Download Arc Length, Inverse Trig Functions and Graphing the Six Trig Functions

Mrs. Turner’s Precalculus – Page 0 Arc Length, Inverse Trig Functions and Graphing the Six Trig Functions Name: ______________________________________________ period: ____________________ Mrs. Turner’s Precalculus – Page 1 4.7 – Inverse Trigonometric Functions Notes ± Today, we are going to learn how to find the inverse of a trig function. Notation you will see today: “arcsin” or “ sin −1 ” “arccos” or “ cos −1 ” “arctan” or “ tan −1 ” ± When we find the value of a regular trig function, our answer is a __________________________________ For example: Find the exact value of sin(π/4) ± When we find the value of an inverse trig function, the answer is an _________________ 2
. For example: Find the angle whose sine value is 2
Can you find another answer on the unit circle? ± Since there are many similar triangles on the unit circle that are perfect reflections, there will be more than one answer. However, we are going to learn that when we take the inverse, we must restrict our answers to just one answer. This is called the principal value. Function Alternate Range/Restriction
Notation Quadrants I and IV arcsin x (must rename angles in IV negative) arccos x
Quadrants I and II arctan x
Quadrants I and IV (must rename angles in IV negative) Mrs. Turner’s Precalculus – Page 2 Find the exact value of each expression without using a calculator. ⎛ 1⎞
⎟ ⎝ 2⎠
⎛ 3⎞
⎟
⎜ 2 ⎟ ⎝
⎠
−1
2. sin ⎜
4. arcsin⎜ −
5. arccos⎜ ⎟ −1
6. cos ⎜ −
−1
8. arcsin (−1) 1. arcsin⎜ −
3. sin −1 2 ⎛
⎜
⎝
2⎞
⎟ 2 ⎟⎠
⎛1⎞
⎝2⎠
7. arccos (1) ⎛
⎜
⎝
3⎞
⎟ 2 ⎟⎠
Mrs. Turner’s Precalculus – Page 3 Find the exact value of the inverse trig function without using a calculator. 9. arctan(1) −1
(− 3 ) 10. tan
11. arctan(0) −1
12. tan ⎜
14. arctan⎜ −
⎛ 3⎞
⎟
⎜ 3 ⎟ ⎝
⎠
13. tan
−1
(− 1) ⎛
⎜
⎝
3⎞
⎟ 3 ⎟⎠
± Evaluating Composition of Functions. Find the exact value of the expression. (No calculator) Hint: Do what is inside the parenthesis first. ⎛
⎝
1. arcsin⎜ sin
3π ⎞
⎟ 2 ⎠
2. tan (arctan( −5) ) (
⎛
⎝
π⎞
4. cos tan
⎛
⎛ π ⎞⎞
⎟ ⎟⎟ ⎝ 2 ⎠⎠
6. sin ⎜ arcsin
−1
3. sin ⎜ cos
⎟ 2⎠
−1
)
3 5. cos −1 ⎜⎜ cos⎜ −
⎝
⎛
⎜
⎝
3⎞
⎟ 2 ⎟⎠
Mrs. Turner’s Precalculus – Page 4 All of the above examples were taken from the unit circle. What if you cannot use the unit circle? You must draw a triangle and label the sides! ⎡
⎛ 3 ⎞⎤
⎝ 5 ⎠⎦
⎛
⎝
2⎞
3⎠
8. cos ⎢arcsin⎜ − ⎟⎥ ⎛
⎝
12 ⎞
⎟ 5⎠
10. sin ⎜⎜ arccos⎜ − ⎟ ⎟⎟ 7. tan⎜ arccos ⎟ ⎣
9. sec⎜ arctan
⎛
⎝
⎛ 5 ⎞⎞
⎝ 8 ⎠⎠
± Use a calculator to approximate the value of the expression. Round your answer to the nearest hundredth. When calculating a trig function of a degree, your calculator should be in ________________ mode When calculating a trig function of a radian, your calculator should be in ________________ mode When calculating an inverse trig function, your calculator should be in _________________ mode 2. tan(18.5°) MODE: ____________ 1. cos( 40°) MODE: ________ ⎛π ⎞
⎟ MODE: _________ ⎝8⎠
3. sec⎜
5. csc(71°) MODE: _________ 7. arctan(−8.45) MODE: _________ 9. arccos 2 MODE: __________ ⎛π ⎞
⎟ ⎝ 16 ⎠
4. cot⎜
6. sec(1.54) MODE: _____________ 8. sin −1 0.2447 MODE: ______________ 10. arcsin1 MODE: _____________ MODE: _____________ Mrs. Turner’s Precalculus – Page 5 Basic Graphs of the Six Trig Functions Quadrantal Angles 0 π/2 π 3π/2 2π sin θ cos θ tan θ csc θ sec θ cot θ y = sin θ y = csc θ y cos θ y sec θ y tan θ y cot θ Mrs. Turner’s Precalculus – Page 6 Transformations of Trigonometric functions The “d” value is the VERTICAL SHIFT. This moves the graph up or down. The “d” can be before or after the trig function but is never inside the parenthesis. Add or subtract this value to the y coordinates of the key points The “b” value is the PERIOD CHANGE. This stretches or shrinks the graph horizontally. Multiply or divide this value to the x coordinates of the key points Changes the y coordinates The “a” value is the AMPLITUDE. It is always the coefficient of the trig function. This stretches or shrinks the graph vertically. Multiply or Divide this value to the y coordinates of the key points IMPORTANT NOTE: When altering the Y COORDINATE, always multiply/divide before you add/subtract. This means do the “a” and then the “d” This follows order of operations: PEMDAS
Changes the x coordinates The “c” value is the HORIZONTAL SHIFT. This will move the graph left or right. Add or Subtract this value to the x coordinates of the key points IMPORTANT NOTE:
When altering the X COORDINATE, always add/subtract before you multiply/divide This means do the “c” and then the “b” This follows reverse order of operations.
Mrs. Turner’s Precalculus – Page 7 Shifting/Transformations of the Key Points of Basic Trig Functions trig ± Amplitude shifts: “a” value Stretches or shrinks the graph vertically Multiply the y coordinates by “a” 1. 2 cos 2. sin 3. tan ± Vertical Shifts: “d” value The d value can be in front or behind the trig function BUT NEVER INSIDE THE PARENTHESIS Moves graph up or down Add or subtract the “d” value from the y coordinates 4. 1 cot 5. cos
2 6. 3 tan ± Amplitude and Vertical shifts combined The order matters! Do the “a” then the “d” 7. 1 2 sin sec
2 8. 9. 1 cot ± Period shifts: “b” value Stretches or shrinks the graph horizontally Multiply or divide the x coordinate by “b” (It’s like you are solving for x) 10. sin 2 11. tan 12. sec 3 Mrs. Turner’s Precalculus – Page 8 ± Horizontal Shift: “c” value Moves the graph left or right Add or subtract the “c” value to the x coordinate (It’s like you are solving for x) 13. cos 14. csc
15. tan ± Period and Horizontal Shifts Combined: The order matters. Pretend you are getting the x by itself. 16. sin 2
17. cot
18. sec 3
± ALL FOUR shifting types combined 19. 2 cos 3
1 20. 2 sin 2 21. 3tan 2
Mrs. Turner’s Precalculus – Page 9 4.5 – Graphs of Sine and Cosine Functions Notes ± Today, we will be learn how to graph the sine and cosine functions and about their properties The sine and cosine graphs repeat infinitely. That is why we call the sine function ______________________. ± PERIOD: The graphs of sine and cosine are periodic. This means that they repeat cycles to form the curve. Each period of the graph includes one “hill” and one “valley”. The period of the basic sine and cosine functions is ___________ Identify the period by analyzing the graph. 2π
(b is the coefficient of x) b
x
3. y = sin 2
Identify the period by analyzing the function. Period = 1. y = sin3x 2. y = sin
x
3
± DOMAIN and RANGE: Domain means all possible ____________________ and Range means all possible ___________________ To find the domain, scan the graph from left to right. What do you notice? To find the range, scan your graph up and down. What’s the highest value? What’s the lowest? Identify the domain and range. Domain: _________________________ Range: ___________________________ Domain: _______________________ Range: _________________________ Mrs. Turner’s Precalculus – Page 10 ± AMPLITUDE: The amplitude is the positive distance from the x‐axis to the _____________ and _____________ points. Changing the amplitude will cause our graphs to stretch or shrink vertically. To find the amplitude of a graph, you can look at the graph or the original function. Identify the amplitude by analyzing the graph. Amplitude: ______________ Amplitude: __________________ Sketch the graph of each function by hand – include two full periods. Identify the domain, range, period, and amplitude. 1. y = 2 sin x 2. y = cos
DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ AMPLITUDE: _________________ θ
x
2
DOMAIN: _____________________ θ
RANGE: _______________________ PERIOD: ______________________ AMPLITUDE: _________________ Mrs. Turner’s Precalculus – Page 11 3. y =
1 ⎛
π⎞
sin ⎜ x − ⎟ 2 ⎝
3⎠
4. y = 2 + 3 cos( 2 x + π ) DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ AMPLITUDE: _________________ θ
⎛x π⎞
− ⎟ − 3 ⎝3 4⎠
5. y = 4 sin ⎜
DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ AMPLITUDE: _________________ θ
DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ AMPLITUDE: _________________ θ
Mrs. Turner’s Precalculus – Page 12 4.6 – Graphs of Other Trig Functions Notes ™
Graphs of Secant 1. y = sec x θ
DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ ASYMPTOTES: ________________ x
2
2. y = − sec DOMAIN: _____________________ RANGE: _______________________ θ PERIOD: ______________________ ASYMPTOTES: ________________ DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ ASYMPTOTES: ________________ π
3. y = 2 sec⎛⎜ x + ⎞⎟ ⎝
4⎠
θ
Mrs. Turner’s Precalculus – Page 13 4. y = 1 + 3 sec(2 x − π ) ™ Graphs of Cosecant 1. y = csc x θ
DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ ASYMPTOTES: ________________ DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ ASYMPTOTES: ________________ DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ ASYMPTOTES: ________________ θ
2. y = csc(2 x − π ) θ
Mrs. Turner’s Precalculus – Page 14 1
2
x
3
3. y = csc DOMAIN: _____________________ RANGE: _______________________ θ PERIOD: ______________________ ASYMPTOTES: ________________ DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ ⎛x π⎞
+ ⎟ ⎝2 4⎠
4. y = −3 + csc⎜
ASYMPTOTES: ________________ θ
™ Graphs of Tangent 1. y = tan x DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ ASYMPTOTES: ________________ θ
Mrs. Turner’s Precalculus – Page 15 2. y = − tan
x
2
DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ ASYMPTOTES: ________________ π
3. y = 3 tan⎛⎜ 2 x + ⎞⎟ DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ ASYMPTOTES: ________________ θ
⎝
4⎠
θ
⎛x
⎞
−π ⎟ ⎝2
⎠
4. y = 2 − tan⎜
θ DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ ASYMPTOTES: ________________ Mrs. Turner’s Precalculus – Page 16 ™ Graphs of Cotangent 1. y = cot x DOMAIN: _____________________ RANGE: _______________________ θ PERIOD: ______________________ ASYMPTOTES: ________________ 2. y = 2 cot
x
3
DOMAIN: _____________________ RANGE: _______________________ θ PERIOD: ______________________ ASYMPTOTES: ________________ 3. y =
1
cot( x + π ) 4
DOMAIN: _____________________ RANGE: _______________________ PERIOD: ______________________ ASYMPTOTES: ________________ θ
Mrs. Turner’s Precalculus – Page 17 π
4. y = − cot⎛⎜ 3x − ⎞⎟ ⎝
4⎠
DOMAIN: _____________________ RANGE: _______________________ θ PERIOD: ______________________ ASYMPTOTES: ________________