Introduction to Trigonometry
The trigonometric circle
Take an x-axis and an y-axis (orthonormal) and let O be the origin. A circle centered in O and with radius = 1, is called a trigonometric circle or unit circle. Turning counterclockwise is the positive orientation in trigonometry.
An angle is the figure formed by two rays that have the same beginning point. That point is
called the vertex and the two rays are called the sides of the angle (also legs). If we call [OA the
initial side of the angle and [OB the terminal side, then we have an oriented angle. This angle is referred to as
An oriented angle is in fact the set of all angles which can be transformed to each other
by a rotation and/or a translation.
The introduction of the trigonometric circle makes it possible to attach a value to each oriented angle
trigonometric circle and let the initial side of this angle coincide with the x-axis.
Then the terminal side intersects the trigonometric circle in point Z.
Then Z is the representation of the oriented angle a on the trigonometric circle.
There are two commonly used units of measurement for angles. The more familiar unit of measurement is that of degrees.
A circle is divided into 360 equal degrees, so that a right angle is 90°. Each degree is subdivided into 60 minutes and each minute into 60 seconds. The symbols °, ‘ and ” are used for degrees, arcminutes and arcseconds.
In most mathematical work beyond practical geometry, angles are typically measured in radians
rather than degrees.
An angle of 1 radian determines on the circle an arc with length the radius of the circle. Because the length of a full circle is 2πR, a circle contains 2π radians. Contrariwise, if one draws in the centre of a circle with radius R an angle of θ radians, then this angle determines an arc on the circle with length θ·R. Subdivisions of radians are written in decimal form.
Conversion between radians and degrees.
when an angle is represented in radians, one does only mention the value, not the term ‘rad’.
The trigonometric numbers.
Sine and cosine in the trigonometric circle
Beside sine and cosine other trigonometric numbers are defined as follows :
Gives a graphical representation of the above trigonometric numbers in terms of distances associated with the unit circle.
The graphical representation of the trigonometric numbers in terms of distances associated with the unit circle.
Consequently, the trigonometric numbers have values which are in the following areas:
Some special angles and their trigonometric numbers
Trigonometric numbers of angles in the other quadrants we shell find through the use of the reference angle (see paragraph 2.6.2.)
Sign variation for the trigonometric numbers by quadrant
Inside a quadrant the trigonometric numbers keep the same sign
sign variation for the trigonometric numbers by quadrant
The basic relationship between the sine and the cosine is the Pythagorean or fundamental trigonometric identity:
This can be viewed as a version of the Pythagorean theorem, and follows from the equation
The triangle OPZ.
Calculation of the trigonometric numbers
Asked: all other trigonometric numbers
Because the sine of this angle is positive, the angle is situated in the first or second quadrant. We
determine the other trigonometric numbers as follows:
from the Pythagorean trigonometric identity:
The two possible solutions for some of the trigonometric numbers correspond with the values of
these numbers according to the quadrant in which the angle is situated.
Proof the following identity
Special pairs of angles
The sines, cosines and tangents, cotangents of some angles are equal to the sines, cosines and tangents, cotangents of other angles.
Special pairs of angles.
The use of reference angles is a way to simplify the calculation of the trigonometric numbers at
Associated with every angle drawn in standard position (which means that its vertex is located at
the origin and the initial side is on the positive x-axis) (except angles of which the terminal side
lies “on” the axes, called quadrantal angles) there is an angle called the reference angle.
The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis.
Angles in quadrant I are their own reference angles. For angles in other quadrants, reference
angles are calculated this way:
The reference angle and the given angle form a pair of angles to which you can apply the properties in the previous paragraph. Due to these properties, the value of a trigonometric number at a given angle is always the same as the value of that angle’s reference angle, except when there is a variation in sign. Because we know the signs of the numbers in different quadrants, we can simplify the calculation of a trigonometric number at any angle to the value of the number at the reference angle for that angle, to be found in the table in paragraph 2.2.
How to find all angles
To find the angle if given a certain trigonometric number, usually there are 2 solutions.
Calculators give the most obvious solution, but in practical situations, there can be a second solution, or the second solution can be the only correct solution.
In this case the user must adjust the solution given by the calculator. The following table gives for positive and negative trigonometric numbers the quadrant in which the solution given by the calculator, is situated, and in the last column the quadrant of the second solution:
The trigonometric functions
If p satisfies this definition, then all positive and negative numbers which are an integer multiple of p also satisfy this definition.
Therefore we call the smallest positive number which satisfies this definition the period P of the function. Graphically this periodicity means that the form of the graph of f(x) repeats itself over subsequent intervals with length P.
Even and odd functions
A function f is called EVEN if:
Consequently two points with opposite x-values must have the same y-value. So the graph must be symmetric about the y-axis.
A function f is called ODD if:
Consequently two points with opposite x-values must have opposite y-values. So the graph is symmetric about the origin.
we consider the argument of trigonometric functions always in terms of radians.
The period of this function is 2p. This function is odd, as opposite angles have opposite sines.
The period of this function is p. This function is odd, as opposite angles have opposite tangents.
The tangent function
The period of this function is p. This function is odd, as opposite angles have also opposite cotangents.
The cotangent function
The secant function
The cosecant function
Orthogonal triangles used to set up the formulas in this paragraph
In a right triangle with a as the right angle, the following formulas apply:
If we draw in the triangle above a circle segment with centre in B and radius a (see the first triangle in fig. 15), then we recognize a segment of a circle with radius a.
The adjacent side of the right angle c and the opposite side b have resp. the following lengths:
In a similar way, by considering a circle segment with centre in C and radius a (see second triangle in fig. 15), we find:
In words :
The cosine of an acute angle is the ratio of the length of the adjacent rectangle side and the length of the hypotenuse.
The sine of an acute angle is the ratio of the length of the opposite rectangle side and the length of the hypothenuse.
By division of the first two formulas we get:
If we do the same with the last two formulas, we get:
In words :
The tangent of an acute angle is the ratio of the length of the opposite rectangle side and the length of the adjacent rectangle side.
The cotangent of an acute angle is the ratio of the length of the adjacent rectangle side and the length of the opposite rectangle side.
First, remember that also for oblique triangles the sum of angles is 180°.
An oblique triangle is any triangle that is not a right triangle. It could be an acute triangle (all three angles of the triangle are less than right angles) or it could be an obtuse triangle (one of the three angles is greater than a right angle).
The sine rules
Apply the same reasoning with the altitude from B to the opposite side b to divide the triangle in two right triangles and derive similar formulas in which occur a and the opposite angle α. Then we get:
The cosine rules
The same expression can be derived if S lies outside side a.
Then, similar expressions can be derived for the other angles.
Summarized, in this way we get:
These statements relate the lengths of the sides of a triangle to the cosine of one of its angles.
For example, the first statement states the relationship between the sides of lengths a , b and c, where α denotes the angle contained between sides of lengths b and c and opposite to the side with length a.
These rules look like the Pythagorean theorem except for the last term, and if you deal with a right triangle, that last term disappears, so these rules are actually a generalization of the Pythagorean theorem.
Solving oblique triangles
One of the most common applications of the trigonometry is solving triangles – finding missing sides and/or angles, given some information about a triangle. The process of solving triangles can be broken down into a number of cases.
In these situations we will use 3 sorts of formulas, applicable in all triangles:
· the sum of all angles is 180°
· the sine rule : relates two sides to their opposite angles
· the cosine rule : relates the three sides of the triangle to one of the angles.
Naturally, the given information must be such that the given elements allow a triangle:
· the sum of the given angles can not be larger than 180°,
· and the sides must meet the triangle inequality which states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
a. If you know one angle and the two adjacent sides.
Then, there is 1 solution:
you can determine the opposite side by using the cosine rule, another angle by using the sine rule and the remaining angle as 180° minus the two already determined angles.
Attention: the sine rule gives two solutions for the second angle (supplementary angles).
Test the solutions by verifying the properties of a triangle (see exercises).
b. If you know one side and the two adjacent angles.
Then there is 1 solution:
the third angle is immediately known as 180° minus the two given angles; the two remaining sides can be determined by using the sine rule.
c. If you know all three sides of a triangle.
Then there is 1 solution:
determine one angle by using a cosine rule, the second angle can be determined by using another cosine rule or by using the sine rule.
The last angle can be determined by the property of triangles that the sum of all angles must be 180°.
d. If you know sides a and b and β (one of the adjacent angles). In this case, there can be 0, 1 or
Determine the angle a by using the sine rule. You will get 0 (if sin a > 1) or 2 solutions (supplementary angles have the same sine). For each solution determine the missing angle g, and then the length of side c by using the sine rule. Finally you test if each solution which you find is acceptable: you can not have negative angles or sides.
Special lines in a triangle
An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side.
This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. The three altitudes intersect in a single point, called the orthocenter of the triangle.
The orthocenter lies inside the triangle if and only if the triangle is acute.
A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas.
The three medians intersect in a single point, the triangle’s centroid. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice the distance between the centroid and the midpoint of the opposite side.
An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point, which always lies inside the triangle.
A perpendicular bisector of a side of a triangle is a straight line passing through the midpoint of the side and being perpendicular to it, i.e. forming a right angle with it.
The three perpendicular bisectors meet in a single point, the triangle’s circumcenter; this point may also lie outside the triangle.
In an isosceles triangle, two sides are equal in length. The unequal side is called its base and the
angle opposite the base is called the “vertex angle”.
The equal sides are called the legs of the triangle. The base angles of an isosceles triangle are always equal.
Property : the altitude and the median from the vertical angle coincide.
In an equilateral triangle all sides have the same length. Therefore all three angles are equal to each other, and thus 60°.
Property : the altitude from a certain angle coincides with the median from that angle. orthocenter and the centroid coincide.
In this paragraph, we discuss formulas involving the trigonometric numbers of a sum or difference of two angles, of a double or half angle, conversions between sums and products of sines and cosines….
As we don’t want you to learn these formulas by heart, it is important to understand their mutual connection, the way how one formula can be derived from another formula.
We also want to emphasize that the knowledge of these formulas facilitates solving integrals of trigonometric functions.
Sum and difference formulas
Let’s start with the addition formula for the sine. Then the other formulas can be derived in an easy way.
You see that the sine formulas keep the plus- or minus sign, but mix the trigonometric functions.
The cosine formulas change the sign but hold the trigonometric functions together.
Trigonometric numbers in terms of tan a/2
By performing the same operation on (7) we find :
Conversions sum/difference of angles into product of angles and vice versa .
These four formulas convert the product of two cosines and/or sines with a different argument into a sum. The reverse formulas we get by bringing factor ½ to the other side and by substitution: