14.1 - Trigonometry



The Cartesian Plane

A cartesian plane is a plane on which an x axis is drawn horizontally and a y axis is drawn vertically. Any point in the plane is 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 is called the rectangular coordinate system (notice the dotted rectangle shown in the picture). Another coordinate system in common use (especially 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









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 subtended 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.

An important property of radians is that since s and r both have the dimensions of length, radians are units without dimensions.






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 a right triangle. (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 geometric proof of Pythagoras’ Theorem is given on wikipedia.




Example: Find the length of the straight line in the Cartesian plane extending from the point (x1y1) = (−1.5, 2) to the point (x2y2) = (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:


The trigonometric functions

For now let us restrict ourselves to right triangles. This means that angle θ must have a value between 0°and 90°.

Notice that if the 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:


Examples:


Example: Find y and r in the triangle.

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




The inverse trigonometric functions

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

One thing we know is that . In the examples above we put an angle θ into the calculator and then pressed the tan button to get a value out for the tangent.

Here we want to do the inverse process - we want to put a value of tangent 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 arcsine, arccosine, and arctangent; abbreviated arcsin, arccos and arctan.)

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:


How to solve 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.


Example: A train is traveling 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 gives 2.500 and solving for the length of the train then gives:
The sentence answer is that to 3 significant figures the length of the train is 1690 m.




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Trigonometric functions of any angle

When we first defined the trigonometric functions (click here to review that) the angle θ was between 0° and 90° and we used the terms adjacent, opposite and hypotenuse to refer to the sides of a triangle.

But we now want to allow angle θ to have values outside this range. One reason is to be able to solve oblique triangles. These triangles can have an angle that is bigger than 90°. Another reason is that we want to be able to describe the angle of a spinning object as it rotates through many revolutions. An example is an aerial skier who jumps and does 2 revolutions in the air. They are “doing a 720”, meaning that they are rotating through 720°.

To allow for angles bigger than 90° we now imagine an arrow (or vector) pointing out from the origin with length r and orientated at angle θ, and with its arrowhead located at (xy).

We construct a triangle by drawing a line vertically from the arrowhead to the x axis and another line horizontally across to the y axis.

We now redefine the six trigonometric functions like this:
If θ is between 0° to 90° then both coordinates, x and y, are positive and the trigonometric functions have the same values as before. But if θ is outside the range 0° to 90° then x or y or both can be negative. (Don’t forget, however, that the arrow’s length r is defined to be positive.) This means that the trigonometric functions can now possibly be negative.





Example: The vector with length r = 2.5 and angle θ = 127° has coordinates x = −1.5 and y = 2.0. Thus:





Here are two ways to remember the quadrants in which the trigonometric functions are positive or negative. The best way, of course, is to simply learn the graphs of the sine, cosine and tangent functions. Click here to see them.




Many students use the saying All Students Take Chemistry to remember the quadrants in which the functions are positive.




Calculating the sine, cosine, or tangent now that the angle can have any value

This is easy. Your calculator is programmed to find the sine, cosine, or tangent of any angle whatsoever (minus sign and all). For example the calculator gives cos(120°) = −0.5.

The answer the calculator gives is unique but it is important to note that several angles have the same sine, cosine, or tangent. For example the calculator gives sin(60°) = 0.866 and also gives sin(120°) = 0.866.

Calculating the angle now that the sine, cosine, or tangent can have any value

Because many angles have the same sine (or cosine, or tangent) and because the calculator only finds one of them, this is not so easy. For example let’s turn the last example around: suppose we know that the sine of some unknown angle equals 0.866 and we want to find the angle. We use the sin−1 button on the calculator and it gives sin−1(0.866) = 60°. It does not give sin−1(0.866) = 120°.

How do we find all the angles whose sine is 0.866? One solution is to use a picture like one of these:







Example: Find all the values of θ between 0° and 360° for which sin(θ) = +0.6293.

Solution: Since , this means that .

Recall that r is defined positive. If we let r = 1, for example, then y = +0.6293. This means that the angles must be in quadrants 1 and 2.

The calculator gives sin−1(0.6293) = 39.0°. The picture shows that this means that the angles are θ = 39.0° and θ = 180° − 39.0° = 141.0°





Example: Find all the values of θ between 0° and 360° for which tan(θ) = −1.24.

Solution: Since , this means that .

The picture shows that there are two ways that this ratio can happen.

If we let x = 1 then y = −1.24. This means that one angle is in quadrant 4.

And if we let x = −1 then y = + 1.24. This means that the other angle is in quadrant 2.

The calculator gives tan−1(−1.24) = −51.1°. The picture shows that this means that the angles are θ = 128.9° and θ = 308.9° (These values are gotten by taking −51.1° and adding 180° once and once again.)





Example: Find all the values of θ between 0° and 360° for which sec(θ) = −4.21

Solution: Since , this is exactly the same as saying that . Since this means that .

If we let r = 1 then x = −0.2375. This means that the angles are in quadrants 2 and 3.

The calculator gives cos−1(−0.2375) = 103.7°
The picture shows that this means that the angles are θ = 103.7° and θ = 360° − 103.7° = 256.3°.



Oblique triangles

An oblique triangle is one that doesn’t contain a right angle.

The naming convention for the angles and sides is that the angle and the side opposite it have the same letter, as shown in the picture to the right.

The sine law and cosine law (derived below) are used to solve oblique triangles

Note the following points regarding the sine and cosine laws:

Derivation of The Sine Law

Break the triangle into 2 right triangles by dropping the perpendicular with length h. Then:
and
Dividing the first equation by the second gives
which is another way to write the sine law.




Derivation of The Cosine Law

Again break the triangle into 2 right triangles. Applying Pythagoras’ Theorem to the two right triangles gives:



Subtracting the second one from the first gives:
This is the cosine law: .



How to use the Sine and Cosine Laws

To solve an oblique triangle, a combination of at least 3 angles and sides must be given. The cases are classified SAS, ASA, ASS, etc. If a side and the angle opposite it are given then you can use the sine law. Otherwise you must use the cosine law.

If you are using the sine law to find an angle you will eventually need to evaluate a sin−1. If the angle you are looking for is acute then the calculator returns the correct value. But if the angle is obtuse then the angle given by the calculator is not the correct one. You need the one in the second quadrant (which can be gotten by subtracting the calculator angle from 180°.) (Recall that an acute angle is an angle between 0° and 90°. An obtuse angle is an angle between 90° and 180°.)





Example: Solve the oblique triangle.

Solution: Since the angles add to 180°:
C = 180° − 32.5° − 49.7° = 97.8°
Now use the sine law to get sides b and c:

The unknown sides and angles are C = 97.8°, b = 321 and c = 417.





Example. Ambiguous case: Solve the oblique triangle with given information about an angle and two sides:
A = 25.3°, c = 152 and a = 95.0
Solution: Note that the given information is not enough to decide whether the triangle is ABC or ABC' shown here:


This is called the ASS ambiguity. If we use the sine law to get angle C, we get:
Notice the two values of sin−1 give the two values of C in the two possible triangles.
In summary there are two possibilities: either C = 43.1°, B = 111.6° and b = 207
or C = 136.9°, B = 17.8° and b = 68.1






Example: Solve the oblique triangle shown.


Solution: Since we don’t know any side - opposite angle pair we must begin with the cosine law:



Now find angle B using the sine law:



From the picture it is clear that B is acute and that the only possibility is that B = 30.2° And therefore A = 127.6°

In summary, c = 91.5, B = 30.2° and A = 127.6°




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Vectors

Definition: A vector is a quantity that has a direction as well as a magnitude. Compare this with a scalar, which is a quantity that has only magnitude. An example of a vector is force. To accelerate an object in a certain direction you need to apply a force in that direction. An example of a scalar is temperature; temperature doesn’t point in any direction.

An arrow can be used to represent a vector. The length of the arrow represents the magnitude of the vector and the direction of the arrow is the direction of the vector.


Vector Addition: To add two vectors A and B geometrically you put the tail of B at the head of A or vice versa. The resulting vector, A + B, shown below in red, points from s (start) to f (finish). This is called parallelogram construction. The vector A + B is called the resultant of adding vectors A and B.


(This tail-to-head addition is an extension of the way that ordinary numbers are added. Click here to see the numbers 2 and 3 and their sum 5.)



The definition of the vector does not say anything about where the vector is located. Vectors are usually drawn with their tail at the origin. This is called the standard position. One advantage of standard position is that complete information about the vector is given by the coordinates of the arrowhead. Here is the same vector addition done with the vectors in standard position.





Vector Subtraction: Vector subtraction can be understood in two ways.

First, the subtraction B − A is equivalent to the addition B + (−A). (The negative of a vector A, namely −A, points in the opposite direction of A.)



The second way is to think of B − A as the difference between B and A. That is, the difference between their arrowheads. Thus it is the vector that points from the arrowhead of A to the arrowhead of B, which is shown in gray, and is then redrawn in standard position in red.




Resolving a vector into its components: Resolving a vector into its x and y components is done by drawing a dotted line from the arrowhead parallel to the y axis until it hits the x axis and another dotted line parallel to the x axis until it hits the y axis, and then drawing vectors along both axes until they touch these lines.


Coordinate Systems for Vectors: Vectors can be expressed in polar coordinates or rectangular coordinates. Resolving a vector expressed in rectangular coordinates into its x and y components is easy. For example:

Adding two vectors expressed in rectangular coordinates is also easy. You add the two x components to get the resultant’s x component and add the two y components to get the resultant’s y component. For example, if A = ( 2, 4 ) and B = ( 3, 1 ), then:
A + B = ( 2, 4 ) + ( 3, 1 ) = ( 5, 5 )
The picture shows that this works because:
A + B = ( 2, 4 ) + ( 3, 1 )
= ( 2, 0 ) + ( 0, 4 ) + ( 3, 0 ) + ( 0, 1 )
= ( 2, 0 ) + ( 3, 0 ) + ( 0, 4 ) + ( 0, 1 )
= ( 5, 0 ) + ( 0, 5 )
= ( 5, 5 )

We only describe 2-dimensional vectors here, but mathematicians use vectors in any number of dimensions.



Converting between Polar and Rectangular Coordinates

Often you must convert a vector expressed in polar coordinates to rectangular coordinates or vice versa.

Your calculator may have polar to rectangular (denoted PR or → x y ) and rectangular to polar conversion (denoted RP or → r θ ) built in. See your calculator manual. If not, you can do the following:



Addition of Polar Vectors

In general there is no simple, direct way to add two polar vectors. For example it turns out that (see example below):
8 ∠ 30° + 10 ∠ −60° = 12.8 ∠ 21.3°

The only exception is if the two vectors lie along the same line. For example,
1 ∠ 50° + 2 ∠ 50° = 3 ∠ 50°

In general, to add two polar vectors you must:



Example: Add the polar vectors 8 ∠ 30° and 10 ∠ −60°.

Solution:



Applications of Vectors

Statics   If an object is in static equilibrium then the forces on it must sum to zero. In other words the sum of the forces to the right equals the sum of the forces to the left and the forces up equal the forces down.






Example: A weight of W = 510 N is suspended by two ropes at the angles shown. (N stands for newtons which are the units of force in the metric system.)
(a) What are the tensions T1 and T2 in the ropes?

(b) If the ropes have a tensile strength of 1000 newtons, what is the maximum weight that can be supported?
Solution: The forces T1 , T2 and W must be in balance:



Substituting 1.049 T1 in for T2 in the second equation gives:

Thus the tensions are T1 = 678 N and T2 = 711 N.

Part (b): Notice that T2 is the bigger tension. The tensions are proportional to the weight and when W = 717 N then T2 = 1000 N. Thus the maximum weight that can be supported is 717 N.








Example: A cart with a weight of W = 1250 newtons is on a plane inclined at an angle θ = 13°. What minimum force F must be applied to keep the cart from rolling downhill, if there is no friction?

Solution: The weight W is due to gravity and so acts vertically downward. The key is to resolve W into components parallel to and perpendicular to the inclined plane. The component of W parallel to the plane is W sin(θ). F must equal at least this much or the cart will roll downhill. Plugging in the numbers we get:
F = W sin(θ) = (1250 N) sin(13°) = 281 N
Thus the minimum force is 281 newtons.




Applications of Radian Measure

We saw above that angles can be measured in radians, with a full circle rotation being 2π or approximately 6.28 radians.

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

For an angle of one full circle the arc length s becomes the circumference C. Since C = 2 π r we get
for an angle of one full circle.

Note that since s and r both have dimensions of length the radian is a unit with no dimensions. This is an important feature of radians.






Example: Express angle θ in the picture in radians.






The equilateral triangle has three angles equal to 60°. Bowing out one side into an arc produces an angle of 1 radian. We see that 1 radian is slightly less than 60°.



It is often useful to express a radian angle as a multiple of π. To do so, multiply by the symbol π and divide by the value of π, namely by 3.14159.



Example: Express 4.22 rads as a multiple of π.






The picture defines the sector, chord and segment.

The area of a circle is:
A = π r2

The area of a sector is a fraction s/C of the area of a circle, where C is the circumference. Thus:

Since C = 2 π r the area of the sector is:
Note the similarity to the formula for the area of a triangle:





Example: What is the distance between a point at latitude North 43.6° and a point due south of it on the equator? (Use the fact that the earth has a diameter of 7920 mi.)

Solution: Solving the equation for the arc length gives s = r θ. This formula requires the radius, , and the angle θ expressed in radians:
Thus the arc length is:
s = r θ = (3960 mi)(0.7610 rads) = 3010 mi,
to 3 sig figs. Notice that the units of s are the same as those of r (miles), since the units of θ (radians) are actually unitless.







Example: What distance does the rack gear move if the pinion gear rotates through an angle of 300°?

Solution: Because the rack and pinion gear mesh, the distance that the rack moves is equal to the arc length s subtended by the angle 300° on the pinion gear, which is given by s = r θ. The radius of the pinion gear is r = 11.25 mm and the angle θ through which it turns expressed in radians is:
Thus:
s = r θ = (11.25 mm)(5.236 rads) = 58.9 mm.
Notice that the units of s are the same as those of r, namely millimeters.




Uniform circular motion

Suppose that an object is moving at a constant velocity v in a circle of radius r and that it moves from A to B, covering a distance s, in time t. Then
since rate = distance traveled / time required.

As this happens angle θ increases. The rate at which θ increases is angle swept out / time required, namely θ / t. This quantity is called the angular velocity ω (the Greek letter omega). Thus:
Angular velocity ω can have any units of angle divided by time but the most useful units are radians / sec.




Example: An object rotates at an angular velocity of ω = 48 π rads / sec. Through what angle does it rotate in 0.25 sec?

Solution: Solving for θ gives:
θ = ω t = (48 π rads / sec)(0.25 sec) = 12 π rads.
(Note that 48 π rads / sec = 24 cycles / sec and that 12 π rads = 6 cycles)










Example: What is the speed in miles per hour of a point on the equator and of a point in Vancouver due to the rotation of the earth? Use the fact that the radius of the earth is 3960 miles, that it rotates once every 24 hours and that Vancouver is at latitude 49.2° north.


Solution: The first picture on the right looks down on the north pole and shows how far a point in Vancouver and a point on the equator move in 1 hr. Both have the same angular velocity ω, namely:
but they have different so-called linear velocities v because they have different radii from the earth’s axis.

To find v use the equation v = r ω, where either r = rE for the equator or r = rV for Vancouver. We were given rE = 3960 mi. To find rV use the side-view picture of the Earth to the right. Trigonometry gives:
Thus the velocities are:
Thus the speed of a point on the equator is 1037 mi / hr and of a point in Vancouver is 677 mi / hr due to the rotation of the earth. (This extra speed is the reason launch sites for orbital rockets are chosen near the equator.)




Graphs of the sine, cosine and tangent functions

Recall that
, and ,
where r is the length of a vector and x and y are its x and y components. Let r = 1. Then sin(θ) = y and cos(θ) = x. In other words sin(θ) is just the y component and cos(θ) is the x component of a vector of length 1.


Note the following features of all three curves:


The Sinusoidal Function y = A sin (ω t + φ)

Waves on the water, vibrations on a string, sound, light, animal population cycles, economic cycles and alternating current can all be described by the sinusoidal function or sinusoidal waveform:
y = A sin (ω t + φ)
where y represents the quantity of interest and t represents time. Because y is proportional to the sin function, it has the characteristic sin wave shape. By changing A, ω and φ we can change the height or width of the wave or shift it left or right to suit the application. Let’s look at each of these parameters, A, ω and φ, in turn:




Sinusoidal Waveform Plus DC Component


In some applications the sinusoidal function discussed in the previous section must be shifted up or down by some amount. To do this we add a constant yDC to the right side of the formula, like this:
y = A sin (ω t + φ) + yDC
The graph compares DC components of 0, 1 and 2.



Finding the Equation of a Sinusoidal Curve

To find the equation of a sinusoidal curve follow these steps:




Example: Find the equation of the sinusoidal wave shown to the right. This represents an alternating voltage. v is the voltage in volts and t is the time in seconds.

Solution:


Drawing the Graph of a Sinusoidal Function

To draw the graph of a sinusoidal function follow these steps:



Example: Graph the function , where v is in volts and t is in seconds.

Solution: Since the first term inside the brackets, namely , is in radians we must change the second term, 60°, to π/3 radians to make the units agree. Thus we will graph .




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