Trigonometric Functions

Let's look at everything to do with trigonometric functions – sine, cosine and tangent functions and their respective graphs. Then let's explore the secant, cosecant, cotangent, arcsine, arccosine and arctangent functions.

What are trigonometric functions?

Trigonometric functions are functions that relate to angles and lengths in a triangle. The most common trigonometric functions are sine, cosine and tangent. However, there are reciprocal trigonometric functions, such as cosecant, secant, cotangent and inverse trigonometric functions such as arcsine, arccosine and arctangent, which we will also explore in this article.

SOH CAH TOA

An easy way to remember the sine, cosine and tangent functions and what sides they correspond to in a right angle triangle is by using SOH CAH TOA. If we have a right angle triangle as below, and we label one angle 𝞱, we must label the three sides of the triangle opposite (for the only side that is opposite the angle 𝞱 and is not in contact with that angle), hypotenuse (for the longest side, which is always the one opposite the 90° angle) and adjacent (for the last side).

Labelling the sides of a right-angled triangle

The sine, cosine and tangent functions relate the ratio of two sides in a right-angled triangle to one of its angles. To remember which functions involve which sides of the triangle, we use the acronym SOH CAH TOA. The S, C and T stand for Sine, Cosine and Tangent respectively and the O, A and H for Opposite, Adjacent and Hypotenuse. So the Sine function involves the Opposite and the Hypotenuse, and so on.

SOH CAH TOA triangles for remembering trigonometric functions

All of the functions sine, cosine and tangent are equal to the sides they involve divided by each other.

\[\sin\theta = \frac{opposite}{hypotenuse}; \space \cos \theta = \frac {adjacent}{hypothenuse}; \space \tan \theta = \frac {opposite}{adjacent}\]

What is the sine function?

As seen above, you can work out the sine of an angle in a right-angled triangle by dividing the opposite by the hypotenuse. The graph for a sine function looks like this (the red curve):

A graphical illustration of the sine function

From this graph, we can observe the key features of the sine function:

Memorising the values of sine

You will need to remember the values of sine for commonly used angles by heart, and although this might sound tricky, there is a way to make it easier to memorise. You will need to know the sine values for the angles 0, \(\frac{\pi}{6}\) (30°), \(\frac{\pi}{4}\) (45°), \(\frac{\pi}{3}\) (60°) and \(\frac{\pi}{2}\) (90°). For this, the easiest way is to start constructing a table for the angle, 𝞱 and sin𝞱:

Now we have to fill out the sine values. For this, we will start by putting the numbers 0 to 4 from left to right:

The next step is to add a square root to all these numbers and divide them by 2:

Now, all we have left to do is simplify what we can:

And that's it!

What is the cosine function?

You can find the cosine value for an angle in a right-angled triangle by dividing the adjacent by the hypotenuse. The graph for the cosine value looks exactly like the sin graph, except that it is shifted to the left by \(\frac{\pi}{2}\) radians (the blue curve):

A graphical illustration of the cosine function

By observing this graph, we can determine the key features of the cosine function:

• The graph repeats every 2𝞹 or 360 °

• The minimum value for cosine is -1

• The maximum value for cosine is 1

• This means that the amplitude of the graph is 1 and its period is 2𝞹 (or 360°)

• The graph crosses the y-axis at \(\frac{\pi}{2}\), and every 𝞹 radians before and after that.

• The cosine function reaches its maximum value at 0 and every 2𝞹 before and after that.

• The cosine function reaches its minimum value at 𝞹 and every 2𝞹 before and after that.

You will also need to remember the values of cosine for commonly used angles by heart, and although this might sound tricky, there is a way to make it easier to memorise. You will need to know the sine values for the angles 0, \(\frac{\pi}{6}\) (30°), \(\frac{\pi}{4}\) (45°), \(\frac{\pi}{3}\) (60°) and \(\frac{\pi}{2}\) (90°). For this, we will use the same method as for sin and start constructing a table for the angle, 𝞱 and cos𝞱:

Now we will fill in the numbers 0 to 4, but this time, we will do this from right to left instead:

The final two steps are the same as before, so we will take the square root of each number and divide it by 2, and we simplify:

As you can see, sine and cosine values for common angles are the same, simply the other way around.

What is the tangent function?

You can work out the tangent of an angle by dividing the opposite by the adjacent in a right-angled triangle. However, the tangent function looks a bit different from the cosine and sine functions. It is not a wave but rather a non-continuous function, with asymptotes:

A graphical illustration of the tangent function

By observing this graph, we can determine the key features of the tangent function:

• The graph repeats every 𝞹 or 180°

• The minimum value for tangent is \(-\infty\)

• The maximum value for tangent is \(\infty\)

• This means that the tangent function has no amplitude and its period is 𝞹 (or 180°)

• The graph crosses the y-axis at 0 and every 𝞹 radians before and after that

• The tangent graph has asymptotes, which are values that the function will get closer to infinity.

• These asymptotes are at \(\frac{\pi}{2}\) and every 𝞹 before and after that.

The tangent of an angle can also be found with this formula:

\[\tan\theta = \sin \theta / \cos \theta \]

Memorising the values of tangent

Similar to before, you will need to remember the tan values for the angles 0, \(\frac{\pi}{6}\) (30°), \(\frac{\pi}{4}\) (45°), \(\frac{\pi}{3}\) (60°) and \(\frac{\pi}{2}\) (90°). For this, we will use the formula above and the tables that we already constructed for sine and cosine and use the fact that \(\tan = \sin /\cos\) to work out the tan𝞱 values:

Note that the value for tan (\(\frac{\pi}{2}\)) cannot be determined as it is equal to 1/0, which cannot be worked out. This will result in an asymptote at \(\frac{\pi}{2}\).

Inverse trigonometric functions

The inverse trigonometric functions refer to the arcsin, arccos and arctan functions, which can also be written as \(\sin^{-1}(x)\), \(\cos^{-1}(x)\) and \(\tan^{-1}(x)\). These functions do the opposite of the sine, cosine and tangent functions, which means that they give back an angle when we plug a sin, cos or tan value into them.

An illustration of the relationship between trigonometric functions and their respective inverse functions

The graphs for these functions look very different to the sin, cos and tan graphs:

An illustration of arcsin, arccos and arctan on the x and y axis

What are the reciprocal trigonometric functions?

The reciprocal trigonometric functions refer to the cosecant, secant and cotangent functions, abbreviated as csc, sec and cot, respectively. We need to look back at our right-angled triangle to understand what these functions represent.

Labelling the sides of a right-angled triangle

We earlier defined sin, cos and tan based on the ratios of the sides of this triangle. The cosecant, secant and cotangent are simply the reciprocals of the sin, cos and tan ratios respectively. This means that to find the equation for cosecant 𝞱, we would flip the equation of sin 𝞱 and so on.

\[\sin\theta = \frac {opposite}{hypothenuse}; \space \cos\theta = \frac {adjacent}{hypothenuse}; \space \tan\theta = \frac {opposite}{adjacent}\]

\[\csc\theta = \frac {hypothenuse}{opposite}; \space \sec\theta = \frac{hypothenuse}{adjacent}; \space \cot \theta = \frac {adjacent}{opposite}\]