Math B is all about trigonometry. Trig, as it is called, uses the functions
sine, cosine, tangent, cotangent, secant, and cosecant. Sine and cosine are
opposites as are tangent and cotangent as well as secant and cosecant. All
these functions apply to a right triangle. To remember the relationship of
three of these functions, simply recall the acronym SOH-CAH-TOA.
SOH => Sine = opposite leg divided by hypotenuse
CAH => Cosine = adjacent leg divided by hypotenuse
TOA => Tangent = opposite leg divided by adjacent leg
We can relate sine and cosine to coordinates on a graph. The ordinate, or y-
value, on a graph is the sine value for a given angle. The quadrant that the
angle is in, therefore, is very important in this matter. Y-values are
positive in the first and second quadrants; therefore, sine values will be
positive in first and second quadrants. Y-values are negative in third and
fourth quadrants; therefore, sine values will be negative i third and fourth
quadrants.
Abscissa, or x-values, would be our cosine values. The value of the abscissa
will determine the value of the cosine of the given angle. Since x-values
are positive in the first and fourth quadrants, cosine values will be
positive in the first and fourth quadrants. Since x-values are negative in
the second and third quadrants, cosine values will be negative in the second
and third quadrants.
Trigonometry refers only to right triangles. All triangles have a total of
180 degrees. If a triangle is located in the second, third, or fourth
quadrant, it needed to rotate there from the first quadrant. Since rotation
forms a circle, and since circles have 360 degrees, we need to determine the
equivalent angle of rotation for a given triangle. For example, if a tire
rotates 450 degrees, the position that it would be in has an equivalent
position between 0 and 360 degrees. To determine that equivalent (coterminal)
angle, we subtract 360 degrees to get the answer 90 degrees.
If the tire reverses itself, it would have negative angles. To determine
the coterminal angle, we add 360 degrees to the negative angle. For example,
if we reverse the tire -632 degrees, we add 360 degrees to see it reversed
-272 degrees. We add 360 degrees again to get an angle of 88 degrees.
To determine which quadrant certain angles lie in after rotations have been
made, refer to the following chart:
Between 0 & 90 degrees = Quadrant I
Between 90 & 180 degrees = Quadrant II
Between 180 & 270 degrees = Quadrant III
Between 270 & 360 degrees = Quadrant IV
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We can identify sine and cosine by the abscissa (x-value) and ordinate (y-
value) for a given co-ordinate. Let's picture a graph with cor-ordinates
(x,y). The opposite leg is the y value,; therefore, the sine of a given
angle theta is equal to the ordinate (y-value). The adjacent leg is the x-
value; therefore, the cosine of a given angle theta is equal to the abscissa
(x-value). In the first quadrant, all values are positive; therefore, sine
and cosine values will always be positive. However, as we move into the
second quadrant, cosine becomes negative since x-values are negative in that
quadrant. The sine values remain positive, though. Moving into the third
quadrant, both the x-values and y-values are negative; therefore, all sine
and cosine values are negative here. In the fourth quadrant, x-values again
become positive so cosine values become positive again. Y-values remain
negative here, so sine values remain negative here. Therefore, if we have a
rotation greater than 90 degrees, we can determine the coterminal angle and
evaluate the sine and cosine values for the coterminal angles. They will be
the same for the angles greater than 90 degrees, except we need to keep in
mind which quadrants have positive and negative sine/cosine values. See
above for the angles that lie in a given quadrant.
Tangent can be defined as opposite over adjacent. Another way of writing
tangent is to use trigonometric functions. The function that refers to the
opposite leg is sine while the function that refers to the adjacen tleg is
cosine. Therefore, tangent is sine/cosine. Since sine and cosine are
positive in the first quadrant, tangent is positive in the first quadrant.
Since sine is positive an dcosine is negative in the second quadrant,
tangent is negative in the second quadrant. Since both sine and cosine are
negative in the third quadrant, tangent will be positive in the third
quadrant. Since sine is negative in the fourth quadrant and cosine is
positive in the fourth quadrant, tangent will be negative in the fourth
quadrant. Just remember, All Students Take Class and you will remember the
positive functions for the various quadrants.
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If we were to draw a circle on a graph with a radius of 1, we would have
coordinates of (1,0), (0,1), (-1,0), and (0,-1). These coordiantes
correspond with the angles of 0 degrees/360 degrees = (1,0); 90 degrees =
(0,1); 180 degrees = (-1,0); and 270 degrees = (0,-1). Using the method above
we can identify the sine, cosine, and tangent values for each of these
angles.
ANGLE SINE (y-value) COSINE (x-value) TANGENT (sine/cosine)
0 degrees 0 1 0/1 = 0
90 degrees 1 0 1/0 = undefined
180 degrees 0 -1 0/-1 = 0
270 degrees -1 0 -1/0 = undefined
360 degrees 0 1 0/1 = 0
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If we were to draw an equilateral triangle on a graph and bisect it, we
would create two right triangles with the following angle measures - 30
degrees, 60 degrees, and 90 degrees. This is a special case. If we use the
Pythagorean formula and plug in the distance of 1 (hypotenuse) and 0.5 (one
leg opposite 30 degrees), we would have a value of radical 3/2 for the leg
adjacent to 30 degrees. Using the values of 1, 0.5, and radical 3/2, we can
determine the sine, cosine, and tangent values for 30 and 60 degrees.
ANGLE SINE (opposite leg) COSINE (adjacent leg) TANGENT (sine/cosine)
30 degrees 0.5 radical 3/2 0.5/(rad 3/2)= rad3/3
60 degrees rad 3/2 0.5 (rad 3/2)/0.5= rad 3
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If we were to draw an isosceles right triangle, we would have angle
measures of 45 degrees, 45 degrees, and 90 degrees. This is a special case.
If our hypotenuse is 1, we can use the Pythagorean formula to determine the
values of the other two legs. They would each measure radical 2/2. The sine,
cosine, and tangent values for 45 degrees can then be determined.
ANGLE SINE (opposite leg) COSINE (adjacent leg) TANGENT (sine/cosine)
45 degrees radical 2/2 radical 2/2 (rad 2/2)/(rad 2/2) = 1
THESE VALUES MUST BE MEMORIZED!!!
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To find the values for any given angle, we simply punch into the calculator
the function value that we are seeking and the angle we want. Then push
enter. This is true even if we are dealing with angles greater than 90
degrees.
However, we still need to know the reference angle in the first quadrant
for any angle in another quadrant. To do this, follow these rules:
QUADRANT ANGLE RANGE FORMULA
II 90-180 degrees 180 degrees - theta
III 180-270 degrees theta - 180 degrees
IV 270-360 degrees 360 degrees - theta
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We can measure angles in other units besides degrees. Another unit we use
is radians. One radian is equal to the measure of the radius of a circle
when there is a central angle drawn in the circle. The angle we are
referring to is the arc intercepted by the central angle. The measure of the
angle in radians can be determined by dividing the length of the intercepted
arc by the length of the radius.
length of the intercepted arc
Angle measure = --------------------------------------
length of the radius
From the above formula, we can determine a relationship between a radian
and a degree. Let's look at a semicircle. A semicircle has an arc equal to
half the circumference of the circle. Therefore, the arc equals (pi)(radius).
The angle of the semicircle is 180 degrees. To find the number of radians,
we can use the formula above:
(pi)(radius) pi
180 degrees = ------------------- => 180 degrees = -----, or 180 degrees = pi
length of radius 1
How many degrees is one radian equal to? Divide 180 degrees by pi and we
get 57.3 degrees.
To change from degrees to radians, we use the proportion:
measure of angle in degrees 180 degrees
---------------------------- = -------------
measure of angle in radians pi
For example, 20 degrees is equal to:
20 degrees 180 degrees 20pi = 180 radians => 20pi
---------- = ---------------- => ---- = radians
radians pi 180
pi
-- = radians
9
To change from radians to degrees, we simply substitute 180 degrees for pi
and do the math.
pi 180
-- radians => ----- = 15 degrees
12 12
The following table should be memorized:
DEGREE RADIAN
0 0
30 pi/6
45 pi/4
60 pi/3
90 pi/2
180 pi
270 3pi/2
360 2pi
To determine the trigonometric function values for sine, cosine, and
tangent for radian measure, we switch our calculators from degrees to
radians and punch in the numbers as we did before. When we see pi, we use
radian. Otherwise, we use degrees.
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Each trigonometric function has a reciprocal function. The reciprocal
function of sine is cosecant (1/sine theta). The reciprocal function of
cosine is secant (1/cosine theta). The reciprocal function of tangent is
cotangent (1/tangent theta). Our calculators do not have these function keys
on them so we need to punch in 1/ whatever the function is we need here for
the correct reciprocal. This is true regardless of whether or not we need
degrees or radians.
In what quadrants are these reciprocals positive? In the same quadrants as
the function sthey are derived from.
QUADRANT FUNCTION VALUE
I all positive
II sine,cosecant positive
III tangent,cotangent positive
IV cosine,secant positive
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So far we have five identities identified. We know three reciprocal
identities - secant, cosecant, and cotangent (see above). We know two
quotient identities - tangent = sine/cosine; cotangent = cosine/sine. Now we
will look at three Pythagorean identities. The Pythagorean formula is a-
squared + b-squared = c-squared. We could substitute x and y for a and b and
get:
x-squared + y-squared = c-squared
Since we use radii of 1, we can rewrite this as:
x-squared + y-squared = 1
We already know that x = cosine and y = sine; therefore:
cosine squared + sine squared = 1
We already proved that true in class.
If we divide this by cosine squared, we get:
1 + tangent squared = secant squared
Why? Because cosine squared divided by itself equals one, sine squared
divided by cosine squared equals tangent squared, and 1 divided by cosine
squared equals secant squared.
If we were to divide the original identity by sine squared, we get:
cotangent squared + 1 = cosecant squared
Why? Because cosine squared divided by sine squared equals cotangent
squared, sine squared divided by itself equals one, and 1 divided by sine
squared equals cosecant squared. These are the three Pythagorean identities.
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If we know one function of an angle, we can solve for the other five using
the identities above. If sine theta degrees equals 12/13, then we plug it
into the first Pythagorean identity and arrive at 5/13. This is cosine.
The reciprocal of 12/13 is 13/12 (cosecant). The reciprocal of 5/13 is 13/5
(secant). Tangent is (12/13)/(5/13), which is 12/5 while cotangent is its
reciprocal, 5/12.
By analyzing the above example, we can now say that sine and cosine are
cofunctions of each other as are secant and cosecant as well as tangent and
cotangent. Therefore, the sine of any angle is equal to the cosine of its
complementary angle. The same is true for secant and cosecant as well as for
tangent and cotangent. For example, sine 20 degrees = cos 70 degrees. Sin 70
degrees = cosine 20 degrees. We can also say the same for secant and
cosecant as well as tangent and cotangent.
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So far we have studied how to find function values for angles in degrees and
radians. We are at the point now where we could graph each function. But,
first, we need to identify pieces of the equation for each function. Sine
and cosine, we have learned, always have values. Therefore, these two
functions would be continous lines on a graph. A standard equation for sine
is y = sin x, where sin x is all values of sine for all angles. It will
increase between the values of 0 and pi/2 because these angles are in
quadrant I. It will decrease from pi/2 to pi because these are in quadrant
II. It will again decrease from pi to 3pi/2 because these values are in
quadrant III. It will increase from 3pi/2 to 2pi because these values are in
quadrant IV. We could change the height (amplitude) of this wave by
multiplying it by a whole number. For example, y = 2 sin x would be twice as
tall because everything is being doubled. We could change the frequancy
(period) of the wave by multiplying the angle by a whole number. For
example, y = sin 2x would have 2 waves because we are doubling all sine
values.
The same argument applies to cosine. However, with tangent, there are
breaks because some values of tangent are undefined.
The last thing to remember about these graphs is the domain and range.
Domain refers to the x values. It usually is between -2pi and 2pi. However,
we could change this to -pi to pi or from 0 to 2pi or anything we want. The
range is the height (amplitude) of the graph.
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When we graph functions onto a graph, we can reflect the graphs through the
line y=x. When we do this, the x- and y-values switch. Therefore, y = sin x
becomes x = sin y (x is the sine value whose angle is y). The same is true
for y = cos x and y = tan x. We could rewrite these equations as y = arc sin
x, y = arc cos x, and y = arc tan x. These equations all mean that when we
have the value of the function we are working with, we could find the angle
measure that corresponds with that value. However, we need to keep in mind
that each function has two angle measures.
The angles we look for could be in degrees or radians. For example, if we
have a sine value of .5, there will be two angles associated here. One is in
the first quadrant because sine is positive there and one is in the second
quadrant because sine is also positive there. The first angle is 30 degrees
and the second is 150 degrees. DO NOT FORGET HOW TO FIND REFERENCE ANGLES!!!
In radian measure, it would be pi/6 and 5pi/6. If it was a negative number,
the answer would be a third and a fourth quadrant angle.
For cosine, the angles are positive in the first and fourth quadrants and
negative in the second and third quadrants. For tangent the values are
positive in the first and third quadrants and negative in the second and
fourth quadrants.
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We can apply laws of exponents to our study of trigonometry. Remember the
following laws when working with exponents:
1) when multiplying numbers or variables with the same base, add the
exponents. For example, (3^3)*(3^2) = 3^5 and (x^2)*(x^4) = x^6
2) when dividing numbers or variables with the same base, subtract the
exponents. For example, (3^3)*(3^2) = 3^1 and (x^2)*(x^4) = x^-2
3) when raising a power to a power, multiply the exponents. For example,
(3^2)^3 = 3^6 and (x^4)^3 = x^12
4) when raising a number to the zero power, the answer is always 1, except
if you have 0^0 (which is undefined). For example, 3^0 = 1 and x^0 = 1
5) when raising a number to a negative exponent, the answer is the
reciprocal of the base raised to a positive power. For example, 3^-2 = 1/
(3^2) = 1/9
We can write powers of 10 in this way. This is called scientific notation.
We can rewrite very large numbers as powers of 10 to the positive powers and
rewrite very small numbers as powers of 10 to the negative powers. For
example, 187,000 can be rewritten as 1.87 * 10^6 while .00035 is 3.5 * 10 ^-4
All we have to do is put the decimal point between the first and second
digits and count how many decimal places it moved. The number of places is
the power of 10 we use. Very large numbers are positive powers while very
small numbers are negative powers.
We can graph exponential functions with our graphing calculator. You will
see that the line curves sharply to the right and up for the equation y = b^x
where b is the number we want and x is the exponent. y = 3^x would curve
right and up and y = 7^x also. A fraction raised to a poweer would curve
down and to the right. For example, y = (1/3)^x would curve down and right
and y = (1/7)^x would also. Both of these forms are in the first and second
quadrants only, and they never touch the x-axis. The equations y = 3^x and y
= (1/3)^x are reflections of each other through the y-axis. We can also
reflect equations through the x-axis. The reflection of y = 3^x is y = -3^x
and will curve to the right and down. The reflection of y = (1/3)^x throught
he x-axis is y = -(1/3)^x and will curve up and to the right. Both of these
equations will sit in the third and fourth quadrants.
We can solve fractional and exponential equations using the following rules.
Solving fractional equations can be done by isolating the variable. Then we
raise both sides to the power of the reciprocal of the fractional exponent.
Lastly, simplify. For example, 2y^(3/2) = 16. First, we divide both sides by
4. That leaves us with y^(3/2) = 8. Now raise both sides to the (2/3) power.
We have y^(3/2)^(2/3) = 8^(2/3). Simplifying, we have y = 8^(2/3) or y = 4
For exponential functions, we need to have the same base on both sides of
the equation. If we don't, we need to rewrite both sides as powers of the
same base. We can then set the exponents on both sides equal to each other
and solve. For example, 2^(x+3) = 2^11 We have the same base so we set both
exponents equal to each other. We now have x+3 = 11 Solve the problem, and
we have x = 8
If the two sides do not have the same base, do the following:
4^3 = 2^(x-7) Set both sides as powers of the same base. We know that 4 = 2^2
so we rewrite the left side of the equation as (2^2)^3 = 2^(x-7) Simplify
the left side, so we have 2^6 = 2^(x-7) Solve the equation since we ahve the
same base. 6 = x-7 or x =13
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