Download NOTE - UGA Math Department

NOTE: In Unit 4 of this class, you may not use a calculator on the test!
Practice avoiding the use of one in your work.
A few HW problems do need a calculator, but those problems would not be on the test. See the instructions
at the top of HW assignments to see which problems can use a calculator.
Section 5.1: Angles and Circular Sectors
An angle is a rotation of one ray, called the initial
side, onto another ray, called the terminal side. We
usually draw angles in standard position, meaning
the initial side is the positive x-axis.
This figure shows three angles α, β, and γ in standard position (Greek letters are common for angles.)
They have the same terminal side but differ in how
many “spins” they make and in direction. Angles with
the same two sides are coterminal.
Measuring Angles
Angles can be measured in degrees or in radians. A full circle measures 360 degrees (360◦ ) or 2π radians.
Units: Degrees use the ◦ superscript, but radians are unitless.
Sample: 1◦ means 1 degree, but 1 means 1 radian. (Radians do not have to use π!)
Signs: Positive angles rotate counter clockwise. Negative angles rotate clockwise (i.e. the opposite way)
Samples:
• 90◦ = a quarter-turn counterclockwise (90 is 1/4 of 360), i.e. one quadrant.
• −π/4 = an eighth-turn clockwise (π/2 is 1/8 of 2π), i.e. half a quadrant.
Coterminal angles: Coterminal values differ by multiples of a full circle (i.e. 360◦ or 2π). Thus, you can add
or subtract 360◦ or 2π to cause no change in position.
Ex 1: For each angle, find one positive and one negative angle coterminal with the given angle.
(a) 145◦
(b) −270◦
(c) π/6
Converting: One circle = 2π radians = 360 degrees, so divide by 2: π radians = 180◦ .
Deg to Rad: Multiply
π
180◦
Rad to Deg: Multiply
180◦
π
Ex 2: Convert the degree measurements into radians and vice versa.
(a) 60◦
(b) 60
(c) −225◦
(d) 2π/3
(e) −5π
Miscellaneous Definitions
Suppose α is an angle. There are many words we can use to describe it.
α is quadrantal if it’s a multiple of 90◦ (or π/2), meaning its terminal side lie on the axes between quadrants.
Note that this means quadrantal angles DON’T have a quadrant!
For angles from 0◦ to 180◦ , there are some special names:
•
•
•
•
α is acute when 0◦ < α < 90◦ (i.e. 0 < α < π/2) in Quadrant I.
A right angle is 90◦ = π/2.
α is obtuse when 90◦ < α < 180◦ (i.e. π/2 < α < π) in Quadrant II.
A straight angle is 180◦ = π.
α’s complement is 90◦ − α or π/2 − α. This is useful because the two acute angles in a right triangle are
complements. In fact, we’re going to see that the “co” in several trig functions really stands for “complement”!
α’s supplement is 180◦ − α = π − α. This is useful because two angles which make up a straight line are
supplements.
Sectors of Circles
A sector is a section of a circle
Assume 0 ≤ θ ≤ 2π. To get arc
swept out by an angle, i.e. a “pizza length and sector area, we look at
slice”. The curved outer side is an the sector as taking up the fraction
arc s which subtends the central θ/(2π) of the circle.
angle θ in this circle of radius r.
Arc length: Multiply circumference 2π by the fraction θ/(2π) and get
s = rθ when θ is in RADIANS
Sector area: Multiply circle area πr2 by θ/(2π) and get
A = θ2 r2 when θ is in RADIANS
These formulas require radians; convert degree values!
Ex 3:
(a) If a sector has central angle 120◦ and radius 5,
find its arc length and area.
(b) If a sector has radius 6 cm and arc length 10 cm,
find its central angle in radians and degrees.
Ex 4: Find the length of the arc s shown in the
figure on the right.
NOTE: The key here is that the angle of the sector is NOT 105◦ ; it’s everything EXCEPT 105◦ . Thus,
θ = 360◦ − 105◦ = 255◦ .
Ex 5: A sector has central angle T (in radians). If the perimeter of the sector is 12, find its radius and its
area as functions of T .
GETTING STARTED: Say the radius is r. A sector has three “sides”: two radial sides of length r, and one
“curved side” whose arc length is s = rT . This means the perimeter is 2r + rT .
Globes
It’s worth noting that globes provide another use for circular sectors. The distance between two points on
a globe is NOT a straight line; it’s an arc! Thus, the s = rθ formula is useful.
Ex 6: Suppose the Earth’s radius is 4000 miles, and Athens, GA has latitude 34.0◦ N. How far north of the
equator is Athens?
(Here, r = 4000, and θ = 34.0◦ , which must be turned into radians.)
See the next page for some coverage of Section 5.2A.
Section 5.2A (part 1): Trigonometry of Right Triangles
Idea: All right triangles with the same acute angle θ have the same side ratios are known. (Any two such
triangles are similar.)
Let’s say the three sides are: the leg opposite θ (“opp”), the leg adjacent to θ (“adj”), and the hypotenuse
(“hyp”). We use these to make six ratios called the trigonometric functions.
The Trig Functions in a Right Triangle:
Name
Definition
opp
Sine
sin(θ) = hyp
adj
Cosine cos(θ) = hyp
Tangent tan(θ) = opp
adj
Name
Cosecant
Secant
Cotangent
Definition
csc(θ) = hyp
opp
sec(θ) = hyp
adj
adj
cot(θ) = opp
The three trigs on the left are the most common ones (also found on most calculators). To remember them,
we have the following famous device:
SOH CAH TOA stands for “Sin is Opp over Hyp, Cos is Adj over Hyp, Tan is Opp over Adj”.
Reciprocal Identities: To get the other three trigs, we often use reciprocals!
csc(θ) =
1
sin(θ)
sec(θ) =
1
cos(θ)
cot(θ) =
1
(unless tan θ is undefined)
tan(θ)
Notation notes:
• Just like with logs, we sometimes omit parentheses around the input if the input is simple.
• sec is the reciprocal of cos, not of sin. Each reciprocal of a “non-co” function is a “co” function and vice
versa!
• Frequently, powers are written on top of the trig function! For instance, sin2 (x) means the same as (sin x)2 .
Ex 7: In the triangle on the right, find sin(A),
csc(A), cos(B), and sec(B).
NOTE: To calculate these, you need the hypotenuse, so find its value first.
Ex 8: If θ is an acute angle in a right triangle with opposite side a and hypotenuse c, find tan(θ) and cot(θ)
in terms of a and c.
Reference Triangles for Converting Trigs to Other Trigs
If you’re given a trig value, and you want to find others, what do you do? Make your OWN triangle using
the ratio of two sides that you’re given! We call that triangle a reference triangle for the angle.
Sample: If you’re told sin(θ) = 3/5, this means√opp/hyp = 3/5, so draw a right triangle with θ in it where
opp is 3 and hyp is 5. Thus, by Pythagoras, adj = 52 − 32 = 4.
Ex 9: If cos(θ) = 15/17 and θ is acute, find the values of sin(θ), tan(θ), and sec(θ).
Getting Everything in Sines and Cosines
We will see, as this unit proceeds, that sin and cos are the easiest trig functions to work with. As a result,
we will need sometimes to write other trig functions in terms of sines and cosines. Here are the main tools we
use for this:
1. sec and csc are written as reciprocals: sec(θ) = 1/ cos(θ) and csc(θ) = 1/ sin(θ).
2. tan and cot can be written as fractions of sine and cosine:
tan(θ) =
Ex 10: Write sec2 (x) −
cos2 (x)
tan2 (x)
sin(θ)
cos(θ)
cot(θ) =
cos(θ)
(i.e. it’s tan’s reciprocal)
sin(θ)
in terms of sines and cosines.
Pythagorean Identities
Now, what if you ONLY want sines or only cosines? We need a new identity. We’ll obtain three new identities
called the Pythagorean Identities which are very popular for manipulating squares of trigs.
To see where these come from, let’s say adj = a, opp = b, and hyp = c. The famous Pythagorean Theorem
says a2 + b2 = c2 . Divide through by c2 to get (a/c)2 + (b/c)2 = 1... this is (adj/hyp)2 + (opp/hyp)2 = 1. You
get...
Pythagorean Identity for sin and cos:
sin2 (θ) + cos2 (θ) = 1
This is the most famous Pythagorean identity, and it is used to write sin in terms of cos and vice versa. For
instance, by subtracting cos2 (θ) to both sides, we get
p
(sin θ)2 = 1 − (cos θ)2 so that sin θ = ± 1 − (cos θ)2
(For Quadrant I, we’ll use the +, as we’ll see more later.) A similar formula writes cos in terms of sin.
Ex 11: Rewrite f (x) = 3 csc2 (x) + 11 tan2 (x) in terms of cos(x).
APPROACH: Like in Ex 11, we write csc2 as 1/ sin2 , and we write tan2 as sin2 / cos2 . That puts everything
into sines and cosines. Next, to get rid of the sines, we use (sin θ)2 = 1 − (cos θ)2 as discussed above.
Ex 12: Assume 6 sin2 (x) − 4 cos2 (x) = 1. Find the numerical value of csc2 (x).
APPROACH: Recall that csc is the reciprocal of sin, so we really need to find sin2 first! To do this, let’s use
our Pythagorean identity to write cos2 in terms of sin2 . Once you have sin2 (x), we know csc2 (x) = 1/ sin2 (x).
There are two more Pythagorean identities. If you take a2 + b2 = c2 and divide a2 instead of c2 , you get
1 + (b/a)2 = (c/a)2 , which becomes
Pythagorean Identity for tan and sec:
1 + tan2 (θ) = sec2 (θ) i.e. tan2 (θ) = sec2 (θ) − 1
(Another way to get this is to take sin2 (θ) + cos2 (θ) = 1 and divide through by cos2 (θ).) Similarly, if you take
a2 + b2 = c2 and divide b2 through, you get 1 + (a/b)2 = (c/b)2 , which becomes
Pythagorean Identity for cot and csc:
1 + cot2 (x) = csc2 (x) i.e. cot2 (θ) = csc2 (θ) − 1
(You can also get this by taking sin2 (θ) + cos2 (θ) = 1 and dividing sin2 (θ).)
NOTE: These identities look almost the same, except the second uses “co” functions in both places. Also,
in contrast to the identity with sin and cos, the 1 does not occur alone! sec and csc are definitely bigger (in
magnitude) than tan and cot.
Sample: Simplify the following formulas.
(a) 2 sec2 (a) − 2 tan2 (a)
(b) sec2 (2a) − tan2 (2a)
With (a), we factor out 2 to get 2(sec2 (a) − tan2 (a)). On the inside, since sec2 (a) = 1 + tan2 (a), we
algebraically rearrange to get sec2 (a) − tan2 (a) = 1. Therefore, part (a) becomes 2(1) = 2.
On the other hand, (b) simplifies sec2 (2a) − tan2 (2a), which equals 1! In this case, the angle is θ = 2a in
both trigs.
MORAL: The trig identity works with any angle, and you cannot factor out a constant like 2 FROM INSIDE
the trig!
NOTE: These techniques work great for sines and cosines, but if you want to write everything in terms of
a different trig (like tangents), this is not what you want. There will be an extra handout online to address a
way to do that.
We continue with Section 5.2 next class.