PLANS AND ELEVATIONS ( PELAN DAN DONGAKAN )

PLANS AND ELEVATIONS ( PELAN DAN DONGAKAN )

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EARTH AS A SPHERE ( BUMI SEBAGAI SFERA)

EARTH AS A SPHERE ( BUMI SEBAGAI SFERA)

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sumber : math4spm.co.cc

Trigonometry II

Trigonometric Functions

The following ratios are for right angle trigonometry. The angle must be acute (angle is less than 90°).

For angles that are obtuse (angle is greater than 90°) or negative, we use the following trigonometric ratios. The x and y variables are the values of the x and y coordinates, respectively. The r variable represents the distance from the origin, to the point (x,y). This value can be found using the Pythagorean theorem.

When negative or obtuse angles are used in trigonometric functions, they will sometimes produce negative values. The CAST graph to the left will help you to remember the signs of trigonometric functions for different angles. The functions will be negative in all quadrants except those that indicate that the function is positive. For example, When the angle is between 0° and 90° (0 and pi/2 radians), the line r is in the A quadrant. All functions will be positive in this region. When the angle is between 90° and 180° (pi/2 and pi radians), the line is in the S quadrant. This means that only the sine function is positive. All other functions will be negative.

Note: For examples of finding trigonometric ratios see questions #2 and #3 in Additional Examples at the bottom of the page.

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Graphs of Trigonometric Functions

Sine Function

Cosine Function

Tangent Function

Note: As illustrated in the graphs above, the sine and cosine functions are defined for all values of x. The tangent function, however, being equal to sin x / cos x, is undefined whenever cos x = 0.

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Periodicity of Trigonometric Function

From the graphs above, you can see that trigonometric functions are periodic. The sine and cosine functions, for example, have a period of 2 pi. In general, for any integer k,

The tangent function, however, has a period of pi. The period of the tangent function is given by

for any integer k.

This allows us to easily graph trigonometric functions. We must only determine the graph of the function over its period.

Examples of periodicity can be easily shown using a calculator. For example, if you were to solve sin(160pi) using your calculator, you would find that it is equal to sin(0)=0. This is because sin(160pi) = sin(0 + 2pi(80)).

Using this concept of periodicity, we can calculate certain trigonometric functions without using our calculators. For example, tan(31pi/3) is equal to tan(pi/3 + pi(10)), or equivalently tan(pi/3). From the table in the section below, we know that tan(pi/3) is equal to the square root of 3. So, tan(31pi/3) is also equal to root 3.

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Special Triangles

I)

II)

Using the "special" triangles above, we can find the exact trigonometric ratios for angles of pi/3, pi/4 and pi/6. These triangles can be constructed quite easily and provide a simple way of remembering the trigonometric ratios. The table below lists some of the more common angles (in both radians and degrees) and their exact trigonometric ratios.

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The Laws of Sines and Cosines

The laws of sines and cosines are useful in determining the sides and angles of oblique triangles. All triangles that are not right angled are classified as oblique triangles. The oblique triangle below will be used in the definitions for the laws of sines and cosines.

The law of sines states:

Note: All the sides and angles of a triangle can be determined using the law of sines if you know the measurements of any 2 angles and any one side.

Note: For an example of solving a triangle using the sine law, see question #4 in the Additional Examples section at the bottom of the page.

The law of cosines states:

Note: All sides and angles of a triangle can be determined using the law of cosines if you know the measurements of 2 sides and the angle enclosed between them, or the lengths of all 3 sides.

Note: For an example of solving a triangle using the cosine law, see question #5 in the Additional Examples section at the bottom of the page.

• | Proof of the cosine law

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Trigonometric Identities

The identities listed below are the basic trigonometric identities. They can be combined with one another to create many more identities.

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Trigonometric Formulas

The trigonometric formulas below can be combined with the identities above to create very complex trigonometric identities. These formulas are often necessary when proving trigonometric identities.

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Bearing

Bearings

A directional compass is shown below. It is used to find a direction or bearing . The four main directions of a compass are known as cardinal points. They are north (N), east (E), south (S) and west (W). Sometimes, the half-cardinal points of north-east (NE), north-west (NW), south-east (SE) and south-west (SW) are shown on the compass. The above compass shows degree measurements from 0° to 360° in 10° intervals with: north representing 0° or 360° east representing 90° south representing 180° west representing 270° When using a directional compass, hold the compass so that the point marked north points directly away from you. Note that the magnetic needle always points to the north. Bearing The true bearing to a point is the angle measured in degrees in a clockwise direction from the north line. We will refer to the true bearing simply as the bearing. For example, the bearing of point P is 065º which is the number of degrees in the angle measured in a clockwise direction from the north line to the line joining the centre of the compass at O with the point P (i.e. OP). The bearing of point Q is 300º which is the number of degrees in the angle measured in a clockwise direction from the north line to the line joining the centre of the compass at O with the point Q (i.e.OQ). Note: The bearing of a point is the number of degrees in the angle measured in a clockwise direction from the north line to the line joining the centre of the compass with the point. A bearing is used to represent the direction of one point relative to another point. For example, the bearing of A from B is 065º. The bearing of B from A is 245º. Note: Three figures are used to give bearings. All bearings are measured in a horizontal plane. Example 10 State the bearing of the point P in each of the following diagrams: Solution: a. Mark the angle in a clockwise direction by indicating the turn between the north line and the line joining the centre of the compass to the point P. The bearing of point P is 048°. b. Mark the angle in a clockwise direction by indicating the turn between the north line and the line joining the centre of the compass to the point P. The cardinal point S corresponds to 180°. It is clear from the diagram that the required angle is 60° larger than 180°. So, the angle measured in a clockwise direction from the north line to the line joining the centre of the compass to point P is 180° + 60° = 240°. So, the bearing of point P is 240°. c. Mark the angle in a clockwise direction by indicating the turn between the north line and the line joining the centre of the compass to the point P. The cardinal point S corresponds to 180°. It is clear from the diagram that the required angle is 40° less than 180°. So, the angle measured in a clockwise direction from the north line to the line joining the centre of the compass to point P is 180° – 40° = 140°. So, the bearing of point P is 140°. d. Mark the angle in a clockwise direction by indicating the turn between the north line and the line joining the centre of the compass to the point P. The cardinal point W corresponds to 270°. It is clear from the diagram that the required angle is 20° larger than 270°. So, the angle measured in a clockwise direction from the north line to the line joining the centre of the compass to point P is 270° + 20° = 290°. So, the bearing of point P is 290°. Direction The conventional bearing of a point is stated as the number of degrees east or west of the north-south line. We will refer to the conventional bearing simply as the direction. To state the direction of a point, write: N or S which is determined by the angle being measured the angle between the north or south line and the point, measured in degrees E or W which is determined by the location of the point relative to the north-south line E.g. In the above diagram, the direction of: A from O is N30ºE. B from O is N60ºW. C from O is S70ºE. D from O is S80ºW. Note: N30ºE means the direction is 30º east of north. Example 11 Describe each of the following bearings as directions. a. 076° b. 150° c. 225° d. 290° Solution: a. The position of a point P on a bearing of 076° is shown in the following diagram. The position of the point P is 76° east of north. So, the direction is N76°E. b. The position of a point P on a bearing of 150° is shown in the following diagram. The position of the point P is 180° – 150° = 30° east of south. So, the direction is S30°E. c. The position of a point P on a bearing of 225° is shown in the following diagram. The position of the point P is 225° – 180° = 45° west of south. So, the direction is S45°W. d. The position of a point P on a bearing of 290° is shown in the following diagram. The position of the point P is 360° – 290° = 70° west of north. So, the direction is N70°W.