Download Elementary Functions Two special triangles The 30 - 60

Two special triangles
Two families of angles often show up in computations on the unit circle
and we use them to practice our understanding of trigonometry. In this
presentation we examine the 30-60-90 triangle and the 45-90-45 triangle
(and all triangles on the unit circle related to these.)
Elementary Functions
The 30-60-90 triangle
The 30-60-90 triangle has a right angle (90◦ ) and two acute angles of 30◦
and 60◦ . We assume our triangle has hypotenuse of length 1 and draw it
on the unit circle:
Part 4, Trigonometry
Lecture 4.2a, Two Special Triangles
Dr. Ken W. Smith
Sam Houston State University
2013
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The 30 − 60 − 90 triangle
The 30 − 60 − 90 triangle
Anytime we consider a 30-60-90 triangle, we imagine that triangle as half
of an equilateral triangle.
For example, we reflect the 30-60-90 triangle about the x-axis and so we
imagine an equilateral triangle with one vertex at the origin and the other
two vertices to the right of the origin, on the unit circle symmetrically
spaced about the x-axis.
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The 30 − 60 − 90 triangle
The 30 − 60 − 90 triangle
This triangle is an equilateral triangle with three equal sides of length 1
unit.
By the Pythagorean Theorem,
x2 + ( 12 )2 = 12
Since this equilateral triangle is symmetric about the x-axis, and since
each side has length 1, then the y-coordinates of the two vertices on the
unit circle are ± 12 .
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The 30 − 60 − 90 triangle
√
(Also cos(−30◦ ) =
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√
3
2
3
4
√
and so the x-value of the points on the unit circle must be x =
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Elementary Functions
3
2 .
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The 30 − 60 − 90 triangle
The two points on the unit circle are P (
we see that the cosine of 30◦ is
so x2 =
√
3
2
3 1
2 , 2)
√
3
1
2 , − 2 ).
30◦ is 12 .
and Q(
and the sine of
From this
and sin(−30◦ ) = − 21 .)
Elementary Functions
A similar argument with equilateral triangles works for an angle of 60◦ .
(We should always think of a 30 − 60 − 90 triangle as half of an equilateral
triangle!) In the figure drawn below we see that the cosine of 60◦ is 12 and
the sine of 60◦ is
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√
3
2
.
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The 30 − 60 − 90 triangle
The 30 − 60 − 90 triangle and its siblings
y
(0, 1)
√
Any angle on the unit circle with a reference √angle of 30◦ or 60◦ will have
coordinates that involve the numbers 21 and 23 .
These angles include
on....
30◦ , 60◦ , 120◦ , 150◦ , 210◦ , 240◦ , 300◦
330◦
3 1
2 ,2
30◦
and so
π
6
(−1, 0)
(1, 0)
x
(0, −1)
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The 30 − 60 − 90 triangle and its siblings
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The 30 − 60 − 90 triangle and its siblings
y
y
(0, 1)
(0, 1)
√ 3
1
,
2 2
π
2
π
3
60◦
(−1, 0)
90◦
(1, 0)
(−1, 0)
(1, 0)
x
x
(0, −1)
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(0, −1)
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The 30 − 60 − 90 triangle and its siblings
− 12 ,
√
3
2
The 30 − 60 − 90 triangle and its siblings
y
y
(0, 1)
(0, 1)
2π
3
√
3 1
2 ,2
−
120◦
5π
6
150◦
(−1, 0)
(1, 0)
(−1, 0)
(1, 0)
x
x
(0, −1)
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(0, −1)
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The 30 − 60 − 90 triangle and its siblings
(−1, 0)
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y
(0, 1)
(0, 1)
(1, 0)
(−1, 0)
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x
√
−
3
1
2 , −2
(0, −1)
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2013
(1, 0)
x
7π
6
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The 30 − 60 − 90 triangle and its siblings
y
π 180◦
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210◦
(0, −1)
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Elementary Functions
The 30 − 60 − 90 triangle and its siblings
The 30 − 60 − 90 triangle and its siblings
y
y
(0, 1)
(0, 1)
(−1, 0)
(1, 0)
(−1, 0)
(1, 0)
x
x
240◦
270◦
4π
3
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− 12 , −
√
3
2
3π
2
(0, −1)
(0, −1)
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The 30 − 60 − 90 triangle and its siblings
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Elementary Functions
y
(0, 1)
(0, 1)
(1, 0)
(−1, 0)
300◦
(0, −1)
11π
6
√
5π
3
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x
330◦
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(1, 0)
x
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The 30 − 60 − 90 triangle and its siblings
y
(−1, 0)
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3
1
2 , −2
√ 3
1
,
−
2
2
(0, −1)
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The 30 − 60 − 90 triangle and its siblings
The 30 − 60 − 90 triangle and its siblings
y
y
(0, 1)
(0, 1)
√
3 1
2 ,2
(−1, 0)
360◦ 2π
(1, 0)
(−1, 0)
(1, 0)
x
(0, −1)
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The 30 − 60 − 90 triangle and its siblings
y
y
(0, 1)
√ 3
1
2, 2
(0, 1)
5π
2
7π
3
420◦
(−1, 0)
450◦
(1, 0)
(−1, 0)
(1, 0)
x
x
(0, −1)
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13π
6
x
(0, −1)
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390◦
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(0, −1)
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cos
π
6
√
=
3
2 ,
sin
π
6
1
2
=
cos
π
6
√
=−
3
2 ,
sin π6 =
1
2
y
y
(0, 1)
(0, 1)
√
3 1
2 ,2
30◦
(−1, 0)
√
−
3 1
2 ,2
π
6
5π
6
150◦
(1, 0)
(−1, 0)
(1, 0)
x
x
(0, −1)
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cos
π
6
√
=−
3
2 ,
(0, −1)
Elementary Functions
sin
π
6
=
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− 12
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cos
π
6
√
=
3
2 ,
Elementary Functions
y
(0, 1)
(0, 1)
(1, 0)
(−1, 0)
√
−
3
1
2 , −2
210◦
330◦
11π
6
√
3
1
2 , −2
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x
(0, −1)
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(1, 0)
x
7π
6
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sin π6 = − 12
y
(−1, 0)
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Elementary Functions
cos
π
6
=
1
2,
sin
√
π
6
=
3
2
cos
π
6
=
− 12 ,
sin
π
6
√
3
2
=
y
y
(0, 1)
√
3
1
2, 2
− 12 ,
√
3
2
π
3
60◦
(−1, 0)
(0, 1)
2π
3
120◦
(1, 0)
(−1, 0)
(1, 0)
x
x
(0, −1)
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cos
π
6
=
− 12 ,
(0, −1)
Elementary Functions
sin
π
6
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√
=−
3
2
cos
π
6
=
1
2,
sin
Elementary Functions
π
6
=−
(0, 1)
(0, 1)
(−1, 0)
x
240◦
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− 12 , −
√
300◦
4π
3
5π
3
3
2
(0, −1)
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(1, 0)
x
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3
2
y
(1, 0)
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√
y
(−1, 0)
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√ 3
1
,
−
2
2
The 30 − 60 − 90 triangle
The 45-90-45 triangle
We collect this information in a table. The first two columns give the
angle, first in degrees and then in radians. The last two columns give the
x and y coordinates.
Another important triangle is the right triangle with both acute angles
equal to 45◦ . Since this triangle has two equal angles then it is isoceles; it
also has two equal sides. We may assume that the hypotenuse has length
1 and so draw this triangle on the unit circle.
θ
θ
cos(θ)
sin(θ)
30◦
60◦
120◦
150◦
210◦
240◦
300◦
330◦
π
6
π
3
2π
3
5π
6
7π
6
4π
3
5π
3
11π
6
3
2
1
2
−√12
− √23
− 23
− 12
1
√2
3
2
1
√2
3
√2
3
2
1
2
−√12
− √23
− 23
− 12
√
I wouldn’t try to memorize this but instead be able to recreate it by proper
use of the 30-60-90 triangle.
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The 45-90-45 triangle
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The 45-90-45 triangle
Since the triangle is isoceles, its height y is equal to the base x. By the
Pythagorean theorem,
1
cos(45◦ ) = sin(45◦ ) = √ .
2
x2 + y 2 = 1 =⇒ x2 + x2 = 1 =⇒ 2x2 = 1.
=⇒ x2 = 12 .
=⇒ x =
√1 .
2
If we wish to get rid of the radical
√ in the denominator, we can multiply
numerator and denominator by 2 so that the equation becomes
√
2
◦
◦
cos(45 ) = sin(45 ) =
.
2
Since y = x then
cos(45◦ ) = sin(45◦ ) =
√1 .
2
Any angle θ√in which the reference angle is 45◦ will have cosine and sine
2
equal to ±
.
2
Here are some examples.
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The 45 − 90 − 45 triangle and its siblings
The 45 − 90 − 45 triangle and its siblings
y
y
(0, 1)
(0, 1)
√
√ 2
2
,
2
2
√
−
2
2
2 , 2
π
4
3π
4
45◦
(−1, 0)
√
135◦
(1, 0)
(−1, 0)
(1, 0)
x
x
(0, −1)
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(0, −1)
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The 45 − 90 − 45 triangle and its siblings
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Elementary Functions
y
(0, 1)
(0, 1)
(1, 0)
(−1, 0)
315◦
5π
4
7π
4
√
√ 2
2
,
−
2
2
√
√ 2
2
,
−
2
2
(0, −1)
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x
225◦
−
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(1, 0)
x
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The 45 − 90 − 45 triangle and its siblings
y
(−1, 0)
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Elementary Functions
The 45 − 90 − 45 triangle and its siblings
The 45 − 90 − 45 triangle and its siblings
y
y
(0, 1)
(0, 1)
√
√ 2
2
,
2
2
√
√
2
2
2 , 2
−
9π
4
405◦
11π
4
495◦
(−1, 0)
(1, 0)
(−1, 0)
(1, 0)
x
x
(0, −1)
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cos
π
4
√
=
2
2 ,
(0, −1)
Elementary Functions
sin
π
4
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√
=
2
2
cos
3π
4
Elementary Functions
√
=−
2
2 ,
sin
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√
π
4
=
2
2
y
y
(0, 1)
(0, 1)
√
√
2
2
2 , 2
√
−
3π
4
45◦
(−1, 0)
√
2
2
2 , 2
π
4
135◦
(1, 0)
(−1, 0)
(1, 0)
x
x
(0, −1)
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(0, −1)
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Elementary Functions
cos
5π
4
√
=−
2
2 ,
sin
√
π
4
2
2
=−
cos
7π
4
√
=
2
2 ,
sin
π
4
√
=−
2
2
y
y
(0, 1)
(0, 1)
(−1, 0)
(1, 0)
(−1, 0)
(1, 0)
x
x
225◦
315◦
5π
4
7π
4
√
−
√ 2
2
,
−
2
2
√
√ 2
2
,
−
2
2
(0, −1)
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The 45-90-45 triangle
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Six Functions on the Unit Circle
All of this information has been provided for us (calculator free!) by our
understanding of two basic triangles. Make sure you can recover these
results just by drawing the appropriate triangle in the appropriate location.
We summarize this in a table.
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θ
θ
cos(θ)
sin(θ)
45◦
135◦
225◦
315◦
π
4
3π
4
5π
4
7π
4
2
2√
2
√2
2
2√
√
− √22
−√ 22
2
2
Elementary Functions
√
− √22
− 22
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Two special triangles
Two special triangles
In the next presentation, we will look at the six trig functions.
(End)
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Six Functions on the Unit Circle
A central angle θ determines a point P (x, y) on the unit circle. The
x-coordinate is cos θ and the y-coordinate is sin θ. There are four other
trig functions, based on this point.
Elementary Functions
Part 4, Trigonometry
Lecture 4.2b, Six Functions on the Unit Circle
The tangent function tan θ is equal to
Dr. Ken W. Smith
y
x
=
sin θ
cos θ .
If the ratio xy looks familiar, this is the “rise” over “run” as we move from
the origin O(0, 0) out to the point P (x, y); that is, tan θ is the slope of
the line joining O to P .
Sam Houston State University
2013
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Six Functions on the Unit Circle
Six Functions on the Unit Circle
There are three more trig functions each of which is the reciprocal of one
of the first three.
1
The secant of θ is the reciprocal of cosine; sec θ = x1 .
2
The cosecant of θ is the reciprocal of sine; csc θ = y1 .
3
The cotangent of θ is the reciprocal of tangent; cot θ = xy .
Since we may write the six trig functions in terms of x and y on the unit
circle then we may also write the six trig functions in terms of cosine and
sine.
We may also write all six trig functions as ratios of the sides of the right
triangle with short legs of length x and y and hypotenuse 1.
The reciprocal of a trig function with the syllable “sine” in it (sine and
cosine) will be a trig function involving “secant” (cosecant and secant.)
Each reciprocal pair has exactly one “co” in the list, so:
the reciprocal of sine is cosecant;
the reciprocal of cosine is secant;
the reciprocal of tangent is cotangent.
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Six Functions on the Unit Circle
The six trig functions on the right triangle
Here is a summary of our results.
x
1 The cosine function cos θ is equal to 1 .
y
2 The sine function sin θ is equal to 1 .
y
sin θ
3 The tangent function tan θ is equal to x = cos θ .
1
1
4 The secant function sec θ is equal to x = cos θ , the reciprocal of
cosine.
1
1
5 The cosecant function csc θ is equal to y = sin θ , the reciprocal of
sine.
x
cos θ
6 The cotangent function cot θ is equal to y = sin θ , the reciprocal of
tangent.
If we draw an angle in general position, intersecting the unit circle at
P (x, y), vertices C(0, 0), P (x, y) and R(x, 0) form a right triangle with a
right angle on the x-axis at R(x, 0).
We can define the six trig functions in terms of the sides of the triangle
4CP R.
We follow tradition: call the side CR as the “adjacent” side (close to the
angle θ), RP as the “opposite” side (far from the angle θ), and CP is the
“hypotenuse.”
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The six trig functions on the right triangle
Six Functions on the Unit Circle
If we abbreviate the lengths “adjacent”, “opposite” and “hypotenuse” by
their first letters, A, O and H, we might draw the triangle like this.
Thousands of trig students have memorized the first three trig functions
as:
O
1 Sin θ = H
A
2 Cos θ = H
O
3 Tan θ = A
and put these together into a chant: SOH-CAH-TOA.
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Six Functions on the Unit Circle
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Six Functions on the Unit Circle
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Elementary Functions
Six Functions on the Unit Circle
Central Angles and Arcs
A side story
I first learned of SOH-CAH-TOA at Central Michigan University when a
number of calculus students wrote the letters at the top of their exams.
Since the Saginaw Chippewa tribe is located near the university and the
university had a course in the Ojibwe language, I wondered if Soh-cah-toa
was an Ojibwe phrase or prayer.
We will look more closely at these right triangle definitions of our trig
functions in the next presentation.
(End)
But one of the students explained that Soh-cah-toa was merely an English
abbreviation for the trig functions. (But they didn’t deny saying a short
prayer as they wrote it at the top of the exam!)
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Six Functions on the Unit Circle
Worked problems.
13π
1 Find all six trig functions of the angle θ = 3 .
Solution. Draw the unit circle and find the point on the circle given
13
1
13π
π
by the angle 13π
3 radians. Since 3 = 4 + 3 then 3 = 4π + 3 and so
13π
π
the angle 3 gives the same point on the unit circle as 3 . This point
is
√
P ( 12 , 23 ).
√
√
3
√
3
13π
1
13π
13π
Therefore cos 3 = 2 , sin 3 = 2 and tan 3 = 21 = 3. The
Elementary Functions
Part 4, Trigonometry
Lecture 4.2c, Practicing With Six Trig Functions
Dr. Ken W. Smith
2
Sam Houston State University
2013
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values of secant, cosecant and tangent are reciprocals of these so the
values of all six trig functions are:
√
3
cos( 13π
)
=
3
2
sin( 13π
= √12
3 )
tan( 13π
=
3
3 )
13π
2
√
sec( 3 ) =
3
csc( 13π
)
=
2
3
13π Functions
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cot(
= √13
3
Six Functions on the Unit Circle
Complementary angles
√
2
Suppose sin θ = − 23 and cos θ is positive. Identify the angle θ and
then find all six trig functions of the angle θ.
◦
Solution. The angle is 5π
3 , equal to 300 . (Or we could choose the
angle −60◦ = − π3 .)
So
√
√
3
1
5π
5π
cos( 5π
)
=
,
sin(
)
=
−
3
2
3
2 , tan( 3 ) = − 3.
Given an angle θ we will talk about the angle we need to add to θ to make
up 180◦ or 90◦ . The angle we need to add to θ to create a 180◦ angle is
the “supplement” of θ. So the supplementary angle of θ is 180◦ − θ or
(in radians) π − θ.
The angle we need to add to θ to “complete” a right angle is the
“complement” of θ; the complementary angle of θ is 90◦ − θ or π2 − θ. In
the figure below, with central angle θ (in black) the complementary angle
of θ is drawn in blue.
The reciprocals of these are
√
2 3
5π
5π
√2
√1
sec( 5π
3 ) = 2, csc( 3 ) = − 3 = − 3 , cot( 3 ) = − 3 = −
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√
3
3 .
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Six Functions on the Unit Circle
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The Pythagorean Theorem on the unit circle
The term “complementary” explains the syllable “co-” in the terms cosine,
cotangent and cosecant.
The cosine of the angle θ is merely the “sine of the complement” of θ;
co-sine is an abbreviation for “sine of the complement.”
O
and the cosine of θ is
From the drawing we see that if the sine of θ is H
π
A
A
then
the
sine
of
the
angle
−
θ
is
also
;
the
2
H
H
“sine-of-the-complement” is merely the “co-sine.”
In a similar way, “co-tangent” and “co-secant” stand for
“tangent-of-the-complement” and “secant-of-the-complement”.
Since the points P (x, y) are on the unit circle then, by the Pythagorean
Theorem, x2 + y 2 = 1. In terms of sine and cosine, this translates into
(cos θ)2 + (sin θ)2 = 1.
(1)
Dividing by (cos θ)2 we have
1 + (tan θ)2 = (sec θ)2 .
(2)
Dividing the top equation by (sin θ)2 we have
(cot θ)2 + 1 = (csc θ)2 .
(3)
These three identities are often called “The Pythagorean Identities” of
trigonometry since they all depend on the Pythagorean theorem on the
unit circle.
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A Worked Problem.
Central Angles and Arcs
Given the cosine or sine of θ we should (by the Pythagorean Theorem) be
able to find all the trig functions of θ. Here is an example.
Find all trig functions of the angle θ if θ is in the second quadrant and
sin(θ) = 23 .
Solution. If sin(θ) = 23 then by the Pythagorean Theorem
In the next presentation, we will look in depth at the sine function and
transformations of it.
cos(θ)2 + ( 23 )2 = 1
=⇒ cos(θ)2 = 1 − ( 23 )2 =
9
9
−
4
9
= 59 .
√
Since cos(θ)2 = 59 then cos(θ) = ± 35 . Since θ is in the second quadrant
we know that the
x-value of the point P (x, y) is negative and so
√
5
x = cos(θ) = − 3 . The point on the unit circle corresponding to the
(End)
√
angle θ is therefore P (− 35 , 23 ) and with this information we can compute
the remaining four trig functions of θ.
√
cos(θ)
sin(θ)
tan(θ)
Smith (SHSU)
=
=
=
−
2
3
5
3
√
− √25 = − 2 5 5
√
Elementary Functions
sec(θ)
csc(θ)
cot(θ)
=
=
=
− √35 = − 3 5 5
3
2√
−
5
2
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Smith (SHSU)
Elementary Functions
2013
70 / 70