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2.1 -- Trig Functions of Acute Angles
Wednesday, August 07, 2013
12:56 PM
Right Triangle Based Definitions of Trig Functions
SOHCAHTOA -- using sohcahtoa to find the trig functions.
B
r
A
x
SOH
CAH
TOA
y
C
csc A, sec A and cot A are just the
reciprocals of the trig ratios above.
Ex1: Finding Trig Function Values of an Acute Angle
Find sin, cos, and tan values for A and B in the given figure.
Find csc, sec, and cot for angles
A and B in the figure above.
sin A =
sin B =
cos A =
cos B =
tan A =
tan B =
csc A =
csc B =
sec A =
sec B =
cot A =
cot B =
Classroom Ex1: find sin, cos, tan, csc, sec, and cot of angles A and B.
Chapter 2 Page 1
Cofunctions: Notice that sin A = cos B and that cos A = sin B
These relations will always be true for the acute angles of a right triangle
tan A = cot B and sec A = csc B.
Since A and B are complementary then A + B = 90
and B = 90 - A, which gives us the cofunction
identities. *these are true for ACUTE angles
Cofunction Identities
sin A = cos (90 - A)
cos A = sin (90 - A)
sec A = csc (90 - A)
csc A = sec (90 - A)
sin and cos are cofunctions
tan and cot are cofunctions
sec and cot are cofunctions
tan A = cot (90 - A)
cot A = tan (90 - A)
The easiest way to look at this is that the angles of the cofunctions add up to 90 and are
therefore complementary.
For ex: Sin A = Cos B
sin 30⁰ = cos 60⁰
sec 40⁰ = csc 50⁰
so A + B = 90
tan 20⁰ = cot 70⁰
cos (2x + 3)⁰ = sin (3x 8)⁰
Ex2: Writing Functions in Terms of Cofunctions
Write each term in terms of its cofunction.
1) cos 52˚
2) tan 71˚
4) sin 9˚
5) cot 76˚
To solve this you would make
2x + 3 + 3x 8 = 90
3) sec 24˚
6) csc 45˚
Ex3: Solving Equations Using Cofunction Identities
Find one solution for each equation. Assume all angles are acute angles.
1) cos (θ + 4)⁰ = sin (3θ + 2)⁰
2) tan (2θ 18)⁰ = cot (θ + 18)⁰
Classroom Ex3:
1) cot (θ 8)⁰ = tan ( 4θ + 13)⁰
Chapter 2 Page 2
2) sec (5θ + 14)⁰ = csc (2θ
8)⁰
As θ increases -- sin θ increases which means the opposite is true for cosθ
As θ increases -- tan θ increases which means the opposite is true for cotθ
As ϴ increases -- sec θ increases which means the opposite for cscθ
All the Cofunctions have smaller values as their angles get bigger.
Ex4: Comparing Function Values of Acute Angles
Determine if the following are true of false.
1) sin 21 > sin 18
2) sec 56 ≤ sec 49
3) tan 25 < tan 23
4) csc 44 < csc 40
6) cos 15 < cos 20
5) cot 32 > cot 23
Special right triangles: 30-60-90 and 45-45-90
The 30-60-90 triangle is half of an equilateral and the 45-45-90 is half of a square.
Solve for a,b,c,and d. Do not use decimals for any values.
Chapter 2 Page 3
2.2 -- Trig Functions of Non-Acute Angles
Wednesday, August 07, 2013
12:57 PM
Reference Angles
reference angles animation
A reference angle, written as θ', is the positive acute angle formed by the terminal side of angle
θ and the x-axis. Basically, you are looking for how far the angle is from the x-axis!
θ' = θ
θ' = 180
θ
θ' = θ
180
θ' = 360
θ
The most important and the most used angles in trig. 45⁰, 30⁰, 60⁰
Ex1: Finding Reference Angles
Find the reference angle for each of the following:
1) 218˚
2) 1387˚
1)
Add or subt 360 until you get an
angle btwn 0 and 360.
2)
Figure out in which quadrant the
angle lies
3)
3) 294˚
4) 883˚
Chapter 2 Page 4
QI: θ' = θ
QII: θ' = 180 - θ
QIII: θ' = θ - 180
QIV: θ' = 360 - θ
4) To find the trig functions
a. Find the trig functions of θ'
b. Determine the correct signs
depending on the quadrant for
the original angle θ.
Ex2: Finding Trig Function Values of a QIII Angle
Find the 6 trig functions for 210˚.
Classroom Ex2: Find the 6 trig values for 225˚
Ex3: Finding Trig Function Values Using Reference Angles
Find the exact value of each expression.
1) Cos ( 240˚)
2) tan 675˚
Classroom Ex3: Find the following:
1) sin ( 150˚)
2) cot 780˚
Ex4: Evaluating an Expression with Function Values of Special Angles
Chapter 2 Page 5
Ex4: Evaluating an Expression with Function Values of Special Angles
Evaluate: cos 120˚ + 2sin2 60˚ tan 30˚.
Classroom Ex4: Evaluate: sin2 45˚ + 3cos2 135˚
2tan 225˚
Ex5: Using Coterminal Angles to Find Function Values
Evaluate each function by first expressing in terms of a function of an angle between 0˚ and 360˚.
1) cos 780˚
2) cot ( 405˚)
3) sin 585˚
4) cot ( 930)
Ex6: Finding Angle Measures, Given an Interval and a Function Value
Find all values of θ, if θ is in the interval [0, 360) (** between 0 and 360) and
1) cos θ =
2) sin θ =
Chapter 2 Page 6
2.3 -- Finding Trig Function Values Using a
Calculator
Wednesday, August 07, 2013
12:57 PM
Finding Function Values Using a Calculator
First you must make sure that your calculator is in degree mode. Never flip the angle by itself -use the reciprocal of the entire trig function when finding cosecant, secant, and cotangent which
are not on the calculator.
** do NOT use the sin-1 or the cos-1 keys except when finding angles (ϴ)
Ex1: Finding Function Values with a Calculator
1) sin 49˚ 12'
2) sec 97.977˚
5) tan 68˚
6) cos 193.622˚
3)
4) sin ( 246˚)
7) csc 35.8471˚
8) sec ( 287˚)
Ex2: Using Inverse Trig Functions to Find Angles
Use a calculator to find an angle θ in the interval [0˚, 90˚] that satisfies each condition:
1) sin θ ≈ 0.967
2) sec θ ≈ 1.054
Ex3: Finding Grade Resistance
Chapter 2 Page 7
3) cos θ ≈ 0.921
Ex3: Finding Grade Resistance
When a car travels uphill or downhill, it experiences a force due to gravity. This force F in pounds
is the grade resistance and is modeled by the equation
F = the force or grade resistance
θ = the grade or the angle formed the hill and a horiz line
W = the weight of the car
1) Calculate F to the nearest 10 lb for a 2500-lb car traveling an uphill grade with θ = 2.5˚.
2) Calculate F to the nearest 10 lb for a 5000-lb truck traveling a downhill grade with θ = -6.1˚.
3) Calculate F for θ = 0˚ and θ = 90˚ (what would you guess about these answers?)
Classroom Ex3:
1) Calculate F to the nearest 10 lb for a 5500-lb truck traveling a uphill grade with θ = 3.9˚.
2) Calculate F to the nearest 10 lb for a 2800-lb truck traveling a downhill grade with θ = -4.8˚.
3) A 2400-lb car traveling uphill has a grade resistance of 288 lb. What is the angle of the grade?
Chapter 2 Page 8
Quiz Review?
Saturday, August 10, 2013
6:10 PM
Chapter 2 Page 9
2.4 -- Solving Right Triangles
Wednesday, August 07, 2013
12:58 PM
Solving a Triangle means finding the measure of all the sides and angles.
Ex1: Solving a Right Triangle Given an Angle and a Side
B
A=
12.7 in.
B=
A
34° 30'
C
a=
b=
C = 90⁰
c = 12.7 in
A=
a = 25.3 cm
B=
b=
C = 90⁰
c=
Classroom Ex1: Solve right Δ ABC, if B = 28° 40' and a = 25.3 cm.
Ex2: Solving a Right Δ Given 2 Sides: Solve right Δ ABC, if a = 29.43 cm and c = 53.58 cm
Classroom Ex2: Solve right Δ ABC, if a = 44.25 cm and b = 55.87 cm
Chapter 2 Page 10
Classroom Ex2: Solve right Δ ABC, if a = 44.25 cm and b = 55.87 cm
Angle of Elevation
Angle of Depression
Ex3: Finding a Length Given the Angle of Elevation
Patty knows that when she stands 123 ft from the base of a flagpole, the angle of elevation to
the top of the flagpole is 26° 40'. If her eyes are 5.3 ft above the ground, find the height of the
flagpole.
Classroom Ex3: The angle of elevation from a point on the ground 15.5 m from the base of a
tree to the top of the tree is 60.4°. Find the height of the tree.
Chapter 2 Page 11
Ex4: Finding an angle of depression
From the top of a 210-ft cliff, David observes a lighthouse that is 430 ft offshore. Find the
angle of depression from the top of the cliff to the base of the lighthouse.
Classroom Ex4: Find the angle of depression from the top of a 97-ft building to the base of a
building across the street located 42 ft away.
Chapter 2 Page 12
2.5 -- Further Applications of Right Triangles
Wednesday, August 07, 2013
12:59 PM
Bearing is a an angle measured from a clockwise direction from due north.
2) 110°
3) 220°
1) 35°
4) 300°
Ex1: Solving a Problem Involving Bearing (Method 1)
Radar stations A and B are on a east-west line, 3.7 km apart. Statin A is east of B. Station A
detects a plane at C, on a bearing of 61°. Station B simultaneously detects the same plane on
a bearing of 331°. Find the distance from A to C.
Classroom Ex1: Radar stations A and B are on a east-west line, 8.6 km apart. Station A is east
of B. Station A detects a plane at C, on a bearing of 53°. Station B simultaneously detects the
same plane on a bearing of 323°. Find the distance from B to C.
Chapter 2 Page 13
Method 2 for expressing bearing starts with a north-south line and uses an acute angle to
show the direction, either east or west from this line.
N 52° W
N 42° E
S 31° E
S 40° W
Ex2: Solving a Problem Involving Bearing (Method 2)
A ship leaves port and sails on a bearing of N 47° E for 3.5 hr. It then it turns and sails on a
bearing of S 43° E for 4 hrs. If the ship's rate of speed is 22 knots (nautical mph), find the
distance that the ship is from port.
** d = rt
Classroom Ex2: A ship leaves port and sails on a bearing of S 34° W for 2.5 hr. It then returns
and sails on a bearing of N 56° W for 3 hr. If the ship's rate of speed is 18 knots. Find the
distance that the ship is from port.
Chapter 2 Page 14
Ex4: Solving a Problem Involving Angles of Elevation
Sam needs to know the height of a tree. From a given point on the ground, he finds that the
angle of elevation to the top of the tree is 36.7°. He then moves back 50 ft. From the 2nd
point, the angle of elevation to the top of the tree is 22.2°. Find the height of the tree to the
nearest tenth.
B
22.2°
D
50
36.7°
A
x
C
Classroom Ex4: Deana needs to know the height of a building. From a given point on the
ground, he finds that the angle of elevation to the top of the tree is 74.2°. He then moves back
35 ft. From the 2nd point, the angle of elevation to the top of the tree is 51.8°. Find the height
of the building to the nearest foot.
Chapter 2 Page 15
Test Review?
Saturday, August 10, 2013
6:10 PM
Chapter 2 Page 16