Question

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Answer 1

@jordanfarar22 can you help me please ?

Answer 2

1 min hold up

Answer 3

what i s ur solections?

Answer 4

@jordanfarar22 theres none... :/

Answer 5

The cartesian plane is a plane on which an x axis is drawn horizontally and a y axis is drawn vertically. Any points in the plane are referred to by using ordered pair notation like this:

This is the point at x = 1.5 and y = 2. It is indicated in the picture to the right.

This method of using an x value and a y value to locate a point in the plane is called the rectangular coordinate system (notice the dotted rectangle shown in the picture). Another coordinate system in common use, for example for complex numbers, is the polar coordinate system; it uses a distance from the origin and a direction to locate a point.

The plane is divided into 4 quadrants. Quadrant 1 has positive x and y values. The other quadrants are reached by going counter-clockwise from the first.





Angles and their Measure

An angle in standard position has its vertex at the origin and one side along the x axis.

The angle is generated when the ray is rotated from the initial to the terminal position.

A counter-clockwise rotation is considered to be a positive rotation.





A clockwise rotation is considered to be a negative rotation.



There are two scales that are commonly used to measure the size of an angle: the degree scale and the radian scale:
Degrees

In this scale an angle is expressed in units called degrees, with a rotation of one full circle being 360 degrees ( symbol ° ).



Radians

In this scale an angle is expressed in units called radians, with a full circle rotation being 2π or approximately 6.28 radians.

The radian scale is formally defined like this:


The size of angle θ expressed in radians equals s (the arc length subtendend by the angle) divided by r (the radius of the circle).


For one full circle s = C (the circumference), and since C = 2 π r we get

for an angle of one full circle.

Note that since s and r have the same units, the radian is a unitless unit!





Example: The angle θ in the picture to the right equals 2 radians since s = 6 and r = 3.


Converting between degrees and radians

To convert between degrees and radians you can use the fact that π radians = 180°. This means that:

These two fractions are called UFOOs (useful forms of one). Multiplying any angle by one of these does not change the size of the angle but it does change the units of the angle.


Example: Convert 118° to radians.

Solution:


Example: Convert 0.438 radians to degrees.

Solution:


The Angles of a Triangle

The sum of the angles in any triangle equals 180° or π radians.


Pythagoras’ Theorem and Lengths

Pythagoras’ theorem is concerned with the lengths of the sides of right triangles. (A right angle is a 90° angle and a right triangle is one that contains a 90° angle.) Consider the right triangle with sides of length a, b and c shown to the right. a is called the altitude, b is called the base and c, the longest side opposite the right angle, is called the hypotenuse. Pythagoras' theorem states that the sides are related by the formula:
c2 = a2 + b2
A nice animated proof of Pythagoras’ theorem can be found at http://www.math.ubc.ca/~morey/java/pyth/



Example: Find the length of the straight line in the Cartesian plane extending from the point (x1, y1) = (−1.5, 2) to the point (x2, y2) = (2, −1).

Solution: Extend the dotted lines horizontally and vertically to create a right triangle. This triangle has base b = 3.5 and altitude a = 3, so Pythagoras' theorem gives:
c2 = 32 + 3.52 = 21.25
Taking the square root of both sides gives the length of the straight line as


This example illustrates the following very useful application of Pythagoras' theorem. If (x1, y1) and (x2, y2) are any two points in the Cartesian plane then the distance c between them can be obtained from the formula:


Definitions of the trigonometric functions

For now let us restrict ourselves to right triangles.

Notice that if a triangle is scaled up in size then the sides get longer, but the ratio of the lengths of any two sides does not change.

It is possible to define six different ratios. Let opposite and adjacent denote the lengths of the sides opposite and adjacent to angle θ and let hypotenuse denote the long side. The six ratios are named sine, cosine, tangent, cotangent, secant and cosecant and are defined as follows:


Note:
The names of the six ratios are abbreviated as sin, cos, tan, cot, sec, csc.

The six ratios depend on (are functions of) the angle θ. Denote this using functional notation like this: sin(θ), cos(θ), etc.

Later when we discuss trigonometric functions of any angle it will be very useful to imagine an arrow (or vector) radiating out from the origin with length r and orientated at angle θ. Then a triangle can be constructed by drawing lines vertically and horizontally to the x and y axes. Suppose that the arrow-head is at coordinate (x, y). Then the six trigonometric functions can be defined like this:

The three trigonometric functions: sin, cos and tan are built into your calculator. Since their values depend on the angle θ, they are found by entering the angle into your calculator in the appropriate mode (degree or radian) and pressing the appropriate button, sin, cos or tan.

The other three trigonometric functions: csc, sec and cot are not built into your calculator. To find them you must instead find the sin, cos or tan on your calculator and then take the reciprocal.


Examples:
sin(30°) = 0.500

tan (57°) = 1.54

cos(0.74 rads) = 0.738

tan (1.1 rads) = 1.96





Example: Find y and r in the triangle.

Solution:
To find y use the fact that .
From the calculator tan(35.8°) = 0.7212. Setting these equal to each other gives which can be solved to give y = 8.94.

To find r use the fact that .
From the calculator cos(35.8°) = 0.8111. Setting these equal to each other gives which can be solved to give r = 15.3.

Answer: The required lengths are y = 8.94 and r = 15.3.



Definitions of the inverse trigonometric functions

Suppose that we wish to find the angle θ in the triangle shown.

One thing we know is that . Previously we would put an angle θ into the calculator and press TAN to get a value out for the tangent.

Here we want to do the inverse process - we want to put a value of tan in and get an angle out. To do this your calculator has the inverse trigonometric functions sin−1, cos−1 and tan−1 built in. (Note: the superscript −1 has NOTHING to do with exponents. It is merely a NOTATION meaning “inverse function”. To avoid this possible confusion these functions are often called the arcsin, arccos and arctan functions respectively.)

For this example tan(θ) = 0.7381 so θ = tan−1(0.7381). The calculator gives tan−1(0.7381) = 36.4°, so θ = 36.4°.

In general the inverse trigonometric functions are defined like this:
If sin(θ) = z, then θ = sin−1(z).

If cos(θ) = z, then θ = cos−1(z).

If tan(θ) = z, then θ = tan−1(z).



Solution of right triangles

Solving a triangle means finding all the missing sides and angles of the triangle. Here are the steps involved in solving right triangles.
Draw a picture (with the angle in standard position, if appropriate).

You will be given at least (a) one side and one angle, or (b) two sides (otherwise the triangle can’t be solved).

In case (a) you can use sin, cos or tan as appropriate to find another side.

In case (b) you can use sin−1, cos−1 or tan−1 as appropriate to find an angle.

Use Pythagoras' theorem to find the final side.

Use the fact that the angles sum to 180° to find the final angle.

Give your answer in a sentence.



Example: A train is travelling along a straight section of track. A pilot flying 4220 meters directly above the caboose at the back end of the train observes that the angle of depression of the locomotive at the front of the train is 68.2°. Find the length of the train.

Solution: The picture on the left describes the situation. Notice that the angle of elevation of the airplane from the locomotive is also 68.2°. The pictures on the right define the terms “angle of depression” and “angle of elevation”, which are used often in surveying work.




Using the tangent function we get:

Evaluating the tan function and solving for the length of the train then gives:

Answer 6

that is all i know about that subject