The following file is for year 12 students covering trigonometric functions sine and cosine. It presents an orderly method to graph functions of the form:

f(x) = a sin[n(x − h)] + k, x₁ ≤ x ≤ x₂ or f(x) = a cos[n(x − h)] + k, x₁ ≤ x ≤ x₂. It assumes knowledge of trig functions (or circular functions) and how to use the unit circle.

Now on to a trig (circular functions) problem. This problem requires that you have covered year 11 trigonometry (called circular functions in some texts).

The water depth in a harbour on a particular day is modelled by the following equation:

So we have the requirement that the sine of “something” has to equal -0.5. When we take a function of something, that something is called the argument of the function. From a table of common sines and knowing that sin(-????) = -sin(????), we get that ????t/6 = -????/6 ⟹ t = -1. We need another angle on the unit circle that has a sine of -0.5:

So ????t/6=7????/6 ⟹ t = 7. The period of this sine function is

Now we add this period to our two core values of t, until we get t > 24:

7 + 12 = 19 -1 + 12 = 11 11 + 12 = 23

So the t values we have that are between 0 and 24 are 7, 11, 19, 23. As the sine function initially increase from 0, the value of t = 7, which is 7 AM, is when the depth is decreasing and is below 8.5 meters. The depth is increasing and above 8.5 at t = 11 which is 11 AM. The depth then decreases below 8.5 at t = 19 which is 7 PM and rises above 8.5 at t = 19 which is 11 PM. So the ship can dock between midnight and 7 AM or between 11 AM and 7 PM or between 11 PM and the following midnight.

I’ve plotted D(t) and the line D(t) = 8.5 to show these solutions.

In fact, doing a rough sketch of this function would help get a feel for the answers before we begin.

We do this by raising the standard sine curve up 10 units and draw 2 cycles from 0 to 24 since we can calculate that the period is 12 hours. Since the sine of anything goes from -1 to +1, the minimums are at 10 – 3 = 7 and the maximums are at 10 + 3 = 13. Drawing a line at D(t) = 8.5, we can see that we should get 4 points that will divide the intervals where the depth is below or above 8.5 meters. Once we have the points at D(t) = 8.5, we can easily see the periods where the depth is above 8.5 meters.

Let’s talk about the sine function, the first graph, though the following discussion can be generalised to the other two.

There are key points on this graph: the peaks and where it crosses the x-axis. The values of x where the sine is zero are 0, ±????, ±2????, ±3????, … . The positive and negative peaks occur at ±????/2, ±3????/2, ±5????/2, … . Also note that the sine goes from -1 to 1, a total distance of 2. The amplitude is defined as half of this distance, so the amplitude of the basic sine wave is 1.

Another aspect to notice is that the curve repeats every 2???? radians. That means that given any x value, adding or subtracting any multiple of 2???? to x will result in the same sine value. This is the definition of a function’s period. The basic sine function has a period of 2????. Mathematically,

sin(x +2n????) = sin(x), where n is any integer.

So let’s look at the solution to sin x = 1/2, 0 ≤ x ≤ 2????. Instead of the unit circle, the sine graph can be used instead:

Noticing that ????/6 has a sine of 1/2, we can use the symmetry of the sine graph to find the other angle with the same sine, ???? – ????/6 = 5????/6. These are the two answers for the given domain.

What if the given domain was -2???? ≤ x ≤ 2????? Because the period of the basic sine wave is 2????, you can add or subtract multiples of 2???? to the basic angles found to get the answers in the required interval. In this case, subtract 2???? from each of the basic angles previously found to get:

If the problem at hand is more complex like 2sin(2x +????) -1 which I solved in my last post, you solve it as before except you use the sine graph instead of the unit circle to find the basic angles, then proceed as before.

Remember what the term argument means? The argument of sin (2x + 7) is 2x + 7, that is an argument of function is the thing the function is operating on. In this case, the sine function is operating on 2x + 7.

So when the argument of a trig function is other than x, the basic method of solving an equation with this function is basically the same as what we did in the last post but an extra bit of algebra is required.

For example, solve 2sin(2x +????) -1 = 0 for 0 ≤ x ≤ 2????. The argument of the sine is 2x + ????. We want to first, solve the equation just as we did before, but in terms of the argument 2x + ????:

2sin(2x +????) -1 = 0 ⟹ sin(2x +????) = 1/2

From our table of common values, we see that the angle ????/6 has a sine of 1/2. So the basic starting point to find all solutions is 2x +???? = ????/6. To correctly limit the number of solutions, let’s convert the given domain 0 ≤ x ≤ 2???? to be in terms of 2x +????. To do this, algebraically change the inequality so that the middle expression is 2x +????:

So we need to find all the angles between ???? and 5???? that have a sine of 1/2. Again, a unit circle diagram will help. I will start with ????/6, but realise that ????/6 is not in the required modified domain:

So starting with the basic angle of ????/6, I add multiples of 2???? until we are outside the given domain. For ????/6, adding 2???? gives 13????/6. Adding 2???? again gives 25????/6. Adding another 2???? gives 37????/6, but 37????/6 is greater than 5???? so it is not a valid solution.

So far, the intermediate solutions are 13????/6 and 25????/6. The other basic angle found from the unit circle is 5????/6. Again, this angle is not in our modified domain so we need to add multiples of 2???? to find the angles within ???? and 5????. So just like we did for ????/6, adding 2???? to 5????/6 gives 17????/6. Adding another 2???? gives 29????/6. Adding another 2???? puts us outside the domain.

To find the final solutions, we need to solve each of these equations for x:

2x + ???? = 13????/6 ⟹ x = 7????/12 2x + ???? = 25????/6 ⟹ x = 19????/12 2x + ???? = 17????/6 ⟹ x = 11????/12 2x + ???? = 29????/6 ⟹ x = 23????/12

A generic method for solving many trig equations is:

Find the basic angles that solve the equation in terms of the argument of the trig function in the equation.

Modify the given domain to be in terms of the argument.

Add or subtract multiples of 2???? to the basic angles to find all angles within the modified domain. Keep in mind that the basic angles themselves may or may not be in the domain.

Set the argument equal to all solutions and solve for the variable used in the problem (usually ???? or x)

There is another way to analyse trig equations which I will show on my next post.

Now let’s turn to solving equations with trig functions. Before we get into it though, let me introduce the inverse trig functions.

Just like square and square root, logs and exponentials, add and subtract, etc are inverse operations, there are corresponding inverse functions for the trig functions. These functions are identified usually by a -1 exponent. For example, sin^{-1}(0.25) is asking the question: what angle has a sine of 0.25? Since sin^{-1}x is an inverse function, then sin^{-1}(sin x) = sin(sin^{-1}x) = x. That is, they undo each other. The same notation is used for the inverse cosine and tangent: cos^{-1}x, tan^{-1}x. Some older calculators may use the notation arcsin(arcsine), arccos(arccosine), and arctan(arc tangent), or they may further abbreviate these as asin, acos, atan. However, these are the same things as the corresponding inverse functions.

Solving trig equations, for the most part, use the same skills you already know for other types of algebraic equations. However, you need to be aware of the cyclic nature of trig functions and the varying signs of these functions in different quadrants, in order to get the correct and complete answers. For example, solve

√2̅cos(x) + 1 = 0, 0 ≤ x ≤ 2????

First notice that frequently, a domain of the equation is specified. This is the 0 ≤ x ≤ 2???? part. In most trig equations, where the unknown is in the trig expression, a domain needs to be specified or there will be an infinite number of solutions. By the way, the expression that is being acted upon by the trig function is called the argument of that function. For example the argument of sin(x² + 7) is x² + 7. In our example, the argument of the cosine is simply x.

So to solve this, you use your algebra skills to get,

So we can take the inverse cosine of both side and use our calculator to find x, and you would find that x =3????/4. However, let’s look at the equation before we take the inverse cosine. You may be asked to solve this without a calculator. You should notice that the cos(x) is equal to an entry in the table of common values I presented in my last post (except for the minus sign). It appears that our answers are associated with the angle ????/4.

Now we have two trig identities that will solve this for us, namely

So, as we want the negative of cos(????/4), then the two angles that will solve this equation within the domain 0 ≤ x ≤ 2???? are ???? – ????/4 = 3????/4 and ???? + ????/4 = 5????/4. Notice that your calculator only gave one of these answers.

Instead of using the trig identities directly, I prefer to use the unit circle to guide me to all correct answers. I highly recommend drawing a unit circle for a particular problem to help guide you to the correct solutions. For this problem:

For practice, draw the unit circle for the same problem, with the provided domain as -???? ≤ x ≤ ????. You should be able to see that the two answers would be -3????/4 and 3????/4.

I will now do some examples of using the trig identities covered in the previous posts. But before I do, I want to show you a table that gives the values of the main trig functions for common angles:

Angle, ????

sin????

cos????

tan????

0

0

1

0

????/6, 30°

1/2

√3̅/2

1/√3̅

????/4, 45°

1/√2̅

1/√2̅

1

????/3, 60°

√3̅/2

1/2

√3̅

????/2, 90°

1

0

Undefined

????, 180°

0

-1

0

3????/2, 270°

-1

0

Undefined

Now let’s do some examples:

If cos(????) = 0.8829 and ???? is in the first quadrant, find cos(3????/2 – ????).

According to the identity developed before, cos(3????/2 – ????) = -sin????. But what is sin(????)? To find this, we need the Pythagorean identity, sin²(????) + cos²(????) = 1:

2. If sin(????) = 0.1736, and ???? is in the first quadrant, find tan(3????/2 + ????).

According to the identity developed before, tan(3????/2 + ????) = -1/tan???? = -cos????/sin????. Again, we need the Pythagorean identity to find cos????:

3. Find the exact value of cos(5????/6) (without a calculator).

The clues here are that I have been developing trig identities and I just gave you a table of exact values of trig functions for common angles. Perhaps we can equate 5????/6 in terms of a common angle. Notice that 6????/6 – ????/6 = 5????/6. That is 5????/6 = ???? – ????/6. We have a trig identity for this: cos(???? – ????) = -cos????. In our case ???? = ????/6 and cos(????/6) = √3̅/2 so cos(5????/6) = -√3̅/2.

By the way, you can find a decimal approximation to √3̅/2, but this will just be an approximation no matter how many decimal places you include since √3̅/2 is an irrational number and has non-repeating decimals. So the exact value is √3̅/2 since we have agreed that √3̅ is the notation for the exact square root of 3.

4. If sin???? = 4/5 and ????/2 < ???? < ????, find the exact value of cos????

The part ????/2 < ???? < ???? means that the angle is in the second quadrant which means the cosine is negative. As before, we can find the cosine via the Pythagorean identity:

The identities shown in my last post showed how measuring ???? from the negative x-axis affected its trig functions. Today’s post shows how measuring ???? from the y-axis affects its trig functions.

Consider the diagram below:

The angle ???? is shown measured from both the x and the y axes. The angle ???? measured from the y-axis is ????/2 – ???? when conventionally measured from the positive x-axis. Notice that the right triangles formed from both measurements are identical – just in different orientations. The x coordinate of the ????/2 – ???? angle on the unit circle, cos(????/2 – ????), is the same as the y coordinate of the ???? angle, sin????. Similarly the y coordinate of the ????/2 – ???? angle on the unit circle, sin(????/2 – ????), is the same as the x coordinate of the ???? angle, cos????. This illustrates the following identities:

I’ve introduced another trig function here, the cotangent, abbreviated cot. The cotangent is the reciprocal of the tangent.

When solving equations involving trig functions (which we will eventually do), these and the following identities can be used to convert sines to cosines and vice versa.

Now let’s measure ???? from the other side of the y-axis. This gives us the conventional angle ????/2 + ????:

The only change here is that the x coordinate on the unit circle is now negative. So the resulting identities are:

Now the angle to the negative y-axis is 3????/2 (270°). An angle measured to the left from the negative y-axis is the conventional angle 3????/2 – ????. The coordinates of this angle on the unit circle are swapped and negative of the angle ???? in the first quadrant. So the picture looks like this:

Now let’s use the unit circle to see some of the common trig identities. These identities (rules) will be used in future posts.

Let’s assume we have an acute angle ????. An acute angle is one that is between 0 and ????/2 (or 0 to 90°). The following identities are valid for any angle, not just acute ones – it is just easier to see the logic in the diagram if we assume this.

The following picture shows the relationship between an angle ???? in the first quadrant, and an angle in the second quadrant which is symmetric with ????:

You can see that to measure this symmetric angle from the postive x-axis, you just subtract it from ????. The coordinates of the intersected point on the unit circle are negative for the x coordinate but the same y coordinate as the original angle ????. So the following identities are evident from this picture:

Now let’s look at a symmetric angle in the third quadrant. To measure this angle from the positive x-axis, you add it to ????. The corresponding coordinates of the intersected point on the unit circle are both the negative of the coordinates for ????. So the following identities are shown in this picture:

As was mentioned before, angles measured clockwise from the positive x-axis are negative. So the following trig identities are shown in the figure above:

This begins many posts on trigonometry. Trigonometry is often called circular functions in some textbooks. I have found that this topic is troublesome for many of my students. I hope you find these posts useful.

Let’s begin with the unit circle. The below picture is from Wolfram MathWorld:

It is very important that you understand this circle as drawing it and analysing the specifics of the problem at hand will greatly add to your understanding of the problem.

The unit circle, as its name implies, is a circle of radius 1, centered at the origin of a cartesian coordinate system. A right triangle can be formed from a point on the unit circle by dropping a vertical line from the point down to the x-axis. The line from the origin to the point completes the triangle with an angle ???? from the positive x-axis. The x coordinate of the point is the length of the base of this triangle. From the basic definition of the trig function cos ????:

Now I will mainly use radians as the angle measure as this is most frequently used in science and engineering. Conventionally, angles are measured from the positive x-axis. They are positive if you go counter-clockwise and negative if you go clockwise. As 2???? radians are a full circle, multiples of 2???? can be added or subtracted from an angle and you get the same angle. For example ????/4 + 2???? = 9????/4, ????/4 – 2???? = -7????/4, ????/4 + 4???? = 17????/4, ????/4 – 4???? = -15????/4. (In degrees, this is the same as adding multiples of 360°). These are all the same angle and they intersect the unit circle at the same place, so their sines are all equal as well as their cosines.

At this point, we can make some observations about the signs of the cosine, sine and tangent of an angle based on what quadrant the angle falls in. Remember that the tangent of an angle, in terms of the sine and cosine, is sin ????/cos ????:

It is very important to be aware of the quadrant you are working in. When solving trigonometric equations, your calculator may give you an answer that is algebraically correct but is in the wrong quadrant based on the information from the problem at hand. I will illustrate this in future posts.

Looking the right triangle in the first figure above, remember the Pythagorean theorem that the sum of the squares of the two right angle sides equals the square of the hypotenuse. Applying this theorem to the right triangle in the figure, you get the most used trig identity, the Pythagorean Identity:

sin²(????) + cos²(????) = 1

You will probably use this enough so that you will automatically remember it.

In my next post, I will develop more trig identities which will be obvious by looking at the unit circle.