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4.2 Trigonometric functions

From Förberedande kurs i matematik 1

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       Theory          Exercises      

Contents:

  • The trigonometric functions cosine, sine and tangent.

Learning outcomes:

After this section, you will have learned:

  • The definition of acute, obtuse and right angles.
  • The definition of cosine, sine and tangent.
  • The values of cosine, sine and tangent for the standard angles 0, 6 , 4 , 3 and 2 by heart.
  • To determine the values of cosine, sine and tangent of arguments that can be reduced to a standard angle.
  • To sketch graphs of cosine, sine and tangent.
  • To solve trigonometric problems involving right-angled triangles.

Trigonometry of right-angled triangles

In the right-angled triangle below, the ratio between the length a of the side opposite the angle and the length b of the adjacent side is called the tangent of the angle u, and is written as tanu.

[Image]

tanu=ba

The value of the ratio ba is not dependent on the size of the triangle, but only on the angle u. For different values of the angle, you can get the value of the tangent either from a trigonometric table or by using a calculator (the relevent button is usually named tan).

Example 1

How high is the flagpole?

[Image]

The flagpole and its shadow form a rectangular triangle where the vertical side is unknown (marked with x below).

[Image]

From the definition of tangent, we have that

tan40=x5 m

and since tan400.84 we get

x=5mtan405m0.84=4.2m.

Example 2

Determine the length of the side designated with the x in the figure.

[Image]

If we call the angle at the far left u there are two ways to construct an expression for tanu.

[Image]

tanu=4022

[Image]

tanu=x60

Equality of the two expressions for tanu gives

4022=x60

which leads to x=604022=33.

There are two other ratios in right-angled triangles that have special names. The first is cosu=bc ("cosine of u") and the second is sinu=ac (" sine of u").

[Image]

cosusinu=cb=ca

Like the tangent the ratios that define the cosine and sine do not depend on the size of the triangle, but only on the angle u.

Example 3

[Image]

In the triangle on the left

cosusinu=54=53

[Image]

From the definition of sine we have

sin38=x5,

and if we know that sin380.616 then

x=5sin3850.6163.1.

[Image]

Cosine is the ratio between the adjacent side and the hypotenuse, so

cos34=x3.

Thus

x=3cos34.

Example 4

Determine sinu in the triangle

[Image]

With the help of the Pythagorean theorem the side on the right can be determined:

[Image]

12=212+x2x=23 

and thus sinu=132=23 .


Some standard angles

For some angles, namely 30°, 45° and 60°, it is relatively easy to calculate the exact values of the trigonometric functions.

Example 5

We start with a square having sides of length 1. A diagonal of the square divides the right angles in opposite corners into two equal parts of 45°.


[Image]


Using the Pythagorean theorem, we can determine the length x of the diagonal:

x2=12+12x=12+12=2. 

Each triangle has the diagonal as the hypotenuse. Thus we can obtain the value of the trigonometric functions for the angle 45:


[Image]

cos45sin45tan45=12=12=11=1

Example 6

Imagine an equilateral triangle where all sides have length 1. The angles of the triangle are all 60°. The triangle can be divided into two halves by a line that divides the angle at the top in equal parts.


[Image]


The Pythagorean theorem allows us to calculate that the length of the vertical side of half-triangle is x=32 . Using the definitions we then get that


[Image]

cos30sin30tan30=132=23=112=21=1232=13cos60sin60tan60=112=21=132=23=1232=3


Trigonometric functions for general angles

For angles less than 0° or greater than 90° the trigonometric functions are defined using the unit circle (that is the circle that has its centre at the origin and has radius 1).

The trigonometric functions cosu and sinu are the x- and y- coordinates of the point on the unit circle reached by turning through the angle u, as shown in the diagram on the right.

[Image]

The tangent function is then defined as

tanu=sinucosu

and the value of the tangent can be interpreted as the slope for the radial line.


Example 7

From the figures below, we obtain the values of cosine and sine.

[Image]

cos104sin104tan104−0.240.970.97−0.24−4.0

[Image]

cos201sin201tan201−0.93−0.36−0.93−0.360.4

Example 8

Which sign do the following have?

  1. cos209

    Since the angle 209 can be written as 209=180+29 the angle corresponds to a point on the unit circle which lies in the third quadrant. The point has a negative x-coordinate, which means that cos209 is negative .

[Image]

  1. sin133

    The angle 133 is equal to 90+43 and gives a point on the unit circle which lies in the second quadrant. The quadrant has points with positive y-coordinate, and therefore sin133 is positive.

[Image]

  1. tan(−40)

    By drawing angle −40 in the unit circle one obtains a radial line which has a negative slope, so that tan(−40) is negative.

[Image]

Example 9

Calculate sin32.

Note that

32=64=63+=2+6

This shows that the point on the unit circle corresponding to the angle 23 is in the the second quadrant and makes the angle 6 with the positive y-axis. If we draw an extra triangle as in the figure below on the right, we see that the 23- point on the unit circle has a y-coordinate which is equal to the adjacent side cos6=32 . So we have that

sin32=23. 

[Image]


Graphs of the trigonometric functions

In the last section we used a unit circle to define the cosine and sine of arbitrary angles, and we will often use the unit circle in the future, for example, to derive trigonometric relationships and solve trigonometric equations. However, there are certain characteristics of the trigonometric functions that are better illustrated by drawing their graphs.


[Image]

The graph of the sine function

[Image]

The graph of the cosine function

[Image]

The graph of the tangent function


In these graphs, we might observe several things more clearly than in the unit circle. Some examples are:

  • The curves for cosine and sine repeat themselves after a change in angle of 2, that is cos(x+2)=cosx and sin(x+2)=sinx. To see why this is true, note that on the unit circle 2 corresponds to a complete revolution, and after a complete revolution we return to the same point on the circle.
  • The curve for the tangent repeats itself after a change in angle of , that is tan(x+)=tanx. Two angles which differ by share the same line through the origin of the unit circle and thus their radial lines have the same slope.
  • Except for a phase shift of 2 the curves for cosine and sine are identical, that is cosx=sin(x+2); more about this in the next section.


The curves can also be important when examining trigonometric equations. With a simple sketch, you can often get an idea of how many solutions an equation has, and where the solutions lie.

Example 10

How many solutions has the equation cosx=x2 ( where x is measured in radians)?

By drawing the graphs y=cosx and y=x2 we see that the curves intersect in two points. So there are two x-values for which the corresponding y-values are equal. In other words, the equation has two solutions.

[Image]


Exercises

Study advice

Basic and final tests

After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.


Keep in mind that...

If you have studied trigonometry, then you should not be afraid to use it in geometric problems. It often produces a simpler solution.

You may need to spend a lot of time understanding how to use a unit circle to define the trigonometric functions.

You should get into the habit of calculating with precise trigonometric values. It is good training in calculating fractions and will eventually help you handle algebraic rational expressions.

Reviews

For those of you who want to deepen your understanding or need more detailed explanations consider the following references:

Learn more about trigonometry from Wikipedia

Learn more about the unit circle from Wikipedia


Useful web sites

Experiment with the sine and cosine functions

Experiment with Euclidean geometry

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