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A triangle is made up of three straight line segments joined together end to end, creating three edges and three vertices. The study of triangles falls under a topic called trigonometry which looks into trigonometric functions such as Sine, Cosine and Tangent. A circle is made of a curved boundary that is closed, forming no corners or edges.
You can safely say that triangles and circles are two very different shapes. Well, did you know that they are more closely related than they appear despite their differences? In this discussion, you shall look into extending the definitions of trigonometric functions onto general angles by introducing the unit circle.
Let's begin this topic by introducing the definition of an angle below.
An angle is a figure formed by a pair of rays that share an endpoint.
This endpoint is called the vertex of an angle. The size of the vertex varies based on how wide or narrow the two rays intersect at their common endpoint.
In a Cartesian coordinate system, let us denote an angle by \(\theta\). If the vertex of \(\theta\) is situated at the origin O and its initial side (first ray) lies in the positive x-axis, then \(\theta\) is in the standard position. The second ray of the angle is called the terminal side.
The standard position of an angle, StudySmarter Originals
The Cartesian coordinate plane is divided into 4 quarters called quadrants. These four quadrants are listed below.
Quadrant I: Upper right quadrant made up of the positive \(x\) and \(y\)-axes.
Quadrant II: Upper left quadrant made up of the negative \(x\)-axis and positive \(y\)-axis.
Quadrant III: Bottom left quadrant made up of the negative \(x\) and \(y\)-axes.
Quadrant IV: Bottom right quadrant made up of the positive \(x\)-axis and negative \(y\)-axis.
The four quadrants, StudySmarter Originals
If the terminal side of an angle falls within a specific quadrant while the angle is in its standard position, then the angle is said to be in that quadrant. Here is an example of an angle in each quadrant.
Angle 56^{o} lies in the first quadrant.
Quadrant I angle, StudySmarter Originals
Angle –174^{o} lies in the second quadrant.
Quadrant II angle, StudySmarter Originals
Angle 203^{o} lies in the third quadrant.
Quadrant III angle, StudySmarter Originals
Angle –47^{o} lies in the fourth quadrant.
Quadrant IV angle, StudySmarter Originals
If the terminal side of an angle (in standard position) lies along the \(x\)-axis or \(y\)-axis, then the angle is called a quadrantal. In other words, an angle is said to be a quadrantal if it is equal to an integer multiple of 90^{o} (or 2π radians). −360^{o}, −270^{o}, −180^{o}, −90^{o}, 0^{o}, 90^{o}, 180^{o}, 270^{o}, and 360^{o }are all quadrantal angles. With that in mind, let's finally define a general angle as follows.
A general angle is any angle formed by two rays, and its measure can be any real value.
The reference angle is an acute angle (less than 90º) between the terminal side and the positive \(x\)-axis, i.e., the initial side lies on the positive \(x\)-axis.
The reference angle is always positive. The size of the reference angle depends on the given standard angle and quadrant it lies on.
The diagram below shows how you can solve for the reference angle in each quadrant. Let's denote the reference angle by r and the given angle by \(\theta\).
Reference angle for each quadrant, StudySmarter Originals
What is the reference angle, r, of the given angle below?
Reference angle example 1, StudySmarter Originals
Solution
The angle above lies in the second quadrant. Thus, given the formula for finding reference angles, you deduce that
\[r=180^{o}-152^{o}=28^{o}\]
What is the reference angle, r, of the given angle below?
Reference angle example 2, StudySmarter Originals
Solution
The angle above lies in the third quadrant. Thus, given the formula for finding reference angles, you deduce that
\[r=224^{o}-180^{o}=44^{o}\]
The reference angle is typically used to find the Sine and Cosine of the standard angle. It is used to simplify trigonometric calculations when a given angle is large or of negative value. You shall observe this in the next section below. Reference angles can also be used to find the coordinates of a point on a (unit) circle.
The unit circle is a circle of radius 1, constructed on a Cartesian plane and centered at the origin. The unit circle is used to describe the trigonometric relationships (such as Sine, Cosine, and Tangent) of a right triangle by linking the measures of its angles and sides to one another. Below is a graphical representation of a unit circle.
Unit circle, StudySmarter Originals
Pythagoras' Theorem and the Unit Circle
Return to the diagram above and observe the right triangle inscribed inside the unit circle where r represents the radius of this circle. By Pythagoras's Theorem, the side lengths of this right triangle are related by the following rule.
\(y^{2}+x^{2}=r^{2}\)
This gives us the equation of the unit circle as
\(y^{2}+x^{2}=1\)
From trigonometric ratios, you know that \(x = cos θ\) and \(y = sin θ\). Substituting this into the equation above, you obtain an important trigonometric identity.
\((sin θ)^{2} + (cos θ)^{2} = 1\)
Let us now label some notable angles corresponding to the coordinates in the unit circle to aid your visual representation. Remember that you need to consider all four quadrants on this plane.
The unit circle is 360^{o}. Thus, each quadrant is equal to 90^{o}. Angles in Quadrant I span from 0^{o} to 90^{o}; angles in Quadrant II range from 90^{o} to 180^{o}; angles in Quadrant III are covered from 180^{o} to 270^{o}; and angles in Quadrant IV go from 270^{o} to 360^{o}_{.}
Reading the unit circle, StudySmarter Originals
The \(x\) and \(y\)-coordinates surrounding this unit circle give us the value of each angle you want to achieve based on the given trigonometric function.
At first glance, this may look rather daunting which probably has discouraged you from memorizing the contents of this unit circle at all. But, fear not!
It actually suffices for you to remember the values of 2 quadrants, Quadrants 1 and 2, from 0 to π. Notice that Quadrants 3 and 4 are reflections of Quadrants 1 and 2 about the \(x\)-axis. This means that the \(y\)-coordinates of Quadrants 3 and 4 are the reciprocal values of the \(y\)-coordinates of Quadrants 1 and 2.
Let us now move on to reading this unit circle. The coordinates on this unit circle are read as
\((x, y) = (cos θ, sin θ)\)
since \(x = cos θ\) and \(y = sin θ\). Look back at the initial diagram of a unit circle and recall the Sine and Cosine Ratio to determine this. With that in mind, for an angle \(θ\), you can find the value of \(sin θ\) and \(cos θ\) just by looking at the corresponding \(x\) and \(y\)-coordinates, respectively.
So how about \(tan θ\)? In this case, you simply use the Tangent Ratio to solve this. Recalling that \(tan\theta=\frac{y}{x}\), you obtain
\(tan\theta=\frac{sin\theta}{cos\theta}\)where the values of \(sin θ\) and \(cos θ\) can be obtained via the unit circle.
Let us now look at some examples that make use of the unit circle.
Find the value of .
Solution
Here, you shall look at the \(y\)-coordinates of each case.
At , the y-coordinate is . Thus, .
At , the y-coordinate is . Thus, .
Calculate the value of .
Solution
Here, you will observe the \(x\)-coordinate for each case.
At , the x-coordinate is . Thus, .
At , the x-coordinate is . Thus, .
Deduce the value of .
Solution
You will use use to solve this.
At , the x-coordinate is while the y-coordinate is . Using the rule above, you obtain
Simplifying this, you get
Similarly, at , the x-coordinate is while the y-coordinate is . From here, you obtain
Simplifying this, you get
Trigonometric functions relate angle measures and the trigonometric ratios of a right-angle triangle. The most common trigonometric functions are Sine, Cosine and Tangent. However, there are reciprocal trigonometric functions, such as Cosecant, Secant and Cotangent as well as inverse trigonometric ratios functions such as Arcsine, Arccosine and Arctangent.
Before continuing with our main topic, it may be helpful to recall the trigonometric ratios mentioned above. These ratios are derived from a right-angle triangle as shown below.
Right angle triangle, StudySmarter Originals
The three sides of this triangle are labelled as the opposite, adjacent and hypotenuse with the angle θ. To remember the Sine, Cosine and Tangent Ratios, you simply use the acronym: SOH CAH TOA. This mnemonic is equivalent to the values below.
Trigonometric ratios, StudySmarter Originals
A more detailed discussion on this topic can be found here: Triangle Trigonometry.
Special angles are angles most commonly used when dealing with trigonometric functions. There are 8 special angle measures to consider, namely, 0^{o}, 30^{o}, 45^{o}, 60^{o}, 90^{o}, 180^{o}, 270^{o} and 360^{o}. The unit of measurement used here is called degrees. You can also define these angles in a ratio form known as a radian.
Special angles may be referred to as standard angles in some textbooks.
There are two ways in which you can represent angles:
Degrees (as mentioned above)
Radians
Measuring an angle in degrees is the most common way to denote an angle. On the other hand, radians use the concept of a circle to represent an angle. Say you have a circle with an angle equal to 1 radian (refer to the diagram below).
Radian, StudySmarter Originals
The circle above describes an arc equal to the length of the radius. This is because the length of a full circle is 2πr. In other words, this circle contains 2π radians. So how do you link the degree and radian of an angle to one another? Here is a general formula you can use to convert an angle in degrees to radians and vice versa.
Degree to Radian | Radian to Degree |
With that in mind, the table below shows the values of our mentioned special angles in degrees and radians.
Angle In Degrees | 0^{o} | 30^{o} | 45^{o} | 60^{o} | 90^{o} | 180^{o} | 270^{o} | 360^{o} |
Angle in Radians | 0 |
Now that we have established our set of special angles, you can use the unit circle to deduce the values of trigonometric functions for these angles. This is shown in the table below.
Angle θ In Degrees | 0^{o} | 30^{o} | 45^{o} | 60^{o} | 90^{o} | 180^{o} | 270^{o} | 360^{o} |
Angle θ in Radians | 0 | |||||||
Sin θ | 0 | 1 | 0 | -1 | 0 | |||
Cos θ | 1 | 0 | -1 | 0 | 1 | |||
Tan θ | 0 | 1 | 0 | 0 |
Indeterminate Form
The value of tan (90^{o}) and tan (270^{o}) cannot be determined as it is equal to \(\frac{1}{0}\) and \(-\frac{1}{0}\) , respectively. This is known as the indeterminate form. You can write this as ∞.
Try plugging this into your calculator. You will find that this will display an error message. In this case, the ends of the curve will have no limits and thus tend to infinity.
In this section, you shall look at the graph properties of the three principle trigonometric functions: Sine, Cosine and Tangent. You can use the unit circle to plot each graph.
Let's begin with the Sine function. Let us now plot these values on the Cartesian plane (note that we are only considering positive \(x\)-values here). It may be helpful to write the \(x\)-axis in the form of radians here.
Sine function, StudySmarter Originals
Let's now extend this graph and consider the negative \(x\)-values.
Extended sine graph, StudySmarter Originals
From here, you can deduce the following main features of a Sine graph.
The sine graph repeats itself after a period of \(2\pi\) (or 360^{o}). This means that the function is periodic with a period of \(2\pi\).
The sine graph is symmetric about the origin, making it an odd function.
The minimum value of the sine function is –1. The sine function reaches its minimum value at \(\frac{3\pi}{2}\) and every \(2\pi\) before and after that.
The maximum value of the sine function is 1. The sine function reaches its maximum value at \(\frac{\pi}{2}\) and every \(2\pi\) before and after that.
The domain of the sine function is the set of all real numbers, (-∞, ∞).
The range of the sine function is [-1, 1].
The sine graph crosses the \(x\)-axis at every multiple of \(\pi\). These are the zeros of the function where \(\sin{\theta}=0\).
The sine function is positive in Quadrants I and II and negative in Quadrants III and IV.
The point above discusses the sign of a particular trigonometric function. For the sine function, it is positive for every [kπ, 2kπ] and negative for every [2kπ, 3kπ], where k is an integer. You can apply the same idea to the cosine and tangent functions below.
Another name for the Sine function is the sinusoidal curve.
Let us now move on to the Cosine Function. We shall now plot these values on the Cartesian plane (note that we are only considering positive \(x\)-values here). It may be helpful to write the \(x\)-axis in the form of radians here.
Cosine function, StudySmarter Originals
Let us now extend this graph and consider the negative \(x\)-values.
Extended cosine function, StudySmarter Originals
By observing this graph, you can determine the key features of the cosine function as below.
The cosine graph repeats itself after a period of \(2\pi\) (or 360^{o}). This means that the function is periodic with a period of \(2\pi\).
The cosine graph is symmetric about the y-axis, making it an even function.
The minimum value of the cosine function is –1. The cosine function reaches its minimum value at \(\pi\) and every \(2\pi\) before and after that.
The maximum value of the cosine function is 1. The cosine function reaches its maximum value at 0 and every \(2\pi\) before and after that.
The domain of the cosine function is the set of all real numbers, (-∞, ∞).
The range of the cosine function is [-1, 1].
The cosine graph crosses the \(x\)-axis at every multiple of \(\frac{\pi}{2}\). These are the zeros of the function where \(\cos{\theta}=0\).
The cosine function is positive in Quadrants I and IV and negative in Quadrants II and III.
Finally, you shall look at the Tangent Function. As before, you shall plot this graph as below.
Tangent graph, StudySmarter Originals
Again, you shall extend the graph and consider the negative \(x\)-values. Notice how the asymptotes get very close to the blue dotted lines but never touch.
Extended tangent graph, StudySmarter Originals
You can thus deduce the key features of the Tangent function as follows.
The tangent graph repeats itself after a period of \(\pi\) (or 180^{o}). This means that the function is periodic with a period of \(\pi\).
The tangent graph does not have a minimum or maximum value due to the presence of the asymptotes.
The tangent graph has asymptotes, which are values that the function will get closer to infinity.
The asymptotes occur at \(\frac{\pi}{2}\) and every \(\pi\) before and after that (this is where the graph tends to infinity at both ends).
The tangent graph crosses the \(x\)-axis at every multiple of \(\pi\). These are the zeros of the function where \(\tan{\theta}=0\).
The domain of the tangent function is the set of all real numbers except the values where cos(x) is equal to 0, that is, the values \(frac{\pi}{2}+\pi n\), for all integers n.
The range of the tangent function is all real numbers.
The tangent function is positive in Quadrants I and III and negative in Quadrants II and IV.
Angle θ In Degrees | 0^{o} | 30^{o} | 45^{o} | 60^{o} | 90^{o} | 180^{o} | 270^{o} | 360^{o} |
Angle θ in Radians | 0 | |||||||
Sin θ | 0 | 1 | 0 | -1 | 0 | |||
Cos θ | 1 | 0 | -1 | 0 | 1 | |||
Tan θ | 0 | 1 | 0 | 0 |
A general angle is any angle formed by two rays on the Cartesian coordinate plane. The rays share an endpoint called the vertex of an angle. If the vertex of an angle lies on the origin O and its initial side lies in the positive x-axis, then it is in the standard position. The second ray of the angle is called the terminal side.
Trigonometric functions are functions that relate 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.
We can use the unit circle to solve trigonometric functions of general angles.
You can follow one of these rules:
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