Trigonometric Ratios and Functions 2.pptx

Trigonometric Ratios and
Functions
1

TRIGONOMETRY
• primarily a branch of mathematics that deals with triangles, mostly right
triangles. In particular the ratios and relationships between the triangle's sides
and angles. It has two main ways of being used:
• In Geometry: In its geometry application, it is mainlyused to solve triangles,
usually right triangles. That is, given some angles and side lengths, we can find
some or all the others.
• Analytically: In a more advanced use, the trigonometric ratios such as Sine and
Tangent, are used as functions in equations and are manipulatedusing algebra.

TRIGONOMETRY
Trigonometry is over three thousand years old. The ancient Egyptians,
Babylonians, and Greeks developed trigonometry to find lengths of the
sides of triangles and the measures of their angles. In Egypt trigonometry
was used to establish land boundaries after the annual flood of the Nile
River. In Babylonia, it was used in astronomy.

ANGLES AND THEIR MEASURE
• A ray is part of the line that is
described as a ‘half- line’ and that
in which one of the one has an end
point on the other side and is goes
on infinitely. The point from which
the ray originates is called the
initial point of the ray.
When two rays share a common
initial point, they form an angle and
the common initial point is calledthe
vertex of the angle. Two examples of
what are commonly thought of as
angles are:

MEASURE OF AN ANGLE
The measure of an angle is a number
which indicates the amount of
rotation that separates the rays of
the angle
Greek Letters are being used to
measure theangles. Some
examples are below:
𝑎 𝖰 𝜽
alpha beta theta

DEGREE MEASURE
Degree Measure – commonly used system to measure
angles. Quantities measured in degrees are denoted by the
symbol “°”. One completerevolution as shown below is
360°. Thus, half of a revolution (a straight angle) measures
180°. A quarter of a revolution (a right angle) measures
90°. “90°”
“360°” “180°”

CONVERTING
DEGREES TO DMS
• In this system, one degree is divided
equally into sixty minutes, and in turn,
each minute is divided equally into sixty
seconds.
• In Illustration:
• 1° = 60’ (read as 60 minutes)
• 1′ = 60” (read as 60 seconds)
• 1° = 3600”
“90°”
“360°” “180°”

CONVERTING DEGREES
TO DMS
1. Convert a measure of 42.125 to DMS System
42.125° = 42° + 0.125°
42.125° = 42° + 0.125°
60°
1′
42.125° = 42° + 7.5′
42.125° = 42° + 7′
+ 0.5°
60°
1′
42.125° = 42° + 7′
+ 30′′
42.125° = 42°7′
30′′

CONVERTING DEGREES
TO DMS
1. Convert a 117°15′
45′′to degree measure
117°15′45′′ = 117° + 15′ + 45′′
117°15′45′′ = 117° + 15
′
1°
60′
+ 45′′
1°
3600°
117°15′45" = 117° + 0.25° + 0.0125
117°15′45" = 117.2626°

CONVERTING DEGREES
TO DMS
3. Convert a 78°8′
14′′to degree measure
78°8′14′′ = 117° + 15′ + 45′′
78°8′14′′= 78° + 8
′
1°
60′
+ 14′′
1°
3600°
78°8′14′′= 78° + 0.13333° + 0.0039°
78°8′
14′′ ≈ 74.137°

CONVERTING DEGREES
TO DMS
4. Convert a 34°. 817 DMS
34.817° = 34° + .817°
34.817° = 34°+.817(60′)
34.817° = 34° + 49.02
34.817° = 34° + 49′ + .02 60′′
34.817° = 34°49′
1.2′′

10.2 Angles and Radian
Measure
After this lesson…
• I can draw angles in standard position.
• I can explain the meaning of radian measure.
• I can convert between degrees and radians.
• I can use radian measure to find arc lengths and the area of a sector.
12

10.2 Angles and Radian Measure
• Angles in Standard Position
• Vertex on origin
• Initial Side on positive x-axis
• Measured counterclockwise

• Coterminal Angles
• Different angles (measures) that
have the same terminal side
• Found by adding or subtracting
multiples of 360°

15
Radian and Degree Measure
Positive and negative angles
Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–4
Section 4.1, Figure 4.3, Positive and
Negative Angles, pg. 248
When sketching angles,
always use an arrow to
show direction.

Find one positive angle and one negative angle that are coterminal with (a)
−45° and (b) 395°.
There are many such angles, depending on what multiple of 360° is added or subtracted.
17

• Draw an angle with the given
measure in standard position. Then
find one positive coterminal angle
and one negative coterminal angle.
• 65°
• 300°
• −900°
−125°

• Radian measure
• Another unit to measure angles
• 1 radian is the angle when the arc
length = the radius
• There are 2π radians in a circle

• To convert between degrees and
radians use fact that
• 180° = π
• Special angles

• Convert the degree measure to radians, or the radian measure to degrees.
• 135°
• -50°
•
5𝜋
4
•
40°

• Sector
• Slice of a circle
• Arc Length
• 𝑠 = 𝑟θ
• 𝜃 must be in radians!
• Area of Sector
• 𝐴 =
1
2
𝑟2𝜃
• 𝜃 must be in radians!

• Find the length of the outfield fence
if it is 220 ft from home plate.
• Find the area of the baseball field.
• Try 535#29 Find the Drone Search
Area

10.1 Right
Triangle
Trigonometry
• I can define the six
trigonometric
functions.
• I can evaluate
trigonometric
functions.
• I can use
trigonometric
functions to find
side lengths of right
triangles.
24

10.1 Right Triangle Trigonometry
• If you have a right triangle, there are
six ratios of sides that are always
constant
• sin 𝜃 =
opposite
hypotenuse
• cos 𝜃 =
adjacent
hypotenuse
• tan 𝜃 =
opposite
adjacent
• csc 𝜃 =
hypotenuse
opposite
• sec 𝜃 =
hypotenuse
adjacent
• cot 𝜃 =
adjacent
opposite
SOH
CAH
TOA

• Evaluate the six trigonometric functions of the angle 𝜃.

• In a right triangle, 𝜃 is an acute angle
and cos 𝜃 =
7
10
. What is sin 𝜃?
• Try 526#9
If sin 𝜃 =
7
11
, what are other trig
functions?

• Special Right Triangles
• 30° - 60° - 90°
• 45° - 45° - 90°

• Use the diagram to solve the right triangle if…
• B = 60°, a = 7
• A = 32°, b = 10
• Try 526#33
A = 43°, b = 31

• Find the distance between Powell
Point and Widforss Point.
• Try 527#37 Find the width of the
river.

10.3 Trigonometric Functions
of Any Angle
• I can evaluate trigonometric functions given a point on an angle.
• I can evaluate trigonometric functions using the unit circle.
• I can find and use reference angles to evaluate trigonometric functions.
• I can solve real-life problems involving projectiles.
31

of Any Angle
• Think of a point on the terminal side
of an angle
• You can draw a right triangle with
the x-axis
• sin 𝜃 =
𝑦
𝑟
csc 𝜃 =
𝑟
𝑦
• cos 𝜃 =
𝑥
𝑟
sec 𝜃 =
𝑟
𝑥
• tan 𝜃 =
𝑦
𝑥
cot 𝜃 =
𝑥
𝑦
• Unit Circle
• r = 1

of Any Angle
• Evaluate the six trigonometric functions of θ.
Try 542#3

of Any Angle
• Quadrantal Angles
• Evaluate the six trigonometric functions of θ.
• θ = 180°

of Any Angle
• Reference Angle
• Angle between terminal side and
x-axis
• Has the same values for trig
functions as 1st quadrant angles
• You just have to add the negative
signs
All
Sin
Cos
Tan

of Any Angle
• Sketch the angle. Then find its
reference angle.
• 150°
• Try 542#15
23𝜋
4

of Any Angle
• Evaluate cos(-60°) without a
calculator
• Try 542#25
sin(−150°)

of Any Angle
• Estimate the horizontal distance traveled by a Red Kangaroo who jumps at an
angle of 8° and with an initial speed of 53 feet per second (35 mph).

10.4 Graphing Sine and
Cosine Functions
• I can identify characteristics of sine and cosine functions.
• I can graph transformations of sine and cosine functions.
39

10.4 Graphing Sine and Cosine Functions
• Work with a partner.
• a. Complete the table for y = sin x, where x is an angle measure in radians.
• b. Plot the points (x, y) from part (a). Draw a smooth curve through the points
to sketch the graph of y = sin x. Make several observations about the graph.
40

• 𝑦 = sin 𝑥
• Starts at 0
• Amplitude = 1
• Period = 2π
• 𝑦 = cos 𝑥
• Starts at 1
• Amplitude = 1
• Period = 2π

• Transformations
• 𝑦 = 𝑎 sin 𝑏 𝑥 − ℎ + 𝑘
• |a| = amplitude = vertical stretch
• If a < 0, then reflect over x-axis
• b = horizontal shrink
• Period 𝑇 =
2𝜋
𝑏
• h = horizontal shift
• k = vertical shift
• Midline 𝑦 = 𝑘

• Graphing sine and cosine
1. Identify the amplitude, period, horizontal shift, and vertical shift
2. Draw the midline, y = k
3. Find the 5 key points (3 zeros, 1 max, 1 min)
4. Draw the graph
43

• Identify the amplitude and period. • Try 551#7
44

• Graph 𝑓 𝑥 = 2 sin 𝑥

• Try 551#13
Graph 𝑦 = sin 2𝜋𝑥

• Graph 𝑦 = 2 sin 𝑥 −
𝜋
2

• Try 552#39
Graph
𝑔 𝑥 = −4 cos 𝑥 +
𝜋
4
− 1

10.5 Graphing Other
Trigonometric Functions
• I can identify characteristics of tangent, cotangent, secant, and cosecant
functions.
• I can graph tangent and cotangent functions.
• I can graph secant and cosecant functions.
49

10.5 Graphing Other Trigonometric Functions
• Work with a partner.
• a. Complete the table for y = tan x, where x is an angle measure in radians.
• b. Plot the points (x, y) from part (a). Then sketch
the graph of y = tan x. Make several observations
about the graph.
50

• 𝑦 = tan 𝑥
• Period = π
• 𝑇 =
𝜋
𝑏
• Asymptotes where tangent
undefined,
𝜋
2
,
3𝜋
2

• 𝑦 = cot 𝑥
• Period = π
• 𝑇 =
𝜋
𝑏
• Asymptotes at 0, π, 2π

• Graph 𝑦 = tan
𝑥
4

• Try 560#7
Graph 𝑔 𝑥 =
1
2
tan 𝜋𝑥

• 𝑦 = csc 𝑥
• Period = 2π
• 𝑇 =
2𝜋
𝑏
• Asymptotes where sine = 0
• 0, 𝜋, 2𝜋

• 𝑦 = sec 𝑥
• Period = 2π
• 𝑇 =
2𝜋
𝑏
• Asymptotes where cosine = 0
•
𝜋
2
,
3𝜋
2

• Graph 𝑦 = 2 csc 𝑥 +
𝜋
2

• Try 560#15
Graph 𝑔 𝑥 = csc 4𝑥

10.6 Modeling with
Trigonometric Functions
• I can write and graph trigonometric functions using frequency.
• I can write trigonometric functions for a given graph.
• I can find a trigonometric model for a set of data using technology.
59

10.6 Modeling with Trigonometric Functions
• Trigonometric functions are periodic
• Useful for modeling oscillating motions or repeated patterns
• Period (T)
• Time of 1 cycles
• Unit: s/cycle = s
• Frequency (f)
• Cycles per 1 second
• Unit: cycles/s = Hz
60

• Find the frequency
𝑦 = 2 cos 3𝑥
• Try 568#3
𝑦 = sin 3𝜋𝑥
61

• Write Trigonometric Models
• Find the amplitude
• Find the period
• If the situation starts at zero, use sine
• If starts increasing +
• If starts decreasing −
• If the situation starts at a maximum or minimum use cosine
• If starts at max +
• If starts at min −
62

• An audiometer produces a pure tone with a frequency f of 1000 hertz (cycles
per second). The maximum pressure P produced by the tone is 20
millipascals. Write a sine model that gives the pressure P as a function of the
time t (in seconds).
63

• Try 568#9
Frequency is measured in hertz, or cycles per second. The lowest frequency of
sounds that can be heard by humans is 20 hertz. The maximum pressure P
produced from a sound with a frequency of 20 hertz is 0.02 millipascal. Write
a sine model that gives the pressure P as a function of the time t (in seconds).
64

• Write a function for the sinusoid
shown.
65

• Try 569#11
Write a function for the sinusoid
shown.
66

• Two people swing jump ropes. The highest point of the middle of each rope is
80 inches above the ground and the lowest point is 2 inches above the
ground. Each rope makes 2 revolutions per second. Write a model for the
height h (in inches) of one of the ropes as a function of the time t (in seconds)
given that the rope is at its lowest point when t = 0.
67

• The tables show the average
monthly low temperatures D (in
degrees Fahrenheit) in Erie,
Pennsylvania, where t = 1 represents
January. Write a model that gives D
as a function of t and interpret the
period of its graph. Use technology.
• Press STAT, Edit… enter points
• Press STAT  CALC, SinReg
68

• Try 569#19
The tables show the average
monthly low temperatures D (in
degrees Fahrenheit) in Las Vegas,
Nevada, where t = 1 represents
January. Write a model that gives D
as a function of t and interpret the
period of its graph. Use technology.
69

10.7 Using Trigonometric
Identities
• I can evaluate trigonometric functions using trigonometric identities.
• I can simplify trigonometric expressions using trigonometric identities.
• I can verify trigonometric identities.
70

10.7 Using Trigonometric Identities
• On unit circle
• 𝑥 = cos 𝜃
• 𝑦 = sin 𝜃
• 𝑥2 + 𝑦2 = 1
• cos2 𝜃 + sin2 𝜃 = 1
• Trigonometric Identity
• Statement showing relationship
between two quantities that are
always =
71

• Reciprocal Identities
• sin 𝑢 =
1
csc 𝑢
csc 𝑢 =
1
sin 𝑢
• cos 𝑢 =
1
sec 𝑢
sec 𝑢 =
1
cos 𝑢
• tan 𝑢 =
1
cot 𝑢
cot 𝑢 =
1
tan 𝑢
• Quotient Identities
• tan 𝑢 =
sin 𝑢
cos 𝑢
cot 𝑢 =
cos 𝑢
sin 𝑢
• Pythagorean Identities
• sin2
𝑢 + cos2
𝑢 = 1
• tan2 𝑢 + 1 = sec2 𝑢
• 1 + cot2 𝑢 = csc2 𝑢

• Even/Odd Identities
• cos −𝑢 = cos 𝑢
• sec −𝑢 = sec 𝑢
• sin −𝑢 = − sin 𝑢
• tan −𝑢 = − tan 𝑢
• csc −𝑢 = − csc 𝑢
• cot −𝑢 = − cot 𝑢
• Cofunction Identities
• sin
𝜋
2
− 𝑢 = cos 𝑢
• cos
𝜋
2
− 𝑢 = sin 𝑢
• tan
𝜋
2
− 𝑢 = cot 𝑢
• cot
𝜋
2
− 𝑢 = tan 𝑢
• sec
𝜋
2
− 𝑢 = csc 𝑢
• csc
𝜋
2
− 𝑢 = sec 𝑢

• Given that sin 𝜃 = −
5
13
and 𝜋 < 𝜃 <
3𝜋
2
, find the values of the other five
trigonometric functions of θ.
• Try 575#3
Given that tan 𝜃 = −
3
7
and
𝜋
2
< 𝜃 <
𝜋, find the values of the other five
trigonometric functions of θ.
74

• Simplify (1 + cos θ )(1 − cos θ ) • Try 575#9
sin 𝑥 cot 𝑥
75

• Verify Trigonometric Identities
• Show that trig identities are true by turning one side into the other side
• Guidelines
1. Work with 1 side at a time. Start with the more complicated side.
2. Try factor, add fractions, square a binomial, etc.
3. Use fundamental identities
4. If the above doesn’t work, try rewriting in sines and cosines
5. Try something!

• Verify
1
sec 𝜃−tan 𝜃
= sec 𝜃 + tan 𝜃 • Try 575#21
cos
𝜋
2
− 𝑥 cot 𝑥 = cos 𝑥
77

10.8 Using Sum and
Difference Formulas
• I can evaluate trigonometric expressions using sum and difference formulas.
• I can simplify trigonometric expressions using sum and difference formulas.
• I can solve trigonometric equations using sum and difference formulas.
78

10.8 Using Sum and Difference Formulas
• Sum and Difference Formulas
• sin 𝑢 + 𝑣 = sin 𝑢 cos 𝑣 + cos 𝑢 sin 𝑣
• sin 𝑢 − 𝑣 = sin 𝑢 cos 𝑣 − cos 𝑢 sin 𝑣
• cos 𝑢 + 𝑣 = cos 𝑢 cos 𝑣 − sin 𝑢 sin 𝑣
• cos 𝑢 − 𝑣 = cos 𝑢 cos 𝑣 + sin 𝑢 sin 𝑣
• tan 𝑢 + 𝑣 =
tan 𝑢+tan 𝑣
1−tan 𝑢 tan 𝑣
• tan 𝑢 − 𝑣 =
tan 𝑢−tan 𝑣
1+tan 𝑢 tan 𝑣

• Find the exact value of sin 75°. • Try 581#1
tan −15°
80

• Find cos(a − b) given that
cos 𝑎 = −
12
13
with
𝜋
2
< 𝑎 < 𝜋 and
sin 𝑏 =
3
5
with 0 < 𝑏 <
𝜋
2
.
• Try 581#9
Find sin 𝑎 + 𝑏 given that cos 𝑎 =
4
5
with 0 < 𝑎 <
𝜋
2
and sin 𝑏 = −
15
17
with
3𝜋
2
< 𝑏 < 2𝜋.
81

• Simplify the expression cos(x − π). • Try 581#17
cos 𝑥 + 2𝜋
82

• Solve cos 𝑥 +
𝜋
4
+ cos 𝑥 −
𝜋
4
= 1
for 0 ≤ 𝑥 < 𝜋.
• Try 581#23 Solve on 0 ≤ 𝑥 < 2𝜋
2 sin 𝑥 − 1 = 0
83

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