Vertical & Horizontal Asymptotes, holes
Holes
1. Algebraically determine the equation of the vertical asymptotes and/or coordinates of the holes for the following functions. Sketch. State the domain, range, positive and negative intervals.
a) [pic] b) [pic] c) [pic]
2. State the domain and range as well as the equations for the vertical and horizontal asymptotes for [pic].
3. Explain the difference, if it exists, between the graphs:
a) [pic] and [pic] b) [pic] and [pic]
ANSWERS:
|1. a) hole at (2, -1) |b) vertical asymptote x = 5 |c) hole at (0, 0) |
|vertical asymptote x = 6 |horizontal asymptote y = 0 |vertical asymptote x = -4 |
|horizontal asymptote y = 1 |no hole |horizontal asymptote y = -3 |
|[pic] |[pic] |[pic] |
|[pic] |[pic] |[pic] |
|positive: [pic] |positive: [pic] |positive: [pic] |
|negative: [pic] |negative: [pic] |negative: [pic] |
|[pic] |[pic] |[pic] |
2. [pic]; [pic]; vertical asymptote: x = 4; horizontal asymptote: y = 0.5
3. a) The two graphs are identical except there is a hole at (b, b - a) for [pic].
b) The two graphs are identical.
Linear Oblique Asymptotes
1. a) Under what conditions does a rational function have an linear oblique asymptote?
b) Explain how to find the equation of the linear oblique asymptote of a rational function
that satisfies the conditions in part (a)?
Find the equation of the linear oblique asymptote of each curve. Use the linear oblique
asymptote and any other asymptotes to sketch the graph.
a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic] f) [pic]
3. A rectangular garden, 21 m2 in area, will be fenced to keep out rabbits and skunks.
The barn already protects one side of the garden.
a) If the side perpendicular to the barn is x metres in length, express the amount
of fencing needed, y, as a function of x.
b) Using a graph that models this situation, find the dimensions (to one decimal place)
that will require the least amount of fencing.
ANSWERS:
1. a) when the degree of the numerator is greater than the degree of the denominator
by 1
b) Divide the numerator of the rational function by the denominator of the rational
function. y = quotient of division is the equation of the linear oblique asymptote.
|2 a) [pic] |b) [pic] |
|linear oblique asymptote y = 3x – 4 |linear oblique asymptote y = x +1 |
|vertical asymptote x = 0 |vertical asymptote x = 1 |
|[pic] | |
| |[pic] |
|c) [pic] |d) [pic] |
|linear oblique asymptote y = 2x - 1 |linear oblique asymptote y = 2x - 1 |
|vertical asymptote x = -2 |vertical asymptote x = 0 |
|[pic] |[pic] |
|e) [pic] |f) [pic] |
|linear oblique asymptote y = -3x + 13 |linear oblique asymptote y = 2x - 2 |
|vertical asymptote x = -5 |vertical asymptote x = -3 |
|[pic] |[pic] |
| | |
|3. a) [pic] |[pic] |
|b) Using the graph, the value for x at the minimum is approximately 3.2 m. | |
|Therefore, the dimension that will require the least amount of fencing are | |
|6.6 m x 3.2 m | |
More Sketching Rational Functions
1. For the following functions:
a) [pic] b) [pic] c) [pic]
i) Determine the x- and y-intercepts, holes, vertical, horizontal and oblique asymptotes.
Use these key features to sketch each function.
ii) State the domain and range.
iii) State the positive and negative interval(s) using interval notation.
iv) State the increasing and decreasing interval(s) using interval notation if they exist.
2. State the equation of a function that has:
a) two vertical asymptotes at x = 1 and x = 3, a horizontal asymptote at y = -1 and
x-intercepts at -2 and 4.
b) a vertical asymptote at x = -3, a hole at x = 4, no horizontal asymptote and
x-intercept at 0.
c) a vertical asymptote at x = 0 and x = -5, a horizontal asymptote at y = 0 and no
x-intercepts.
3. Find the equation of the vertical, horizontal and linear oblique asymptote if they exist.
a) [pic] b) [pic] c) [pic]
4. For each function, find the intercepts, vertical, horizontal and oblique asymptotes if
they exist. Using these key features, sketch the graphs. Identify any holes if they
exist.
a) [pic] b) [pic]
c) [pic] d) [pic]
Answers:
|1 a) |[pic] |[pic] |
| |Hole at (2, 0) | |
| |no x-intercept y-intercept is -2 | |
| |vertical asymptote none | |
| |horizontal asymptote none | |
| |linear oblique asymptote none | |
| |[pic] [pic] | |
| |positive: [pic] negative: [pic] | |
| |increasing:[pic] decreasing: none | |
| | | |
| | | |
| b) |[pic] |[pic] |
| |x-intercept none y-intercept is [pic] | |
| |vertical asymptote x = -3 | |
| |horizontal asymptote y = 0 | |
| |linear oblique asymptote none | |
| |[pic] [pic] | |
| |positive: [pic] negative: [pic] | |
| |increasing: none decreasing: [pic] | |
| | | |
| | | |
| c) |[pic] |[pic] |
| |Hole at (3, -1) | |
| |x-intercept is 0 y-intercept is 0 | |
| |vertical asymptote x = -9 | |
| |horizontal asymptote y = -4 | |
| |linear oblique asymptote none | |
| |[pic] | |
| |[pic] | |
| |positive: [pic] | |
| |negative: [pic] | |
| |increasing: none | |
| |decreasing: [pic] | |
2. ANSWERS MAY VARY.
a) [pic] b) [pic] OR [pic] c) [pic]
3. a) vertical asymptote: [pic], no horizontal asymptote, linear oblique asymptote: [pic]
b) vertical asymptote: [pic], no horizontal asymptote, linear oblique asymptote: [pic]
c) vertical asymptotes: [pic] and [pic], no horizontal asymptote, linear oblique asymptote: [pic]
|4. a) |[pic] |b) |[pic] |
| |Hole at (1, 4) | |For [pic] |
| |x-intercepts are -3 and [pic] | |x-intercepts are -5 and -1 |
| |y-intercept is [pic] | |y-intercept [pic] |
| |vertical asymptote x = 4 | |vertex is (-3, 4) |
| |horizontal asymptote none | |For [pic] |
| |linear oblique asymptote y = 2x + 9 | |x-intercept none y-intercept is [pic] |
| | | |vertical asymptotes x = -5, x = -1 |
| |[pic] | |horizontal asymptote y = 0 |
| | | |linear oblique asymptote none |
| | | | |
| | | |[pic] |
| c) |[pic] | d) |[pic] |
| |Hole at (-1, 1) | |x-intercept is [pic] and 3 y-intercept is [pic] |
| |x-intercept none | |vertical asymptote x = 5 |
| |y-intercept is [pic] | |horizontal asymptote none |
| |vertical asymptote x = -2 | |linear oblique asymptote y = -3x -10 |
| |horizontal asymptote y = 0 | | |
| |linear oblique asymptote none | | |
| | | |[pic] |
| |[pic] | | |
Solving Rational Inequalities
Solve for x. State the solution using interval notation.
1) [pic] 2) [pic] 3) [pic]
4) [pic] 5) [pic] 6) [pic]
ANSWERS:
|1. | |2. | |
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|3. | |4. | |
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| |[pic] | |[pic] |
|5. | |6. | |
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More Solving Rational Inequalities
Solve for the unknown. State your answer using interval notation.
1) [pic] 2) [pic] 3) [pic]
4) [pic] 5) [pic] 6) [pic]
ANSWERS:
|1. | |2. | |
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|3. | |4. | |
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| |[pic] | |[pic] |
|5. | |6. | |
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Unit #5 Review – Rational Functions
Sketch the following rational functions. Label all key features. State the:
▪ domain and range
▪ positive and negative interval(s)
▪ increasing and decreasing interval(s)
a) [pic] b) [pic]
c) [pic] d) [pic]
Without sketching, state the equation(s) of the vertical, horizontal and linear oblique asymptotes, if they exist. Also, state the coordinates of any holes:
a) [pic] b) [pic]
c) [pic] d) [pic]
Sketch. Label all key features.
|a) Sketch the reciprocal function of the following graph. |[pic] |
|b) [pic] |c) [pic] |
4. Solve the inequality. State your solution using interval notation.
a) [pic] b) [pic] c) [pic] d) [pic]
5. Create a function that has a graph with:
a) vertical asymptotes at x = 5, x = -3; no horizontal asymptote; x-intercepts of 1 and -2
b) vertical asymptote at x = 0; horizontal asymptote [pic]; x-intercepts of 3 and -2
c) vertical asymptote at x = -8, horizontal asymptote y = 0; hole at x = -4, x-intercept of 6
d) no horizontal or vertical asymptotes; hole at x = -5; y-intercept of -3
ANSWERS:
|1. a) | |b) | |
| |[pic] | |[pic] |
| | | | |
| |[pic] | |[pic] |
| |[pic] | |[pic] |
| |positive: [pic] | |positive: [pic] |
| |negative: [pic] | |negative: [pic] |
| |increasing: none | |increasing: [pic] |
| |decreasing: [pic] | |decreasing: none |
|c) | |d) |[pic] |
| |[pic] | | |
| | | |[pic] |
| |[pic] | |[pic] |
| |[pic] | |positive: [pic] |
| |positive: [pic] | |negative: [pic] |
| |negative: [pic] | |increasing: [pic] |
| |increasing: [pic] | |decreasing: none |
| |decreasing: none | | |
2. a) vertical asymptote [pic], x = -4 b) vertical asymptote x = -2
horizontal asymptote [pic] horizontal asymptote none
linear oblique asymptote none linear oblique asymptote [pic]
no hole hole [pic]
2. c) vertical asymptotes x = -7, [pic] d) vertical asymptote x = 4
horizontal asymptote [pic] horizontal asymptote none
linear oblique asymptote none linear oblique asymptote [pic]
no hole no hole
|3. a) |[pic] |b) |[pic] |
| | | | |
| |[pic] | |[pic] |
| c) |[pic] |[pic] |
| | | |
| |hole at (-2, -15) | |
| | | |
| |linear oblique asymptote y = x - 5 | |
| | | |
4. a) [pic] b) [pic] c) [pic] d) [pic]
5. Answers may vary.
a) [pic] or [pic] b) [pic]
c) [pic] d) [pic]
MHF 4UI Unit 6
Worksheet 11: Radian MeasureRadian Measure
1. Convert each degree to radian measure in terms of π:
2.
|a) 18º |b) - 72º |c) 870º |d) 1200º |
|e) 135º |f) 540º |g) - 315º |h) -225º |
3. Convert each radian measure to radian degrees. If the answer is not exact, round correctly to one decimal place.Convert each degree to radian measure in terms of (.
4.
|a) 18° |b) -72° |c) 870° |d) 1200° |
|e) 135° |f) 540° |g) -315° |h) -225° |
|a) 18°[pic] |b) -72°[pic] |c) [pic]870° |d) 1200°[pic] |
|e) [pic]135° |f) 540°1.75 |g) -315°[pic] |h) -225°2.31 |
23. Convert each radian measure to degrees. If the answer is not exact, round correctly to one decimal place.
|a) [pic] rad |b) [pic] rad |c) [pic] rad |d) 5( rad |
|e) [pic] |f) 1.75r |g) [pic] |h) 2.31 r |
|a) [pic] rad |b) [pic] rad |c) [pic] rad |d) 5( rad |
|e) [pic] |f) 1.75r |g) [pic] |h) 2.31 r |
3. The earth rotates on its axis once every 24 hours.
a) How long does it take the Earth to rotate through an angle of [pic]?
b) How long does it take the Earth to rotate through an angle of 120°?
4. The length of any arc, a, can be found using the formula a = r(, where r is the radius of the circle, and ( is the radian measure of the angle subtended by the arc at the centre of the circle. Find the length of the arc for each of the following. State the exact answer and the answer accurate to 3 decimal places. (The central angle is the angle subtended at the centre of the circle by the arc.)
a) radius of 12 cm, central angle 75°
b) radius of 8 m, central angle 185°
c) radius of 18 mm, central angle 30°
5. If an object moves along a circle of radius r units, then its linear velocity, v, is given by [pic], where [pic] represents the angular velocity in radians per unit of time. Find the linear velocity for each, to three decimal places.
a) A pulley of radius 8 cm turns at 5 revolutions per second.
b) A bicycle tire of diameter 26 inches rotates 3 revolutions per second.
6. The formula for the area of a sector of a circle (“pie wedge”) is given as [pic], where r is the radius and ( is the measure of the central angle, expressed in radians. Find the area of each sector. State the exact answer and the answer accurate to 3 decimal places.
a) ( = 315°, diameter is 20 cm
b) ( = 135°, radius is 16 feet
7. When is it beneficial to work with angles in radians? When is it beneficial to work in degrees?
8. Explain how to convert between radians and degrees.
9. In a circle with radius 5 cm, the arc length is [pic] cm. Determine the measure of the angle subtended at the centre of the circle. Leave your answer as an exact value.
10. In a circle, the arc length is 30 cm and the angle subtended at the centre is [pic]. What is the circumference of the circle?
Answers:
1. a) [pic] b) [pic] c) [pic] d) [pic]
e) [pic] f) [pic] rad g) [pic] rad h) [pic]
The earth rotates on its axis once every 24 hours.
a) How long does it take Earth to rotate through an angle of [pic]?
b) How long does it take Earth to rotate through an angle of 120º?
4. The length of any can be found using the formula, where r is the radius of the circle, and [pic] is the radian measure of the central angle that creates the arc. Find the length of the arc for each, to 3 decimal places:
a) radius of 12cm, central angle 75º
b) radius of 8m, central angle of 185º
c) radius of 18mm, central angle of 30º
2. a) 60° b) 270° c) 660° d) 900°
e) 10° f) 100.3° g) 128.6° h) 132.4°
3. a) 16 hours b) 8 hours
4. a) 5( cm, 15.708 cm b) [pic] m, 25.831 m c) 35( mm, 9.425 mm
5. a) 251.327 cm / sec b) 245.044 in / sec
6. a) [pic] cm2, 274.889 cm2 b) 96( ft2, 301.593 ft2
7. It is more beneficial to work in radians if the formula given calls for radians and if working with professionals with a mathematics background. It is more beneficial to work in degrees if the formula given calls for degrees and if working with the general population.
8. To convert radians to degrees, multiply by [pic] and simplify.
To convert from degrees to radians, multiply by [pic] and simplify.
9. [pic] rad 10. The circumference is 180 cm.5. If an object moves along a circle of radius r units, then its linear velocity, v, is given by [pic], where [pic] represents the angular velocity in radians per unit of time. Find the angular velocity for each:
a) a pulley of radius 8cm turns at 5 revolutions per second.
b) A bike tire of diameter 26 inches 3 revolutions per second.
6. The formula for the area of a sector of a circle (“pie wedge”) is given as [pic], where r is the radius and [pic] is the measure of the central angle, expressed in radians. Find the area of each sector described:
a) [pic] = 315º, diameter is 20cm.
b) [pic] = 135º, radius is 16 ft.
7. When is it beneficial to work with angles measured in radians? Degrees?
8. Explain how to convert between radians and degrees.
9. In a circle with radius 5 cm, the arc length is [pic] cm. Determine the measure of the angle subtended at the centre of the circle.
10. In a circle, the arc length is 30 cm and the angle subtended at the centre is [pic]. What is the circumference of the circle?
1. [pic] 2. [pic] 3. [pic] 4. [pic] 5. [pic] 6. [pic] 7. [pic]
8. [pic]
9. a. 16h
b. 8h
10. a. 15.708cm
b. 25.831m.
c. 9.425mm
11. a. 150.796cm/sec
b. 245.044 in/sec
12. a. 274.889cm2
b. 7.069 ft2
13. It is more beneficial to work in radians if the formula given calls for radians and if working with professionals with a mathematics background. It is more beneficial to work in degrees if the formula given calls for degrees and if working with the general population.
14. To convert radians to degrees, multiply by [pic] or substitute π = 180º and simplify.
To convert from degrees to radians, multiply by [pic] or cross multiply using equivalent fractions.
MHF 4UI Unit 6 Worksheet 51: Radian MeasureTransformations (1)
5. Convert each degree to radian measure in terms of π:
6.
|a) 18º |b) - 72º |c) 870º |d) 1200º |
|e) 135º |f) 540º |g) - 315º |h) -225º |
7. Convert each radian measure to radian degrees. If the answer is not exact, round correctly to one decimal place.For each of the following, state the amplitude, period, phase shift and vertical shift.
8.
a) y = 20sin3x + 10 b) [pic]
c) [pic] d) [pic]
e) [pic]
9. Write the equation of a sine function having the indicated properties.
a) amplitude 3, period (, reflected in the x-axis
b) amplitude 1.5, phase shift -(, vertical shift -2, period 4(
c) amplitude 20, vertical shift 25, period 8(, phase shift (
10. For each of the following functions,
i) determine the amplitude, period, phase shift and vertical shift;
ii) graph at least one complete period; and
iii) state the domain and range.
a) y = 3 cos x + 1 b) [pic]
c) y = -4 sin 3x d) [pic]
e) y = [pic] f) [pic]
g) [pic]
Answers:
| |Amplitude |Period |Phase Shift |Vertical Shift |
|1. (a) |20 |k = 3, per = [pic] |0 |10 |
|1. (b) |[pic] |k = 4, per = [pic] |-[pic] |-3 |
|1. (c) |1 |k = 2, per = ( |[pic] |0 |
|1. (d) |5 |k = 2, per = [pic] |[pic] |1 |
|1. (e) |[pic] |k = 3, per = [pic] |[pic] |[pic] |
| |Amplitude |Period |Phase Shift |Vertical Shift |
|1 (a) |20 |K=3, per=[pic] |0 |10 |
|1 (b) |[pic] |K=4, per=[pic] |-[pic] |-3 |
|1 (c) |1 |K=2, per=( |[pic] |0 |
|1 (d) |5 |K=2, per=( |( |1 |
|1 (e) |[pic] |K=3, per=[pic] |-[pic] |-[pic] |
11. Part of a trigonometric function is shown on the graphs below. For each:
i) determine the amplitude, period, and vertical shift;
ii) if the graph represents a sine function, determine the phase shift and write a sine equation to represent the function;
iii) if the graph represents a cosine function, determine the phase shift and write a cosine equation to represent the function.
a)
b)
Answers:Answers (continued)
2. a) y = -3sin2x b) [pic] c) [pic]
3. a) amp = 3, period = 2(, ps = 0, vs = 1, Domain=[pic], Range=[pic]
3. b) amp = 2, period = 2(, ps =[pic], vs = -3, Domain=[pic], Range=[pic]
3. c) amp = 4, period = [pic], ps = 0, vs = 0, Domain=[pic], Range=[pic]
|a) [pic] |b) [pic] |c) [pic] |d) [pic] |
|e) [pic] |f) 1.75 |g) [pic] |h) 2.31 |
3. d) amp = 2, period = 2(, ps = -[pic], vs = 0, Domain=[pic], Range=[pic]
3. e) amp = 3, period = 4(, ps = 0, vs = -2, Domain=[pic], Range=[pic]
3. f) amp = 1, period = 3(, ps = 0, vs = 4, Domain=[pic], Range=[pic]
3. g) amp = 2.5, period = (, ps =[pic], vs = 1, Domain=[pic], Range=[pic]
4. a) (i) amp = 2, period = (, vs = 0 (ii) ps = 0, y = 2sin2x (iii) ps = [pic], y = 2cos2(x-[pic][pic])
4. b) (i) amp = 1, period = 2(, vs = 2 (ii) ps = -[pic][pic], y = sin(x + [pic][pic])+2
(iii) ps = [pic], y= cos(x - [pic][pic])+2
3. The earth rotates on its axis once every 24 hours.
a) How long does it take Earth to rotate through an angle of [pic]?
b) How long does it take Earth to rotate through an angle of 120º?
4. The length of any can be found using the formula, where r is the radius of the circle, and [pic] is the radian measure of the central angle that creates the arc. Find the length of the arc for each, to 3 decimal places:
a) radius of 12cm, central angle 75º
b) radius of 8m, central angle of 185º
c) radius of 18mm, central angle of 30º
5. If an object moves along a circle of radius r units, then its linear velocity, v, is given by [pic], where [pic] represents the angular velocity in radians per unit of time. Find the angular velocity for each:
a) a pulley of radius 8cm turns at 5 revolutions per second.
b) A bike tire of diameter 26 inches 3 revolutions per second.
6. The formula for the area of a sector of a circle (“pie wedge”) is given as [pic], where r is the radius and [pic] is the measure of the central angle, expressed in radians. Find the area of each sector described:
a) [pic] = 315º, diameter is 20cm.
b) [pic] = 135º, radius is 16 ft.
7. When is it beneficial to work with angles measured in radians? Degrees?
8. Explain how to convert between radians and degrees.
9. In a circle with radius 5 cm, the arc length is [pic] cm. Determine the measure of the angle subtended at the centre of the circle.
11. In a circle, the arc length is 30 cm and the angle subtended at the centre is [pic]. What is the circumference of the circle?
1. [pic] 2. [pic] 3. [pic] 4. [pic] 5. [pic] 6. [pic] 7. [pic]
8. [pic]
9. a. 16h
b. 8h
10. a. 15.708cm
b. 25.831m.
c. 9.425mm
11. a. 150.796cm/sec
b. 245.044 in/sec
12. a. 274.889cm2
b. 7.069 ft2
13. It is more beneficial to work in radians if the formula given calls for radians and if working with professionals with a mathematics background. It is more beneficial to work in degrees if the formula given calls for degrees and if working with the general population.
14. To convert radians to degrees, multiply by [pic] or substitute π = 180º and simplify.
To convert from degrees to radians, multiply by [pic] or cross multiply using equivalent fractions.
More Transformations
12. Determine the equation of a sine function with maximum value 10, minimum value 4,
period [pic] and phase shift [pic].
13. Determine the equation of a cosine function with a minimum at ((,-4) and a maximum
at (3(, 10).
14. What would your answer to #2 be if the graph was a sine function?
15. A cosine function with x ( 0 has a vertical shift of -4, a period of [pic], and its first
minimum point at [pic]. Determine an equation to represent the function.
For each of the following trigonometric equations:
( Write the amplitude, period, phase shift (horizontal shift) & vertical shift.
← Sketch the function on a grid with an appropriate horizontal scale.
Show at least 1 complete cycle.
5. [pic][pic] 6. [pic] 7. [pic]
8. [pic] 9. [pic] 10. [pic]
For each of the following trigonometric equations:
( Write the amplitude, period, phase shift (horizontal shift) & vertical shift.
( Write the equation of the function.
11. Sine Function Equation: __________________________
Amplitude = _____
Period = _____
Phase Shift = _____
Vertical Shift = _____
12. Cosine Function Equation: __________________________
Amplitude = _____
Period = _____
Phase Shift = _____
Vertical Shift = _____
13. Sine Function Equation: ____________________________
Amplitude = _____
Period = _____
Phase Shift = _____
Vertical Shift = _____
14. Cosine Function Equation: _____________________________
Amplitude = _____
Period = _____
Phase Shift = _____
Vertical Shift = _____
15. Sine Function Equation: ________________________
Amplitude = _____
Period = _____
Phase Shift = _____
Vertical Shift = _____
Answers:
1. [pic] 2. [pic] OR [pic] OR [pic]
3. [pic] OR [pic] 4. [pic] OR [pic]
5. amp=3, k=2, period=(, vs=-2, ps=0 6. amp=2, k=[pic], period=4(, vs=0, ps=0
7. amp=5, k=3, period=[pic], vs=1, ps=[pic] 8. amp=2.5, k=[pic], period=3(, vs=1.5, ps=[pic]
9. amp=2, k=3, period=[pic], vs=-4, ps=[pic] 10. amp=3, k=[pic], period=4(, vs=-2, ps=[pic]
11. amp=2, k=3, period=[pic], vs=-3, ps=0, [pic] 12. amp=4, k=[pic], period=4(, vs=-1, ps=0,[pic]
OR [pic]
13. amp=[pic], k=2, period=[pic], vs=3, ps=0, [pic]
14. amp=3,k=2,period=[pic],vs=-1,ps=[pic],[pic]OR[pic] OR [pic]
15. amp=1,k=[pic],period=[pic], vs=3, ps=[pic], [pic] OR [pic]
MHF 4UI Unit 6
Worksheet 21: Radian MeasureTrigonometric Ratios and Special Triangles
16. Convert each degree to radian measure in terms of π:
17.
|a) 18º |b) - 72º |c) 870º |d) 1200º |
|e) 135º |f) 540º |g) - 315º |h) -225º |
1.
Convert each radian measure to radian degrees. If the answer is not exact, round correctly to one decimal place.Determine the related acute angle associated with each of the following standard position angles. A sketch may help!
|a) [pic] |b) [pic] |c) [pic] |d) [pic] |
|e) [pic] |f) [pic] |g) [pic] |h) 6.54 |
|a) [pic] |b) [pic][pic] |c) [pic] |d) [pic] |
|e) [pic] |f) [pic]1.75 |g) [pic] |h) 2.316.54 |
23. Evaluate the following, correct to 4 decimal places.
|a) tan (11.5) |b) sin [pic] |c) cos[pic] |
|d) csc (5) |e) cot [pic] |f) sec (-1) |
|a) tan (11.5) |b) sin [pic] |c) cos[pic] |
|d) csc (5) |e) cot [pic] |f) sec (-1) |
3. The coordinates of a point P on the terminal arm of standard position angle ( are given, where 0 < ( < 2(. Determine the exact valuesEXACT VALUES of sin(, cos(, and tan(. Include a clearly labelled sketch.
a) P(-2, -5) b) P(3,4) c) P(4, -8)
4. ( is in standard position with its terminal arm in the stated quadrant, and 0 < ( < 2(.
A trigonometric ratio is given. Determine the exact valueEXACT VALUE of the indicated trigonometric ratios. Include a clearly labelled sketch.
a) sin( = [pic], quadrant II, determine cos( and cot(
b) tan( = [pic] , quadrant IV, determine sin( and csc(
c) cos( = [pic], quadrant III, determine sec( and tan(
d) cos( = [pic], determine tan(
e) sin( = [pic], determine sec( and cot(
5. For each of the following,
(i) sketch the standard position angle,
(ii) determine the related acute angle, and
(iii) determine the exact value of the
specified trigonometric ratio. Do not
use a calculator.
|a) tan [pic] |b) sin [pic] |c) cos[pic] |d) tan [pic] |
|e) cos [pic] |f) sin [pic] |g) sec [pic] |h) csc [pic] |
|a) tan [pic] |b) sin [pic] |c) cos[pic] |d) tan [pic] |
|e) cos [pic] |f) sin [pic] |g) sec [pic] |h) csc [pic] |
The earth rotates on its axis once every 24 hours.
a) How long does it take Earth to rotate through an angle of [pic]?
b) How long does it take Earth to rotate through an angle of 120º?
4. The length of any can be found using the formula, where r is the radius of the circle, and [pic] is the radian measure of the central angle that creates the arc. Find the length of the arc for each, to 3 decimal places:
a) radius of 12cm, central angle 75º
b) radius of 8m, central angle of 185º
c) radius of 18mm, central angle of 30º
6. Complete the following chart. Do not use a calculator.
| |Standard Position Angle |sin( |cos( |tan( |csc( |
4.
| 4. |a) |cos( = |b) |sin( = |c) |
| | |[pic] | |[pic] | |
| | |cot( = | |csc( = | |
| | |[pic] | |[pic] | |
5. a) [pic] b) [pic] c) [pic] d) -1 e) [pic] f) [pic] g) - 2 h) [pic]
6.
|Standard. Positionn. Angle |sin( |cos( |tan( |csc( |sec( |cot( | |a) |0 |0 |1 |0 |undefinedundef |1 |undefinedUndef | |b) |[pic] |1 |0 |undefinedundef |1 |undefinedundef |0 | |c) |( |0 |-1 |0 |undefinedundef |-1 |undefinedUndef | |d) |[pic] |-1 |0 |undefinedundef |-1 |undefinedundef |0 | |e) |2( |0 |1 |0 |undefinedundef |1 |undefinedUndef | |
| |sin( |cos( |tan( |csc( |sec( |cot( | |a) |0 |0 |1 |0 |undef |1 |undef | |b) |[pic] |1 |0 |undef |1 |undef |0 | |c) |( |0 |-1 |0 |undef |-1 |undef | |d) |[pic] |-1 |0 |undef |-1 |undef |0 | |e) |2( |0 |1 |0 |undef |1 |undef | |1. [pic] 2. [pic] 3. [pic] 4. [pic] 5. [pic] 6. [pic] 7. [pic]
8. [pic]
9. a. 16h
b. 8h
10. a. 15.708cm
b. 25.831m.
c. 9.425mm
11. a. 150.796cm/sec
b. 245.044 in/sec
12. a. 274.889cm2
b. 7.069 ft2
13. It is more beneficial to work in radians if the formula given calls for radians and if working with professionals with a mathematics background. It is more beneficial to work in degrees if the formula given calls for degrees and if working with the general population.
14. To convert radians to degrees, multiply by [pic] or substitute π = 180º and simplify.
To convert from degrees to radians, multiply by [pic] or cross multiply using equivalent fractions.
Solving Trigonometric Equations
1. Solve each of the following equations to two decimal places, [pic].
a) sin( = 0.3124 b) cos( = 0.7315 c) tan( = 3.1571
d) sin( = -0.8135 e) cos( = -0.1476 f) tan( = -0.3541
2. For parts d, e, and f of question 1, rewrite your answers if ( [pic] R.
3. Solve each of the following equations. Give exact answers only, [pic].
a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic] f) [pic]
4. Solve each of the following equations. If an exact answer is not possible, write your answers correct to two decimals. Be careful with the domain of the solutions.
a) [pic], [pic]
b) [pic], [pic]
c) [pic], [pic]
d) [pic], [pic]
e) [pic], [pic]
5. Solve each of the following equations. State exact answers if possible. Otherwise,
round your answers correctly to two decimal places.
a) [pic], [pic]
b) [pic], [pic]
c) [pic], [pic]
6. Solve for x, [pic]. Give exact answers.
a) [pic] b) [pic]
Answers:
1. a) 0.32, 2.82 b) 0.75, 5.53 c) 1.26, 4.41 d) 4.09, 5.33 e) 1.72, 4.56
1. f) 2.80, 5.94 2. d) [pic]
2. e) [pic] f) [pic]
3. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
4. a) 1.37, 4.51 b) [pic] c) no solution d) 2.42 e) 3.58
5. a)[pic] b) 0.67, 3.81 c)[pic] 6. a)[pic] b)[pic]
-----------------------
D21
y = 1
x = 6
(
y = 0
x = 5
y = -3
x = -4
(
D22
y = 3x - 4
x = 0
x = 1
y = x + 1
y = 2x - 1
x = -2
D23
y = 2x - 1
x = 0
x = -5
y = -3x + 13
y = 2x - 2
x = -3
x = 0
y = 2x
D24
D25
HOLE
(2, 0)
(
y = 0
x = -3
y = -4
x = -9
(
HOLE
(3, -1)
y = 2x + 9
x = 4
HOLE
(1, 4)
(
D26
y = 0
x = -5
x = -1
HOLE
(-1, 1)
y = 0
x = -2
(
y = -3x -10
x = 5
D27
y = 1
x = 1
x = 5
y = -1
x = -3
y = -2
x = 5
y = 4
x = [pic]
y = [pic]
x = [pic]
y = [pic]
D28
y = 0
x = 4
x = -2
x = -4
y = 0
x = 3
x = -2
y = -1
x = -1
x = 0
y = 0
x = 1
x = -3
y = 1
x = -1
[pic]
[pic]
y = 0
x = 1
x = -3
x = -1
D29
D30
y = 2
x = -2
x = -4
y = 0
HOLE
[pic]
(
x = -3
9
D30
y = 2
x = -2
x = -4
y = 0
HOLE
[pic]
(
x = -3
y = 1
y = 2
x = -7
(
HOLE
(0, 0)
D31
y = 0
x = -1
x = -5
y = 0
x = 5
x = -1
HOLE
(-2, -15)
(
y = x - 5
D32
D33
r
θ
a
D34
D35
[pic]
[pic]
r
θ
a
D36
D37
D38
D39
r
θ
a
D40
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