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Chapter 12: Trigonometric Functions
12.1: Trigonometric Functions in Right Triangles
trigonometry: the study of relationships among the angles and sides of a right triangle
trigonometric ratio: compares the side lengths of a right triangle
trigonometric function: has a rule given by a trigonometric ratio
𝑠𝑖𝑛 𝜃 =
𝑐𝑜𝑠 𝜃 =
𝑡𝑎𝑛 𝜃 =
𝑜𝑝𝑝
𝑐𝑠𝑐 𝜃 =
ℎ𝑦𝑝
𝑎𝑑𝑗
𝑠𝑒𝑐 𝜃 =
ℎ𝑦𝑝
𝑜𝑝𝑝
𝑐𝑜𝑡 𝜃 =
𝑎𝑑𝑗
30o-60o-90o
ℎ𝑦𝑝
𝑜𝑝𝑝
ℎ𝑦𝑝
𝑎𝑑𝑗
𝑎𝑑𝑗
𝑜𝑝𝑝
=
1
sin 𝜃
1
=
=
cos 𝜃
1
tan 𝜃
45o-45o-90o
𝑠𝑖𝑛 30° =
1
𝑐𝑜𝑠 30° =
√3
2
2
𝑡𝑎𝑛 30° = √3
𝑠𝑖𝑛 60° =
𝑐𝑜𝑠 60° =
√3
2
𝑠𝑖𝑛 45° =
√2
2
1
𝑐𝑜𝑠 45° =
√2
2
2
𝑡𝑎𝑛 60° = √3
𝑡𝑎𝑛 45° = 1
Inverse Trigonometric Ratios: used to find the angle of a right triangle (sin−1 𝜃 , cos −1 𝜃 , tan−1 𝜃)
angle of elevation: angle formed by the line of sight when looking upward, to a line parallel to the horizon
angle of depression: angle formed by the line of sight when looking downward, to a line parallel to the horizon
angle of elevation = angle of depression
****Note To Teacher: Day I (#1a, 1b, 3a.3b, 4a, 4b, 5)
Day II (# 2a, 2b, 3c, 4c, 6)
Examples:
1. Find the values of the six trig functions for angle 𝜃.
a.
𝜃
8
6
b.
16
𝜃
12
2. In a right triangle, ∠A is acute. Find the values of the five remaining trig functions.
a. 𝑐𝑜𝑠 𝐴 =
4
b. 𝑡𝑎𝑛 𝐴 =
7
20
21
3. Use a trig function to find the value of x. Round to the nearest tenth.
a.
b.
c.
60o
52o
x
6
7
35o
x
x
22
4. Find the value of x. Round to the nearest tenth.
a.
b.
c.
6
o
x
14
6
xo
8
xo
15
16
5. Christian found two trees directly across from each other in a canyon. When he moved 100 feet from the tree on his
side (parallel to the edge of the canyon), the angle formed by the tree on his side and the tree on the other side was
70o. Find the distance across the canyon.
6. The recommended angle of elevation for a ladder used in fire fighting is 75o. At what height on a building does a 21foot ladder reach if the recommended angle of elevation is used? Round to the nearest tenth.
The unit circle:
( x , y)
(cos, sin)
Tan=
𝑠𝑖𝑛
𝑐𝑜𝑠
12.2 Day 1
30
Sin
Cos
Tan
45
60
12.2: Angles and Angle Measure
y
standard position: the vertex is at the origin and one ray is on the positive x-axis
initial side: the ray on the x-axis
x
terminal side: the ray that rotates about the center
coterminal angles: two or more angles in standard position with the same terminal side
radian: 𝜋 radians = 180o
convert degrees to radians: multiply the number of degrees by 𝜋 radians
degrees
convert radians to degrees: multiply the number of radians by degrees
𝜋 radians
central angle: an angle with a vertex at the center of the circle
arc length: s, equals the product of r and 𝜃
NOTE:𝜽 Must be in radians
Examples:
1. Draw an angle with the given measure in standard position:
a. 140o
b. – 60o
c. 390o
2. Find an angle with a positive measure and an angle with a negative measure that are coterminal with each
angle.
a. 25o
b. 175o
c. – 100o
3. Rewrite each degree measure in radians and each radian measure in degrees.
𝜋
a. 4
b. 225o
c. – 45o
4. A tennis player’s swing moves along the path of an arc. If the radius of the arc’s circle is 4 feet and the angle
of rotation is 100o, what is the length of the arch?
12.3: Trigonometric Functions of General Angles
Trig Functions of General Angles:
(recommended: draw
a picture!!)
𝑠𝑖𝑛 𝜃 =
𝑐𝑠𝑐 𝜃 =
𝑂
𝑐𝑜𝑠 𝜃 =
𝐻
𝐻
𝑂
SOHCAHTOA
,𝑂 ≠ 0
𝑠𝑒𝑐 𝜃 =
𝐻
𝐴
𝐴
𝑡𝑎𝑛 𝜃 =
𝐻
,𝐴 ≠ 0
𝑐𝑜𝑡 𝜃 =
𝐴
𝑂
𝑂
𝐴
,𝑂 ≠ 0
quadrantal angle: if the terminal side 𝜃 in standard position lies on the x- or y-axis
reference angle: the acute angle formed by the terminal side and the x-axis
Examples: DAY I
1. Sketch each angle, then find its reference angle.
a. 300o
b. 115o
c. −
3𝜋
4
2. The terminal side 𝜃 in standard position contains each point. Find the exact values of the six trig functions
of 𝜃.
a. (1, 2)
b. (–8, –15)
c. (0, –4)
d. (5, 0)
e. (2, -5)
f. (-2,6)
12.3 Day II Use Unit Circle and reference angles from chart!
30
45
60
Sin
Cos
Tan
3. Find the exact value of each trig function.
a. 𝒔𝒊𝒏
e. tan
i. tan
𝝅
𝟒
−𝝅
𝟒
𝟑𝝅
𝟒
b. 𝒕𝒂𝒏
f. sin
𝟓𝝅
𝟑
𝟓𝝅
𝟑
k. cos
−𝟕𝝅
𝟐
c. sec 120o
d. sin 300o
g. cos π
h. Cos
j. cos 240ᵒ
k. sin
−𝝅
𝟑
−𝟏𝟏𝝅
𝟔
l. csc
𝝅
𝟐
m. cot π
n. sec
−𝟓𝝅
o. cot 2π
𝟒
1
4. Alexandria opens her portable DVD player so that it forms a 125 o angle. The screen is 5 2 inches long.
a. Find the reference angle. Then write and solve a trig function that can be used to find the distance to
the wall d that she can place the DVD player.
12.4: Law of Sines
law of sines: in ∆ABC, if sides with lengths a, b, and c are opposite angles with measures A, B, and C, then the
sin 𝐴
sin 𝐵
sin 𝐶
following is true: 𝑎 = 𝑏 = 𝑐
Area of a Triangle: the area of a triangle is one half the product of the lengths of two sides, and the sine of
1
1
1
their included angle; Area = 2 𝑏𝑐 ∙ sin 𝐴 = 2 𝑎𝑐 ∙ sin 𝐵 = 2 𝑎𝑏 ∙ sin 𝐶
solving a triangle: using given measures to find all unknown side lengths and angle measures of a triangle
Possible Number of Solutions of a Triangle: none, one, or two
Examples:
1. Find the area of ∆ABC to the nearest 3 decimal places, if necessary.
a.
A
b.
o
86 7 mm
4 yd
B
8mm
A 42o
B
30o
3 yd
108o
C
C
2. Solve each triangle. Round side lengths to the nearest 3 decimal places and angle measures to the nearest
degree.
a.
b.
3. Solve ∆FGH if G = 80o, H = 40o, g = 14.
Check h 
𝒉 = 𝒃 𝐬𝐢𝐧 𝑨
4. Determine whether each ∆ABC has no solution, one solution, or two solutions. Then solve the triangle.
Round side lengths and angles to the nearest 3 decimal places.
a. A = 95o, a = 19, b = 12
b. A = 60o, a = 15, b = 24
c. A = 34o, a = 8, b = 13
12.5: Law of Cosines
Law of Cosines:
a2 = b2 + c2 – 2bc · cos A
b2 = a2 + c2 – 2ac · cos B
c2 = a2 + b2 – 2ab · cos C
Examples:
1. Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.
a.
c. a = 5, b = 8, c = 12
b.
d. B = 110o, a = 6, c = 3
2. Determine whether each triangle should be solved by beginning with the Law of Sines or the Law of Cosines.
Then solve the triangle.
a.
b.
c. In ∆RST, R = 35o, s = 16, and t = 9.
3. In a football game, the quarterback is 20 yards from Receiver A. He turns 40o to see Receiver B, who is 16
yards away. How far apart are the two receivers?
12.6 Circular and Periodic Function
unit circle: a circle with a radius of 1 unit centered at the origin on the coordinate plane
circular functions: functions defined using a unit circle (such as x = sin 𝜃 and 𝑥 = cos 𝜃)
periodic function: y-values that repeat at regular intervals
cycle: one complete pattern
period: the horizontal length of one cycle
Examples:
1. The terminal side of angle 𝜃 in standard position intersects the unit circle at each point P.
Find cos 𝜃 and sin 𝜃.
15
8
a. 𝑃 (17 , 17)
2. Determine the period of each function.
a.
b.
b. 𝑃 (−
√2
2
,
√2
)
2
c.
3. Find the exact value of each expression.
13𝜋
a. sin 6
b. sin −60°
c. cos 540°
4. The height of a swing varies periodically as the function of time. The swing goes forward and reaches its
high point of 6 feet. It then goes backward and reaches 6 feet again. Its lowest point is 2 feet. The time it
takes to swing from its high point to its low point is 1 second.
a. How long does it take for the swing to go forward and back one time?
b. Graph the height of the swing h as a function of time t.
5. Find the exact value of each expression.
a. cos 60° − cos 45°
𝜋
b. 2 sin 3 − 3 cos
c.
(cos 45°)(sin 135°)
cos 315°
2𝜋
3
12.7: Graphing Trigonometric Functions
amplitude: half the difference between the minimum and maximum values
frequency: the number of cycles in a given unit of time
Examples:
Day I
1. Find the amplitude and period of each function. Then graph the function.
1
a. y = 4 sin 𝜃
b. y = sin 3𝜃
c. 𝑦 = 2 sin 3 𝜃
1
d. 4 cos 𝜃
e. 𝑦 = cos 2𝜃
f. y =
g. 𝑦 = 3 tan 𝜃
h. 𝑦 = 2 tan 4𝜃
i. 𝑦 = 4 tan 2 𝜃
j. 𝑦 = 6 sin 2𝜃
k. 𝑦 = 4 cos 3𝜃
1
2
cos 3𝜃
1
l. 𝑦 = 5 tan 2𝜃
DayII
2. Find the period of each function. Then graph the function.
a. 𝑦 = 2 sin 𝜃
b. 𝑦 = 2 csc 𝜃
c. 𝑦 = 2 csc 2𝜃
d. 𝑦 = 4 cos 𝜃
e. 𝑦 = 4 sec 𝜃
f. 𝑦 = 2 sec 𝜃
g. 𝑦 = 2 tan 𝜃
h. 𝑦 = cot 2 𝜃
k. 𝑦 = 2 cot 𝜃
12.8: Translations of Trigonometric Graphs
phase shift: a horizontal translation of a periodic function
**the phase shift of the function 𝑦 = 𝑎 sin 𝑏(𝜃 − ℎ), 𝑦 = 𝑎 cos 𝑏(𝜃 − ℎ) and 𝑦 = 𝑎 tan 𝑏(𝜃 − ℎ) is h,
where b > 0; if h> 0, the shift is h units to the right; if h < 0, the shift is |h| units to the left
vertical shift: graphs of the trigonometric functions can be translated vertically
**the vertical shift of the functions 𝑦 = 𝑎 sin 𝑏𝜃 + 𝑘, 𝑦 = 𝑎 cos 𝑏𝜃 + 𝑘, and 𝑦 = 𝑎 tan 𝑏𝜃 + 𝑘 is k; if k > 0,
the shift is k units up; if k < 0 the shift is |k| units down
midline: the new horizontal axis is the line y = k about which the graph oscillates and can be used to help draw
vertical translations
amplitude: = a
period:
360°
|𝑏|
phase shift: h
Day 1 Examples:
1. State the amplitude, period, and phase shift for each function. Then graph the function.
a. 𝑦 = 𝑠𝑖𝑛(𝜃 − 180°)
𝜋
b. 𝑦 = 𝑡𝑎𝑛 (𝜃 − 4 )
1
c. 𝑦 = 2 cos(𝜃 + 90°)
2. State the amplitude, period, vertical shift, and equation of the midline for each function. Then graph the
function.
1
a. y = cos 𝜃 + 4
b. y = 2 tan 𝜃 + 1
c. y = sec 𝜃 – 5
Day II
3. State the amplitude, period, vertical shift, and equation of the midline for each function. Then graph the
function.
a. 𝑦 = cos(𝜃 + 𝜋) − 3
𝜋
b. 𝑦 = tan(𝜃 + 30°) + 4
c. 𝑦 = sin (𝜃 − 4 ) + 6
4. State the amplitude, period, phase shift and vertical shift for each function. Then graph the function.
a. 𝑦 = 2 cos(𝜃 − 𝜋) + 2
1
c. 𝑦 = 5 tan(𝜃 + 90°) + 2
𝜋
b. 𝑦 = 3 sec (𝜃 + 2 ) − 4
𝜋
d. 𝑦 = 2 csc (𝜃 − 2 ) − 3
Day III
5. State the amplitude, period, phase shift and vertical shift for each function. Then graph the function.
a. 𝑦 = cos 2(𝜃 + 90°) − 2
𝜋
c. 𝑦 = tan 2 (𝜃 + 2 ) − 3
𝜋
b. 𝑦 = sin 3 (𝜃 − 6 ) + 3
d. 𝑦 = csc 4(𝜃 + 𝜋) − 2
Day IV
6. State the amplitude, period, phase shift and vertical shift for each function. Then graph the function.
a. 𝑦 = −𝑠𝑖𝑛(𝜃 − 180°)
b. 𝑦 = − cos 𝜃 + 4
c. 𝑦 = 2 cos(𝜃 − 𝜋) + 2
Day V
7. State the amplitude, period, phase shift and vertical shift for each function. Then graph the function.
a. 𝑦 = cos 3 (𝜃 − 𝜋) − 4
1
𝜋
b. 𝑦 = 4sin 2 (𝜃 − 2 ) + 5
12.9: Inverse Trigonometric Functions
principal values: the values in a restricted domain
Arcsine: the inverse of sine
Arccosine: inverse of cosine
Arctangent: inverse of tangent
Examples:
1. Find each value. Write angle measures in degrees and radians.
1
a. Sin– 1 2
b. Arctan(−√3)
c. Arccos(–1)
2. Find each value. Round to the nearest hundredth if necessary.
4
a. cos(Arcsin 5 )
b. tan(Cos– 1 1)
c. 𝑠𝑖𝑛 (𝑆𝑖𝑛−1
√3
)
2
3. Solve each equation. Round to the nearest tenth if necessary. 0ᵒ < θ < 360ᵒ
a. Sin 𝜃 = 0.422
b. Cos 𝜃 = 0.9
c. Sin 𝜃 = – 0.46
d. Tan 𝜃 = 2.1
4. A cross section of a superpipe for snowboarders is shown. Write an inverse trig function that can be used
to find 𝜃, the angle that describes the steepness of the superpipe. Then find the angle to the nearest degree.
𝜃
18 ft
6.2 ft