Download Review of REQUIRED Knowledge of Triangle Trigonometry Spring

Review of REQUIRED Knowledge of Triangle Trigonometry
Spring 2014 Dave Parker MATH 201
FORM ING REFERENCE TRIANGLES AND ANGLES
Angles in standard position begin on the positive Xaxis, and rotate a fixed radius R in a positive direction
(counter-clockwise) or in a negative direction (clockwise),
dropping a perpendicular to the X-axis from the far end of the
radius, thereby form ing a triangle on the X-axis. For exam ple,
if the result of so doing results in the form ation of the reference
triangle to the right, where " is a positive angle, 0 < " < 90
degrees, then we label the three sides of this triangle X, Y, and
R. Note that either X or Y could be positive or negative,
depending upon the coordinates of the point (X,Y).
(In this case we ended up with both X and Y positive.)
R is always positive, and if R is 1,
we have unit-circle trigonom etry.
The angle " form ed above, with vertex at the origin, with one side on the X-axis, with one side
vertical and perpendicular to the X-axis, and with hypotenuse intersecting the origin, is called the
reference angle for the angle through which we rotated. Note that we would have the sam e reference angle
if we had rotated the radius several tim es around the origin prior to ending up where we did at the point (X,Y).
The trigonom etric functions of the reference angle are the sam e as the trigonom etric functions for the angle
through which we rotated. In the exam ple above, the triangle lies in the first quadrant as does the reference
angle. The reference angle is ALWAYS located at the origin and is ALWAYS opposite the side of the
reference triangle which is vertical to the X-axis. There are three other quadrants in which the reference
triangle (and reference angle) could be located, with X and Y either positive or negative, depending upon the
coordinates for the point (X,Y), as will be demonstrated in class. AGAIN NOTE: R is ALWAYS positive.
DEFINITIONS (AND IM PORTANT TRIVIALITIES) USING THE REFERENCE TRIANGLE ABOVE:
Notation: relative to the reference angle, X is the adjacent (horizontal) side or Adj, Y is the
opposite (vertical) side or Opp, and R is the hypotenuse or Hyp.
Definition
Triviality
(sine of ")
sin " = Y'R
=
Opp'Hyp
=
1'csc "
(cosine of ")
cos " = X'R
=
Adj'Hyp
=
1'sec "
(tangent of ")
tan " = Y'X
=
Opp'Adj
=
1'cot "
(cosecant of ")
csc " = R'Y
=
Hyp'Opp
=
1'sin "
(secant of ")
sec " = R'X
=
Hyp'Adj
=
1'cos "
(cotangent of ")
cot " = X'Y
=
Adj'Opp
=
1'tan "
NOTATION: sin n x (exam ple sin 2 x) m eans (sin x) n except when n is &1, and in that case n denotes the
inverse sine of x, NOT the reciprocal of the sine of x. Sim ilarly we have cos n x, tan n x, etc. NOTE: NONE
of these can be written as sin x n, cos x n, tan x n, etc., so sin 2x and sin x 2 are DIFFERENT!
ADDITIONAL IM PORTANT TRIVIALITIES:
tan " = sin " ' cos "
NOTE that this is complete nonsense: tan = sin ) cos
cot " = cos " ' sin "
From the Pythagorean Theorem we know that X 2 + Y 2 = R 2, so
(1)
sin 2"+ cos 2" = 1
again noting that sin 2" = (sin ") 2, etc.
because (Y'R) 2 + (X'R) 2 = (X 2 + Y 2)'R 2 = R 2'R 2 = 1
(2)
tan 2" + 1 = sec 2"
by dividing (1) by cos 2" and using the definitions and trivialities above.
2
(3)
1 + cot " = csc 2"
by dividing (1) by sin 2" and using the definitions and trivialities above.
ANGLE M EASURE:
Instead of m easuring angels in degrees (as done above), in m athem atics, engineering, and science
we will usually use radians. A com plete revolution of 360 degrees is 2B radians, or as it is usually stated: 180
degrees is B radians. From now on, unless degrees are explicitly stated, ALWAYS assume angle measures
are all in radians! For exam ple, sin 30 m eans the sine of 30 radians, not the sine of 30 degrees!
STANDARD, USEFUL RIGHT TRIANGLES:
B'4 B'4 B'2 (In degrees: 45 45 90) Has sides of length 1, 1, and %2.
B'6 B'3 B'2 (In degees: 30 60 90) Has sides of length 1, 2, and %3.
B'2 B'2
0
(In degrees: 90 90 0) Has sides of length 0, 1, and 1. (A degenerate
triangle!)
Note: Draw the sides of each reference triangle so that they are approximately in proportion
to each other. In each case the longest side is the hypotenuse, not necessarily the
last number listed! ALW AYS Check your reference triangles using the Pythagorean
Theorem !
S I G N S
O F
TRIG O N O M ETRIC
F U N C T IO N S IN
T H E
F O U R
QUADRANTS
In other words,
•
ALL of the trigonom etric functions are positive if the reference triangle is in the first quadrant,
•
The Sine and thus (W hy?) the Cosecant functions are positive (and the rest are negative) if
the reference triangle is in the second quadrant,
•
The Tangent and thus (W hy?) the Cotangent functions are positive (and the rest are
negative) if the reference triangle is in the third quadrant, and
•
The Cosine and thus (W hy?) the Secant functions are positive (and the rest are negative) if
the reference triangle is in the fourth quadrant.
GRAPHS TO KNOW
(NOT SHOW N HERE BUT ILLUSTRATED IN CLASS AND FOUND IN THE TEXTBOOK):
y = sin x
y = cos x
y = tan x
The other three are easily constructed from these.
If p is the smallest number such that f(x + p) = f(x) for all real x, then p is called the period of f(x).
The Sine and Cosine functions have periods of 2B and the Tangent function has a period of B .
Multiplying a periodic function f(x) by a constant scales the function’s graph. Multiplying x by a
constant greater than one m akes the period shorter; m ultiplying x by a constant less than one m akes
the period longer. Details are in the textbook and will be dem onstrated in class as necessary.
-----------------------------------------------------------------------------SOM E TRIGONOM ETRY NEEDED IN M ATH 202 (and elsew here):
STANDARD FORM ULAE:
sin (A + B) = sin A cos B + cos A sin B
sin (A & B) = sin A cos B & cos A sin B
cos (A + B) = cos A cos B & sin A sin B
cos (A & B) = cos A cos B + sin A sin B
EASILY DERIVED FROM THE ABOVE USING ALGEBRA (So KNOW the above standard formulae!):
If B = A,
sin (A + A) = sin (2A) = 2 sin A cos A
and
cos (A + A) = cos (2A) = cos 2 A & sin 2 A
sin 2(x) = ( 1 & cos(2x) ) ) 2
cos 2(x) = ( 1 + cos(2x) ) ) 2