Download M19500 Precalculus Chapter 6.3 Trigonometric functions of angles

Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
Every real number is the radian measure of some angle
Every real number, viewed as a number of
radians, represents an angle. The initial
side of the angle is a horizontal arrow
pointing right from the origin. Rotate that
arrow counterclockwise (or clockwise) with
each full turn counting as 2π (or −2π)
radians. The final position of the arrow is
the terminal side of the angle.
θ
initial side
terminal side
In the example at the right, the initial side
is rotated through 43 of a complete circle.
The rotation is suggested by the red
dashed arc. The angle θ corresponds to 34
of a complete trip around the circle, and so
◦
θ = 34 · 2π = 3π
2 rad = 270 .
y
More examples are on the next slide.
Stanley Ocken
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
x
Radian measure
Trig functions of general angles
Radians
Degrees
y
Finding the reference angle
2π
360
y
π
180
x
3π
rad
4
= 135◦
π/3
60
y
π/4
45
x
y
−πrad = −180◦
Stanley Ocken
π/6
30
x
5π
rad
4
= 330◦
x
= 225◦
y
x
− π6 rad = −30◦
Quiz Review
y
y
x
Calculating trig functions
x
11π
rad
6
πrad = 180◦
y
− 5π
rad = −225◦
4
π/2
90
Basic trig identities
x
- 3π
rad = −135◦
4
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
Trig functions of angles in Quadrant 1
Let θ be any angle. Place its initial side on the
positive x-axis. Suppose the arrow tip of the
terminal side of θ ispat point P (x, y) in the
x,y-plane. Let r = x2 + y 2 be the distance
from P (x, y) to the origin. r is positive!
y
5
4
P (x, y)
(4, 3)
3
The trig functions of angle θ
x
r
y
sin(θ) = r
tan(θ) = xy
cos(θ) =
r=5
2
1
r
cos θ = x
1
r
csc(θ) = sin θ = y
1
x
cot(θ) = tan
θ = y
sec(θ) =
3
1
θ
0
-5
-4
-3
-2
-1
0
1
4
2
3
4
5
-1
In the picture at the right, the arrow tip of the
terminal side is atp
P (4, 3). Thus
x = 4, y = 3, r = x2 + y 2 = 5 and so
x
4
r = 5
sin(θ) = yr = 35
tan(θ) = xy = 43
cos(θ) =
-2
-3
-4
r
5
x = 4
csc(θ) = yr = 53
cot(θ) = yr = 43
sec(θ) =
-5
Stanley Ocken
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
x
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
Trig functions of angles in Quadrants 2,3,4
On this slide we are concerned not with the values of trig functions, but with their
signs. Remember that r is always positive, while x and y can be positive or negative.
5 y
• cos(θ) = x/r has the same sign as x, which is
positive in Q1 and Q4, negative in Q2 and Q3.
• sin(θ) = y/r has the same sign as y, which is
positive in Q1 and Q2, negative in Q3 and Q4.
• tan(θ) = y/x is positive in Q1 and Q3,
negative in Q2 and Q4.
4
3
Q2: x Neg; y Pos
Q1: x Pos; y Pos
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
Q3: x Neg; y Neg
Q4: x Pos; y Neg
-3
-4
-5
Stanley Ocken
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
x
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
Trig functions of angles in Quadrants 2,3,4
On this slide we are concerned not with the values of trig functions, but with their
signs. Remember that r is always positive, while x and y can be positive or negative.
5 y
• cos(θ) = x/r has the same sign as x, which is
positive in Q1 and Q4, negative in Q2 and Q3.
• sin(θ) = y/r has the same sign as y, which is
positive in Q1 and Q2, negative in Q3 and Q4.
• tan(θ) = y/x is positive in Q1 and Q3,
negative in Q2 and Q4.
You may choose to derive these facts from
ASTC, but it’s better to think of the picture.
The quadrant by quadrant results are:
Quadrant 1: All: cos, sin, tan are +.
Quadrant 1
4
P (x = 4, y = 3)
3
2
1
θ
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
-4
-5
Stanley Ocken
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
x
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
Trig functions of angles in Quadrants 2,3,4
On this slide we are concerned not with the values of trig functions, but with their
signs. Remember that r is always positive, while x and y can be positive or negative.
5 y
• cos(θ) = x/r has the same sign as x, which is
positive in Q1 and Q4, negative in Q2 and Q3.
• sin(θ) = y/r has the same sign as y, which is
positive in Q1 and Q2, negative in Q3 and Q4.
• tan(θ) = y/x is positive in Q1 and Q3,
negative in Q2 and Q4.
You may choose to derive these facts from
ASTC, but it’s better to think of the picture.
The quadrant by quadrant results are:
Quadrant 1: All: cos, sin, tan are +.
Quadrant 2: Sin is +, cos and tan are −.
Quadrant 2
4
P2 (x = −4, y = 3)
3
θ2
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
-4
-5
Stanley Ocken
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
x
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
Trig functions of angles in Quadrants 2,3,4
On this slide we are concerned not with the values of trig functions, but with their
signs. Remember that r is always positive, while x and y can be positive or negative.
5 y
• cos(θ) = x/r has the same sign as x, which is
positive in Q1 and Q4, negative in Q2 and Q3.
• sin(θ) = y/r has the same sign as y, which is
positive in Q1 and Q2, negative in Q3 and Q4.
• tan(θ) = y/x is positive in Q1 and Q3,
negative in Q2 and Q4.
You may choose to derive these facts from
ASTC, but it’s better to think of the picture.
The quadrant by quadrant results are:
Quadrant 1: All: cos, sin, tan are +.
Quadrant 2: Sin is +, cos and tan are −.
Quadrant 3: Tan is +, cos and sin are −.
4
3
θ3
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
P3 (x = −4, y = −3)
-4
Quadrant 3
-5
Stanley Ocken
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
x
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
Trig functions of angles in Quadrants 2,3,4
On this slide we are concerned not with the values of trig functions, but with their
signs. Remember that r is always positive, while x and y can be positive or negative.
5 y
• cos(θ) = x/r has the same sign as x, which is
positive in Q1 and Q4, negative in Q2 and Q3.
• sin(θ) = y/r has the same sign as y, which is
positive in Q1 and Q2, negative in Q3 and Q4.
• tan(θ) = y/x is positive in Q1 and Q3,
negative in Q2 and Q4.
You may choose to derive these facts from
ASTC, but it’s better to think of the picture.
The quadrant by quadrant results are:
Quadrant 1: All: cos, sin, tan are +.
Quadrant 2: Sin is +, cos and tan are −.
Quadrant 3: Tan is +, cos and sin are −.
Quadrant 4: Cos is +, sin and tan are −.
4
3
θ4
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
P4 (x = 4, y = −3)
-4
Quadrant 4
-5
Stanley Ocken
x
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
y
5
We are now going to calculate trig functions of
closely related angles. Start with angle θ from
before. It will reappear in the next 3 pictures.
Quiz Review
4
P(4,3)
3
cos(θ) = xr = 45
sin(θ) = yr = 53
tan(θ) = xy = 43
2
1
r=
0
In this example, θ is a Quadrant 1 angle.
This means: θ is an angle whose terminal side
lies in Quadrant 1.
All trig functions of θ are positive. θ will
reappear in the next 3 slides as the reference
angle of angles in Quadrants 2,3, and 4.
-5
-4
-3
-2
-1
0
1
5
θ
2
3
4
5
x
-1
-2
-3
-4
-5
Definition
The reference angle of a given angle is the
acute ( < 90◦ ) angle between the x-axis and
the given angle’s terminal line.
Stanley Ocken
The reference angle of angle θ is colored
yellow on these slides. It is a positive angle,
between 0 and 90◦ . All of its trig functions
are positive.
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
◦
The Quadrant 2 angle θ2 shown below has reference angle θ = 180
p − θ2 The endpoint
of the terminal side of θ2 is P2 (−4, 3) Thus x = −4, y = 3, r = (−4)2 + 32 = 5.
4
5 y
cos(θ2 ) = xr = −4
5 = −5
y
3
sin(θ2 ) = r = 5
4
3
P2 (−4, 3)
tan(θ2 ) = xy = −4
= − 43
3
Since θ2 is a Quadrant 2 (Q2) angle,
cos(θ2 ) is negative since x is positive in Q2.
sin(θ2 ) is positive since y is positive in Q2.
tan(θ2 ) is negative since y/x is negative in Q2.
It’s easy to compute the trig functions of θ2 if
you already know the trig functions of its
reference angle θ. For θ2 in Quadrant 2:
cos(θ2 ) = − cos(θ).
sin(θ2 ) = + sin(θ).
tan(θ2 ) = − tan(θ).
Stanley Ocken
θ2
2
r=
θ
-5
-4
-3
-2
5
1
0
-1
0
1
2
3
4
-1
-2
-3
-4
-5
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
5
x
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
◦
The Quadrant 3 angle θ3 shown below has reference angle θ3 − 180
p . The endpoint of
the terminal side of θ3 is P3 (−4, −3) Thus x = −4, y = −3, r = (−4)2 + (−3)2 = 5.
4
cos(θ3 ) = xr = −4
5 = −5
y
3
sin(θ3 ) = r = −3
5 = −5
y
−3
3
tan(θ3 ) = x = −4 = 4
y
5
4
3
Since θ3 is a Quadrant 3 (Q3) angle,
cos(θ3 ) is negative since x is positive in Q3.
sin(θ3 ) is negative since y is negative in Q3.
tan(θ3 ) is positive since y/x is positive in Q3.
It’s easy to compute the trig functions of θ3 if
you already know the trig functions of its
reference angle θ. For θ3 in Quadrant 3:
cos(θ3 ) = − cos(θ)
sin(θ3 ) = − sin(θ);
tan(θ3 ) = + tan(θ).
Stanley Ocken
θ3
2
1
-5
-4
-3
-2
-1
0
1
2
3
4
0
θ
r=
5-1
-2
-3
P3 (−4, −3)
-4
-5
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
5
x
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
◦
The Quadrant 4 angle θ4 shown below has reference angle θ = 360
p − θ4 . The endpoint
of the terminal side of θ4 is P4 (4, −3) Thus x = 4, y = −3, r = 42 + (−3)2 = 5.
cos(θ4 ) = xr = 45
5 y
3
sin(θ4 ) = yr = −3
=
−
5
5
4
3
tan(θ4 ) = xy = −3
4 = −4
3
Since θ4 is a Quadrant 4 (Q4) angle,
cos(θ4 ) is positive since x is positive in Q4.
sin(θ4 ) is negative since y is negative in Q4.
tan(θ4 ) is negative since y/x is negative in Q4.
It’s easy to compute the trig functions of θ4 if
you already know the trig functions of its
reference angle θ. For θ4 in Quadrant 4:
cos(θ4 ) = + cos(θ)
sin(θ4 ) = − sin(θ);
tan(θ4 ) = − tan(θ).
Stanley Ocken
θ4
2
1
-5
-4
-3
-2
-1
0
1
2
3
4
0
-1
r=
θ
5
-2
-3
P4 (4, −3)
-4
-5
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
5
x
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
Finding the reference angle of a given angle.
We abbreviate ”the reference angle of θ2 ” by Ref(θ2 ). It is the angle θ shown in yellow
on the previous slides. Ref(θ2 ) is the smaller angle between the terminal line of θ2 and
the x-axis.
The reference angle is always positive: 0◦ <Ref(θ2 ) < 90◦
Similar statements hold for the angles θ3 and θ4 .
Summary: How to find the reference angle of an angle given in degrees
• Quadrant 1: If 0◦ < θ1 < 90◦ then Ref(θ1 ) = θ1 .
• Quadrant 2: If 90◦ < θ2 < 180◦ then Ref(θ2 ) = 180◦ − θ2 .
• Quadrant 3: If 180◦ < θ3 < 270◦ then Ref(θ3 ) = θ3 − 180◦ .
• Quadrant 4: If 270◦ < θ4 < 360◦ then Ref(θ4 ) = 360◦ − θ4 .
• Angles 0, ±90◦ , ±180◦ .... that are a multiple of 90◦ have undefined reference angle.
• If θ < 0 or θ > 360◦ , add or subtract a multiple of 360◦ to/from θ to get an angle θ0
with 0◦ ≤ θ0 < 360◦ . Then Ref(θ) = Ref(θ0 ), computed according to the above rules.
Stanley Ocken
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
Summary: How to find the reference angle of an angle given in radians
• Quadrant 1: If 0 < θ1 < π2 then Ref(θ1 ) = θ1 .
• Quadrant 2: If π2 < θ2 < π then Ref(θ2 ) = π − θ2 .
• Quadrant 3: If π < θ3 < 3π
2 then Ref(θ3 ) = θ3 − π.
• Quadrant 4: If 3π
<
θ
<
2π then Ref(θ4 ) = 2π − θ4 .
4
2
• Angles 0, ± π2 , ±π, ± 3π
....
that
are a multiple of π2 have undefined reference angle.
2
• If θ < 0 or θ > 2π, add or subtract a multiple of 2π to/from θ to get an angle θ0
with 0 ≤ θ0 < 2π. Then Ref(θ) = Ref(θ0 ), computed according to the above rules.
Example 1: Find the cosines of the following angles: 150◦ , 225◦ , 300◦
Solution:
• 150◦ : Terminal line in Q2: Ref(150◦ ) = 180◦ − 150◦ = 30◦ .
√
Since cosine is negative in Q2, cos(150◦ ) = − cos(30◦ ) = − 23
• 225◦ : Terminal line in Q3: Ref(225◦ ) = 225◦ − 180◦ = 45◦ .
Since cosine is negative in Q3, cos(225◦ ) = − cos(45◦ ) = − √12
• 300◦ : Terminal line in Q4: Ref(300◦ ) = 360◦ − 300◦ = 60◦ .
Since cosine is positive in Q4, cos(300◦ ) = + cos(60◦ ) =
Stanley Ocken
1
2
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
The following summary need not be memorized if you understand the previous slides.
Given any angle θ, let θ be the reference angle of θ.
Trig functions of angles in terms of the reference angle
•
•
•
•
•
•
cos(θ) = cos(θ) for angles θ ending in Q1 and Q4.
cos(θ) = − cos(θ) for angles θ ending in Q2 and Q3.
sin(θ) = sin(θ) for angles θ ending in Q1 and Q2.
sin(θ) = − sin(θ) for angles θ ending in Q3 and Q4.
tan(θ) = tan(θ) for angles θ ending in Q1 and Q3.
tan(θ) = − tan(θ) for angles θ ending in Q2 and Q4.
Example 2: Find the reference angle of 920◦ .
Solution: 920◦ − 360◦ − 360◦ = 200◦ . Since 180◦ < 200◦ < 270◦ ,
angle 200◦ is in Q3. Its reference angle is 200◦ − 180◦ = 20◦ .
Example 3: Find the reference angle of 19π
3 .
Solution: Method 1: Convert to degrees: 19( π3 ) = 19(60◦ ) = 1140◦ .
Then 1140◦ − 360◦ − 360◦ − 360◦ = 60◦ is in Q1, and so its reference angle is 60◦ .
19π
19π
18π
π
Method 2: Keep radians. 19π
3 − 2π − 2π − 2π = 3 − 6π = 3 − 3 = 3 is in Q1 and so its
reference angle is π3 .
Stanley Ocken
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
The Pythagorean trigonometric identities
If the terminal side of angle θ goes from theporigin to P (x, y)
• The distance from P to the origin is r = x2 + y 2 , and
x2 + y 2 = r2 .
• cos(θ) = xr
• sin(θ) = yr
• tan(θ) = xy
r
r
• sec(θ) = x
• csc(θ) = y
• cot(θ) = xy
In what follows, recall that powers of trig functions are
abbreviated somewhat strangely:
(cos(θ))2 is written cos2 (θ), for instance.
x2
Divide
x 2
+
r
+
y 2
r
y2
r2
r2
=
by
= 1 and so
y
5
4
P(4,3)
3
r=5
2
y=3
1
θ
0
0
1
x=4
2
3
4
to get
(cos(θ))2 + (sin(θ))2 = 1. This is usually written as
cos2 (θ) + sin2 (θ) = 1 .
Similarly, dividing x2 + y 2 = r2 by x2 gives the identity 1 + tan2 (θ) = sec2 (θ) .
Similarly, dividing x2 + y 2 = r2 by y 2 gives the identity 1 + cot2 (θ) = csc2 (θ) .
These identities should be memorized for success in this course and/or calculus.
Stanley Ocken
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
5
x
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
Each of the Pythagorean identities relates two trig functions.
You can express each one in terms of the other. For example, you could solve
cos2 (θ) + sin2 (θ) = 1 for cos(θ) as follows:
cos2 (θ) = 1 − sin2 (θ)
p
cos(θ) = ± 1 − (sin2 θ). The choice of plus or minus depends on the quadrant:
p
First recallpthat (like any other radical) 1 − (sin2 θ) is positive. Therefore
cos(θ) = 1 − sin2 (θ) in Quadrants 1 and 4, where cos(θ) is positive, while
p
cos(θ) = − 1 − sin2 (θ) in Quadrants 2 and 3, where cos(θ) is negative.
Example 4: If θ is a Quadrant 3 angle, express tan(θ)
in terms of sin(θ).
p
Solution: From above, we know that cos(θ) = − 1 − sin2 (θ) in Quadrant 3.
Therefore
tan(θ) =
sin(θ)
cos(θ)
= √sin(θ) 2
−
1−sin (θ)
and so in Q3: tan(θ) = − √ sin(θ)2
1−sin (θ)
Check your answer: In Quadrant 3, sin(θ) is negative in Q3 while tan(θ) is positive.
Therefore both sides of the boxed answer have the same sign. If they would have
different signs, you would need to go back and check for an error.
Stanley Ocken
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
Evaluating trigonometric functions
r=
Example 5: If sec(θ) = − 53 and θ is in Quadrant 2, find the other 5 trig functions of θ.
Solution: If the terminal line of θ goes from the origin to P (x, y), located distance r
from the origin, then sec(θ) = xr = − 35 . Choose any r and x that satisfy this equation,
but remember that r is positive. The easiest solution is r = 5 and x = −3. But
x2 + y 2 = r2 . Thus
(−3)2 + y 2 = 52
9 + y 2 = 25 and so
y = ±4. But (x, y) is in quadrant 2, where y is positive, and so y = 4.
5
checks with the given
Therefore x = −3, y = 4, r = 5. Note that sec(θ) = xr = −3
information. The requested five trig functions are
y
5
3
P (−3, 4)
cos(θ) = xr = −3
5 = −5
4
sin(θ) = yr = 45
3
4
tan(θ) = xy = −3
= − 34
θ
2
x
−3
3
cot(θ) = y = 4 = − 4
1
csc(θ) = yr = 54
0
x
5
-5
Stanley Ocken
-4
-3
-2
-1
0
1
2
3
4
5
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
How your calculator figures out cosines
Example 6: If θ is the radian measure of an angle, it is shown in Calculus III that if
−π/2 ≤ θ ≤ π/2, then an approximate formula for cosine is
cos(θ) ≈ 1 −
θ2 θ4 θ6
+
−
2!
4!
6!
You know that cos(π/3) = 21 .
Test the accuracy of the above formula with θ = π/3 ≈ 1.047.
Solution: Compute by hand with 3 decimal place accuracy, or (if you must) use a
calculator to check that
1.0472 1.0474 1.0476
+
−
≈ 0.500
1−
2
24
720
Stanley Ocken
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
Radian measure
Trig functions of general angles
Finding the reference angle
2
4
Basic trig identities
6
The functions y = cos(θ) and y = 1 − θ2! + θ4! − θ6!
Note that the graphs look identical for − π2 ≤ θ ≤ π2
Calculating trig functions
Quiz Review
are graphed below.
1
0
−π
− π2
0
π
π
2
y = cos(θ)
-1
y =1−
Stanley Ocken
2
θ
2!
+
4
θ
4!
−
6
θ
6!
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
How close to identical, really? The difference of the functions y = cos(θ) and
2
4
6
y = 1 − θ2! + θ4! − θ6!
is graphed below. Look at the scale to the left of the graph!
When θ = π/3, the difference equals ≈ 0.00003543467. In other words the formula for
cos(π/3) = 0.5 gives the correct answer with better than 99.99 per cent accuracy!
0.001
y =1−
−π
0
θ2
2!
+
− π2
θ4
4!
−
0
θ6
6!
− cos(θ)
π
2
π
-0.001
Stanley Ocken
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles
Radian measure
Trig functions of general angles
Finding the reference angle
Basic trig identities
Calculating trig functions
Quiz Review
Quiz Review
Example 1: Find the cosines of the following angles: 150◦ , 225◦ , 300◦
Example 2: Find the reference angle of 920◦ .
Example 3: Find the reference angle of
19π
3 .
Example 4: If θ is a Quadrant 3 angle, express tan(θ) in terms of sin(θ).
Example 5: If sec(θ) = − 35 and θ is in Quadrant 2, find the other 5 trig functions of θ.
Stanley Ocken
M19500 Precalculus Chapter 6.3 Trigonometric functions of angles