that takes its basis in visuals and critical thinking.
Those who can visualize angles and triangles and circles in their minds while performing the calculations will find learning it easier than those who cannot.
Because trigonometry is very visual, it’s harder for some individuals to learn.
9. Relating Functions To Sides Of Triangles
One of the first things that students will learn when beginning trigonometry is the six functions used in trigonometric equations.
Sine, Cosine, and Tangent all represent different sides of a triangle.
The common moniker used to remember the sides they represent is SohCahToa.
Sine represents the opposite length and hypotenuse length.
Cosine represents the adjacent length and hypotenuse length.
Tangent represents the opposite length and adjacent length.
When students first start studying trigonometry, it’s easy to get these functions mixed up.
Sine and Cosine are the easiest to mix up since they sound similar and have only one real difference between them.
This becomes even more complex when you start bringing in arc functions, too.
Students will then have to remember what arcsine represents and what arccosine represents.
If they do forget and use the incorrect lengths, then all the math they do afterward is going to be wrong.
Even if they calculate everything perfectly, because they used the incorrect lengths, the final answer is going to be wrong.
Since trigonometry can take a long time, that’s a lot of wasted effort and time.
Trigonometry is hard because it’s easy to use the incorrect lengths for the functions when first starting out.
10. Length Of Calculations
The final reason trigonometry is hard is that calculating its equations can become extremely long.
It also uses several different variables and symbols to represent various functions within the calculation.
If the student forgets what even one symbol represents, then it can completely ruin the rest of the calculation.
The sheer length of time it takes to complete a complex trigonometric equation means there are more opportunities for something to go wrong.
Whether it involves converting from degrees to radians or radians to degrees, the more steps involved, the more likely it is that they’re going to have an incorrect answer.
That’s because when several steps are required to find the solution to a problem, there’s a greater chance that the student might write down the wrong number, accidentally skip a step, or even type the wrong number into their calculator.
Little mistakes made through human error become more probable with lengthier math problems.
When it comes to trigonometry, which is a very precise type of math, even the slightest mistake can result in a completely wrong answer.
Longer problems also exhaust the mind further.
When students have several long trigonometric equations to solve, their minds become tired.
This further increases the risk of error.
Trigonometry is hard because the solutions can take several steps and a lot of time to reach.
Is Trigonometry Harder Than Calculus?
While trigonometry is hard to learn, you may wonder if it’s more complex than calculus.
In most cases, mathematicians will tell you that calculus is more complex than trigonometry.
That’s because calculus brings together several different branches of math together.
It uses geometry, algebra, and trigonometry.
Since it marries all these different types of arithmetic together, it’s a lot more involved and complex than trigonometry.
Trigonometry, at most, uses geometry and algebra for its calculations.
Calculus is another step beyond that.
It uses concepts like derivations, inversions, and imaginary numbers.
The math gets really weird once you start getting into deep calculus.
Trigonometry is a bit easier to grasp because it’s rooted in reality.
You’re working with real distances and angles.
Calculus can sometimes stretch into mathematical theory, which requires deep critical thinking to understand.
While trigonometry may seem hard, calculus is even harder.
Who Invented Trigonometry?
Modern trigonometry has the Ancient Greeks to thank for its existence.
While many confuse Pythagoras as the inventor of trigonometry, it was actually Hipparchus.
He believed that every triangle was inscribed on a circle.
It didn’t matter if the triangle was planar or spherical.
They all had lines that connected to form a circle.
The first few centuries of trigonometric research revolved around determining the lengths of triangles and how they related to the triangle’s angles.
During Hipparchus’s time, geometry represented the functions.
It was in the 17th century that they developed the modern symbols for these functions used today.
The first major work that used trigonometry at its center was The Almagest.
This was a mathematical treatise written by the ancient mathemetician, Ptolemy.
It included a demonstration, using trigonometry, that introduced the concept of the Earth remaining still while the other planets revolved around it.
Ptolemy used basic trigonometry and geometry for his argument.
Nicolaus Copernicus would later introduce the idea of the heliosphere in which the planets, including Earth, revolved around the sun instead.
Trigonometry is an ancient branch of math that has its roots as far back as Ancient Greece.
Although it’s been around for a long time, understanding this type of math is still complex.
From conversions to memorization to relying on visuals makes this particular branch of math especially complex.
However, it isn’t as complex as calculus which includes trigonometry and several other branches of math.