Miyerkules, Oktubre 15, 2014

trigonometric functions

Right-angled triangle definitions[edit]

(Top): Trigonometric function sinθfor selected angles θ, π − θ, π + θ, and 2π − θ in the four quadrants. (Bottom) Graph of sine function versus angle. Angles from the top panel are identified.

The notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides. That is, for any similar triangle the ratio of thehypotenuse (for example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is these ratios that the trigonometric functions express.

To define the trigonometric functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows:

The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.

The opposite side is the side opposite to the angle we are interested in (angle A), in this case side a.

The adjacent side is the side having both the angles of interest (angle A and right-angle C), in this case side b.

In ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180° (π radians). Therefore, in a right-angled triangle, the two non-right angles total 90° (π/2 radians), so each of these angles must be in the range of (0°,90°) as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range. They can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin θ for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle (top) and as a graph (bottom). The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, this behavior repeats periodically with a period 2π.

The trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram.

Function Abbreviation Description Identities (using radians)

sine sin opposite / hypotenuse $\sin \theta = \cos \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\csc \theta}$

cosine cos adjacent / hypotenuse $\cos \theta = \sin \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\sec \theta}\,$

tangent tan (or tg) opposite / adjacent $\tan \theta = \frac{\sin \theta}{\cos \theta} = \cot \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cot \theta}$

cotangent cot (or cotan or cotg or ctg or ctn) adjacent / opposite $\cot \theta = \frac{\cos \theta}{\sin \theta} = \tan \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\tan \theta}$

secant sec hypotenuse / adjacent $\sec \theta = \csc \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cos \theta}$

cosecant csc (or cosec) hypotenuse / opposite $\csc \theta = \sec \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\sin \theta}$

The sine, tangent, and secant functions of an angle constructed geometrically in terms of a unit circle. The number θ is the length of the curve; thus angles are being measured in radians. The secant and tangent functions rely on a fixed vertical line and the sine function on a moving vertical line. ("Fixed" in this context means not moving as θ changes; "moving" means depending on θ.) Thus, as θ goes from 0 up to a right angle, sin θ goes from 0 to 1, tan θgoes from 0 to ∞, and sec θ goes from 1 to ∞.

The cosine, cotangent, and cosecant functions of an angle θ constructed geometrically in terms of a unit circle. The functions whose names have the prefix co-use horizontal lines where the others use vertical lines.

Sine, cosine and tangent [edit]

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. (The word comes from the Latin sinus for gulf or bay,^[1] since, given a unit circle, it is the side of the triangle on which the angle opens.) In our case

$\sin A = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {a} {h}.$

This ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles aresimilar.

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse: so called because it is the sine of the complementary or co-angle.^[2] In our case

$\cos A = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac {b} {h}.$

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side: so called because it can be represented as a line segment tangent to the circle, that is the line that touches the circle, from Latin linea tangens or touching line (cf. tangere, to touch).^[3] In our case

$\tan A = \frac {\textrm{opposite}} {\textrm{adjacent}} = \frac {a} {b}.$

The acronyms "SOHCAHTOA" ("Soak-a-toe", "Sock-a-toa", "So-kah-toa") and "OHSAHCOAT" are commonly used mnemonics for these ratios.

Reciprocal functions[edit]

The remaining three functions are best defined using the above three functions.

The cosecant csc(A), or cosec(A), is the reciprocal of sin(A); i.e. the ratio of the length of the hypotenuse to the length of the opposite side:

$\csc A = \frac {1}{\sin A} = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac {h} {a}.$

The secant sec(A) is the reciprocal of cos(A); i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:

$\sec A = \frac {1}{\cos A} = \frac {\textrm{hypotenuse}} {\textrm{adjacent}} = \frac {h} {b}.$

It is so called because it represents the line that cuts the circle (from Latin:secare, to cut).^[4]

The cotangent cot(A) is the reciprocal of tan(A); i.e. the ratio of the length of the adjacent side to the length of the opposite side:

$\cot A = \frac {1}{\tan A} = \frac {\textrm{adjacent}} {\textrm{opposite}} = \frac {b} {a}.$

Slope definitions[edit]

Equivalent to the right-triangle definitions, the trigonometric functions can also be defined in terms of the rise, run, andslope of a line segment relative to horizontal. The slope is commonly taught as "rise over run" or $rise ⁄ run$ . The three main trigonometric functions are commonly taught in the order sine, cosine, tangent. With a line segment length of 1 (as in aunit circle), the following mnemonic devices show the correspondence of definitions:

"Sine is first, rise is first" meaning that Sine takes the angle of the line segment and tells its vertical rise when the length of the line is 1.

"Cosine is second, run is second" meaning that Cosine takes the angle of the line segment and tells its horizontal run when the length of the line is 1.

"Tangent combines the rise and run" meaning that Tangent takes the angle of the line segment and tells its slope; or alternatively, tells the vertical rise when the line segment's horizontal run is 1.

This shows the main use of tangent and arctangent: converting between the two ways of telling the slant of a line, i.e.,angles and slopes. (The arctangent or "inverse tangent" is not to be confused with the cotangent, which is cosine divided by sine.)

While the length of the line segment makes no difference for the slope (the slope does not depend on the length of the slanted line), it does affect rise and run. To adjust and find the actual rise and run when the line does not have a length of 1, just multiply the sine and cosine by the line length. For instance, if the line segment has length 5, the run at an angle of 7° is 5 cos(7°)

Unit-circle definitions[edit]

The unit circle

Signs of trigonometric functions in each quadrant. The mnemonic "AllScience Teachers (are) Crazy" lists the functions which are positive from quadrants I to IV.^[5] This is a variation on the mnemonic "All Students Take Calculus".

The six trigonometric functions can also be defined in terms of theunit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles.

The unit circle definition does, however, permit the definition of the trigonometric functions for all positive and negative arguments, not just for angles between 0 and π/2 radians.

It also provides a single visual picture that encapsulates at once all the important triangles. From the Pythagorean theorem the equation for the unit circle is:

$x^2 + y^2 = 1. \,$

In the picture, some common angles, measured in radians, are given. Measurements in the counterclockwise direction are positive angles and measurements in the clockwise direction are negative angles.

Let a line through the origin, making an angle of θ with the positive half of the x-axis, intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively.

The triangle in the graphic enforces the formula; the radius is equal to the hypotenuse and has length 1, so we have sin θ = y/1 and cos θ = x/1. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the lengths of their hypotenuses equal to 1.

These values (sin 0°, sin 30°, sin 45°, sin 60° and sin 90°) can be expressed in the form

$\frac{1}{2}\sqrt{0},\quad \frac{1}{2}\sqrt{1},\quad \frac{1}{2}\sqrt{2},\quad \frac{1}{2}\sqrt{3},\quad \frac{1}{2}\sqrt{4},$

but the angles are not equally spaced.

The values for 15°, 18°, 36°, 54°, 72°, and 75° are derived as follows:

$\sin 15^\circ = \cos 75^\circ = \tfrac14(\sqrt6 - \sqrt2)\,\!$
$\sin 18^\circ = \cos 72^\circ = \tfrac14(\sqrt5 - 1)$
$\sin 36^\circ = \cos 54^\circ = \tfrac14\sqrt{10 - 2\sqrt5}$
$\sin 54^\circ = \cos 36^\circ = \tfrac14(\sqrt5 + 1 )\,\!$
$\sin 72^\circ = \cos 18^\circ = \tfrac14\sqrt{10 + 2\sqrt5}$
$\sin 75^\circ = \cos 15^\circ = \tfrac14(\sqrt6 + \sqrt2).\,\!$

From these, the values for all multiples of 3° can be analytically computed. For example:

$\sin 3^\circ = \cos 87^\circ = \tfrac1{16}\left(\sqrt{30} + \sqrt{10} + \sqrt{20 + 4 \sqrt5} - \sqrt6 - \sqrt2 - \sqrt{60 + 12 \sqrt5}\right)\,\!$
$\sin 6^\circ = \cos 84^\circ = \tfrac18\left(\sqrt{30 - 6 \sqrt5} - \sqrt5 - 1 \right)\,\!$
$\sin 9^\circ = \cos 81^\circ = \tfrac1{32}\left(\sqrt{90} + \sqrt{18} + \sqrt{10} + \sqrt2 - \sqrt{20 - 4 \sqrt5} - \sqrt{180 - 36 \sqrt5}\right)\,\!$
$\sin 84^\circ = \cos 6^\circ =\tfrac18\left(\sqrt{10 - 2\sqrt5} + \sqrt{15} + \sqrt3\right)$
$\sin 87^\circ = \cos 3^\circ =\tfrac1{16}\left(\sqrt{60 + 12\sqrt5}+\sqrt{20 + 4\sqrt5}+\sqrt{30}+\sqrt2-\sqrt6-\sqrt{10}\right)$

Though a complex task, the analytical expression of sin 1° can be obtained by analytically solving the cubic equation

$\sin 3^\circ = 3\sin 1^\circ - 4\sin^3 1^\circ$

from whose solution one can analytically derive trigonometric functions of all angles of integer degrees.

The sine and cosine functions graphed on the Cartesian plane.

Animation showing the relationship between the unit circle and the sine and cosine functions.

For angles greater than 2π or less than −2π, simply continue to rotate around the circle; sine and cosine are periodic functions with period 2π:

$\sin\theta = \sin\left(\theta + 2\pi k \right),\,$
$\cos\theta = \cos\left(\theta + 2\pi k \right),\,$

for any angle θ and any integer k.

The smallest positive period of a periodic function is called theprimitive period of the function.

The primitive period of the sine or cosine is a full circle, i.e. 2π radians or 360 degrees.

Above, only sine and cosine were defined directly by the unit circle, but other trigonometric functions can be defined by:

$\begin{align} \tan\theta & = \frac{\sin\theta}{\cos\theta},\ \cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{1}{\tan\theta} \\ \sec\theta & = \frac{1}{\cos\theta},\ \csc\theta = \frac{1}{\sin\theta} \end{align}$

So :

The primitive period of the secant, or cosecant is also a full circle, i.e. 2π radians or 360 degrees.

The primitive period of the tangent or cotangent is only a half-circle, i.e. π radians or 180 degrees.

Trigonometric functions: Sine, Cosine, Tangent,Cosecant (dotted), Secant (dotted), Cotangent (dotted)

The image at right includes a graph of the tangent function.

Its θ-intercepts correspond to those of sin(θ) while its undefined values correspond to the θ-intercepts of cos(θ).

The function changes slowly around angles of kπ, but changes rapidly at angles close to (k + 1/2)π.

The graph of the tangent function also has a vertical asymptoteat θ = (k + 1/2)π, the θ-intercepts of the cosine function, because the function approaches infinity as θ approaches (k + 1/2)π from the left and minus infinity as it approaches (k + 1/2)π from the right.

All of the trigonometric functions of the angle θ can be constructed geometrically in terms of a unit circle centered at O.

Alternatively, all of the basic trigonometric functions can be defined in terms of a unit circle centered at O (as shown in the picture to the right), and similar such geometric definitions were used historically.

In particular, for a chord AB of the circle, where θ is half of the subtended angle, sin(θ) is AC (half of the chord), a definition introduced in India^[6] (see history).

cos(θ) is the horizontal distance OC, and versin(θ) = 1 − cos(θ) is CD.

tan(θ) is the length of the segment AE of the tangent line through A, hence the word tangent for this function. cot(θ) is another tangent segment, AF.

sec(θ) = OE and csc(θ) = OF are segments of secant lines(intersecting the circle at two points), and can also be viewed as projections of OA along the tangent at A to the horizontal and vertical axes, respectively.

DE is exsec(θ) = sec(θ) − 1 (the portion of the secant outside, orex, the circle).

From these constructions, it is easy to see that the secant and tangent functions diverge as θ approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as θ approaches zero.

For the angle  pictured in the figure, we see that

There are a few angles for which all trigonometric functions may be found using the triangles shown in the following Figure 2.

Figure 2

This list may be extended with the use of reference angles (see Example 2 below).

EXAMPLE 1: Find the values of all trigonometric functions of the angle  .

Solution: From Figure 2, we see that the angle of  corresponds to the point  on the unit circle, and so

EXAMPLE 2: Find the values of all trigonometric functions of the angle  .

Solution: Observe that an angle of  is equivalent to 8 whole revolutions (a total of  ) plus  , Hence the angles  and intersect the unit circle at the same point Q(x,y), and so their trigonometric functions are the same. Furthermore, the angle of  makes an angle of  with respect to the x-axis (in the second quadrant). From this we can see that  and hence that

We call the auxiliary angle of  the reference angle of  .

EXAMPLE 3 Find all trigonometric functions of an angle  in the third quadrant for which  .

Solution: We first construct a point R(x,y) on the terminal side of the angle  , in the third quadrant. If R(x,y) is such a point, then and we see that we may take x=-5 and R=6. Since  we find that (the negative signs on x and y are taken so that R(x,y) is a point on the third quadrant, see Figure 3).

Figure 3

It follows that

Here are some Exercises on the evaluation of trigonometric functions.

ASSESSMENT

Name ___________
Date _

Trigonometry
(Answer ID # 0241143)
Complete. Round to the nearest hundredth.

1.   Find cos U

QG

  =   14.2

GU

  =   5.6

QU

  =   15.26

2.   Find tan B

BM

  =   3.6

VM

  =   5.1

BV

  =   6.24

3.   Find sin T

AT

  =   0.41

AX

  =   0.28

TX

  =   0.3

4.   Find sin T

UT

  =   8.62

UN

  =   8

TN

  =   3.2

5.   Find cos Y

YJ

  =   4.55

YR

  =   3.8

RJ

  =   2.5

6.   Find tan R

PA

  =   0.76

AR

  =   2.5

PR

  =   2.61

7.   Find tan Y

VY

  =   20.94

VJ

  =   10.7

YJ

  =   18

8.   Find cos R

KR

  =   27.5

SR

  =   29.26

SK

  =   10

9.   Find sin X

KX

  =   4.12

KE

  =   0.97

XE

  =   4

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