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4.2 Trigonometrische Funktionen

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Contents:

The trigonometric functions cosine, sine and tangent.

Learning outcomes:

After this section, you will have learned :

The concepts of acute, obtuse and right angles.
The definition of cosine, sine and tangent in the unit circle.
The values of cosine, sine and tangent for the standard angles \displaystyle 0, \displaystyle \pi/6 , \displaystyle \pi/4 , \displaystyle \pi/3 and \displaystyle \pi/2 by heart.
To determine the values of cosine, sine and tangent of arguments that can be reduced to a standard angle in a quadrant of the unit circle.
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 opposite side \displaystyle a and the adjacent side \displaystyle b is called the tangent of the angle \displaystyle u and is written as \displaystyle \tan u.

4.2 - Figur - Rätvinklig triangel med vinkeln u och kateterna a och b

\displaystyle \tan u = \displaystyle \frac{a}{b}

The value of the ratio \displaystyle \frac{a}{b} is not dependent on the size of the triangle, but only on the angle \displaystyle u. For different values of the angle, you can get the equivalent 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?

4.2 - Figur - Flaggstång

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

4.2 - Figur - Flaggstångstriangel

From the definition of tangent, we have that Vorlage:Displayed math

and since \displaystyle \tan 40^\circ \approx 0\textrm{.}84 we get Vorlage:Displayed math

Example 2

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

4.2 - Figure - A double triangle

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

4.2 - Figure - A double triangle with the small triangle highlighted

\displaystyle \tan u = \displaystyle \frac{22}{40}

4.2 - Figure - A double triangle with the larger triangle highlighted

\displaystyle \tan u = \dfrac{x}{60}

Equality of the two expressions for \displaystyle \tan u gives Vorlage:Displayed math

which leads to \displaystyle x=60 \cdot \displaystyle \frac{22}{40} = 33.

There are two other ratios in right-angled triangles that have special names, and one is \displaystyle \cos u = b/c ("cosine of \displaystyle u") and the other \displaystyle \sin u = a/c (" sine of \displaystyle u").

4.2 - Figur - Rätvinklig triangel med vinkeln u och sidorna a, b och c

\displaystyle \begin{align*} \cos u &= \frac{b}{c}\\[8pt] \sin u &= \frac{a}{c} \end{align*}

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 \displaystyle u.

Example 3

		4.2 - Figur - Rätvinklig triangel med vinkeln u och sidor 3, 4 och 5		In the triangle on the left Vorlage:Displayed math

		4.2 - Figur - Rätvinklig triangel med vinkeln 38° och sidor x och 5		The definition of sine gives that Vorlage:Displayed math and if we know that \displaystyle \sin 38^\circ \approx 0\textrm{.}616 then we get Vorlage:Displayed math

		4.2 - Figur - Rätvinklig triangel med vinkeln 34° och sidor 3 och x		Cosine is the ratio between the adjacent side and the hypotenuse Vorlage:Displayed math Thus Vorlage:Displayed math

Example 4

Determine \displaystyle \sin u in the triangle

4.2 - Figur - Rätvinklig triangel med vinkel u och sidor ½ och 1

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

4.2 - Figur - Rätvinklig triangel med vinkel u och sidor ½, x och 1

\displaystyle 1^2= \bigl( \tfrac{1}{2} \bigr)^2 + x^2 \quad\Leftrightarrow\quad x = \frac{\sqrt{3}}{2}

and thus \displaystyle \sin u = \frac{\sqrt{3}/2}{1} = \frac{\sqrt{3}}{2}.

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°.

4.2 - Figure - Two unit squares

Using the Pythagorean theorem, we can determine the length \displaystyle x of the diagonal, Vorlage:Displayed math

Each triangle has the diagonal as the hypotenuse, thus we can obtain the value of the trigonometric functions for the angle \displaystyle 45^\circ.

4.2 - Figur - Enhetskvadrat vars halva är en rätvinklig triangel

\displaystyle \begin{align*} \cos 45^\circ &= \frac{1}{\sqrt{2}}\\[8pt] \sin 45^\circ &= \frac{1}{\sqrt{2}}\\[8pt] \tan 45^\circ &= \frac{1}{1}= 1\\ \end{align*}

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.

4.2 - Figure - Two equilateral triangles

The Pythagorean theorem shows that the vertical side of either half-triangle is \displaystyle x=\sqrt{3}/2. From one of these half-triangles we get that

4.2 - Figure - A half of an equilateral triangle

\displaystyle \begin{align*} \cos 30^\circ &= \frac{\sqrt{3}/2}{1} = \frac{\sqrt{3}}{2}\,;\\[8pt] \sin 30^\circ &= \frac{1/2}{1} = \frac{1}{2}\,;\\[8pt] \tan 30^\circ &= \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}\,;\\ \end{align*} \qquad\quad \begin{align*} \cos 60^\circ &= \frac{1/2}{1} = \frac{1}{2}\\[8pt] \sin 60^\circ &= \frac{\sqrt{3}/2}{1} = \frac{\sqrt{3}}{2}\\[8pt] \tan 60^\circ &= \frac{\sqrt{3}/2}{1/2}=\sqrt{3}\\ \end{align*}

Trigonometric functions for general angles

For angles of 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 a radius 1).

The trigonometric functions \displaystyle \cos u and \displaystyle \sin u are x- and y- coordinates of the intersection between the unit circle and the radial line that forms the angle \displaystyle u with the positive x-axis.

4.2 - Figur - Enhetscirkeln med vinkeln u och punkten (cos u, sin u)

Tangent function is defined as

Vorlage:Displayed math

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.

	4.2 - Figure - The unit circle with the angle 104° and the point (-0.24,0.97)		\displaystyle \begin{align} \cos 104^\circ &\approx -0{,}24\\[8pt] \sin 104^\circ &\approx 0{,}97\\[8pt] \tan 104^\circ &\approx \dfrac{0{,}97}{-0{,}24} \approx -4{,}0\\ \end{align}
	4.2 - Figur - Enhetscirkeln med vinkeln 201° och punkten (-0,93; -0,36)		\displaystyle \begin{align} \cos 201^\circ &\approx -0{,}93\\[8pt] \sin 201^\circ &\approx -0{,}36\\[8pt] \tan 201^\circ &\approx \dfrac{-0{,}36}{-0{,}93} \approx 0{,}4\\ \end{align}

Example 8

Which sign do the following have?

\displaystyle \cos 209^\circ Since the angle \displaystyle 209^\circ can be written as \displaystyle 209^\circ = 180^\circ + 29^\circ 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 \displaystyle \cos 209^\circ is negative .		4.2 - Figur - Enhetscirkeln med vinkeln 209° och linjen x = cos 209°
\displaystyle \sin 133^\circ The angle \displaystyle 133^\circ is equal to \displaystyle 90^\circ + 43^\circ and gives a point on the unit circle which lies in the second quadrant. The quadrant has points with positive y-coordinate, and therefore \displaystyle \sin 133^\circ is positive.		4.2 - Figure - The unit circle with the angle 133° and the line y = sin 133°
\displaystyle \tan (-40^\circ) By drawing angle\displaystyle -40^\circ in the unit circle one obtains a radial line which has a negative slope, i.e. \displaystyle \tan (-40^\circ) is negative.		4.2 - Figure - The unit circle with the angle -40° and the line with slope tan -40°

Example 9

Calculate \displaystyle \,\sin\frac{2\pi}{3}.

Rewriting Vorlage:Displayed math

shows that the angle \displaystyle 2\pi/3 lands in the the second quadrant and makes the angle \displaystyle \pi/6 with the positive y-axis. If we draw an extra triangle as in the figure below on the right, we see that the \displaystyle 2\pi/3- point on the unit circle has a y-coordinate, which is equal to the adjacent side \displaystyle \cos \frac{\pi}{6} = \sqrt{3}/2. So we have that Vorlage:Displayed math

4.2 - Figure - Two unit circles with angle 2π/3 (angle π/6 with the y-axis)

The trigonometric functions graphs

In the last section, we used a unit circle to define cosine and sine of arbitrary angles and we often will 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.

4.2 - Figure - A sine curve The graph of the sine function 4.2 - Figure - A cosine curve The graph of the cosine function 4.2 - Figure - A tangent curve 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 \displaystyle 2\pi, that is the \displaystyle \cos (x+2\pi) = \cos x and \displaystyle \sin (x+2\pi) = \sin x. For the unit circle \displaystyle 2\pi corresponds to a revolution, and after a complete revolution angles return to the same location on the unit circle and therefore have the same coordinates.

The curve for the tangent repeats itself after a change in angle of \displaystyle \pi, that is \displaystyle \tan (x+\pi) = \tan x. Two angles which differ by \displaystyle \pi 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 \displaystyle \pi/2 the curves for cosine and sine are identical, that is \displaystyle \cos x = \sin (x+ \pi/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 \displaystyle \cos x = x^2 ( where \displaystyle x is measured in radians)?

By drawing the graphs \displaystyle y=\cos x and \displaystyle y=x^2 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.

4.2 - Figur - Kurvorna y = cos x och y = x²

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 on understanding how to use a unit circle to define the trigonometric functions.

Get into the habit of calculating with precise trigonometric values. It provides a good training in calculating fractions and eventually algebraic rational expressions.

Reviews

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

Learn more about trigonometry in Per Edström "Interactive Mathematics"

Learn more about trigonometry in the English Wikipedia

Learn more about the unit circle in the English Wikipedia

Useful web sites

Experiment with the sine and cosine in the unit circle

Experiment with Euclidean geometry

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/4.2_Trigonometrische_Funktionen“

3. Wurzeln und Logarithmen