WEEK 10
Posted: Fri Nov 19, 2021 5:56 pm
Trigonometric Ratios of Angles Between 0º and 360º.
The following summary shows how to find the sign and magnitude of the Sine, Cosine, and Tangent of angles in the four quadrants of the Cartesian plane.
First Quadratic 0 < θ < 90º
In the figure above.
Sin θ = + sin θ
Cos θ = + cos θ
Tan θ = +tan θ
2nd quadrant, 90º < θ < 180º
In the figure above.
Sin θ = + sin (180-θ)
Cos θ = - cos (180-θ)
Tan θ = -tan (180-θ)
3rd quadrant, 180º < θ < 270º
In the figure above
Sin θ = - sin (θ - 180)
Cos θ = - cos (θ -180)
Tan θ = + tan (θ - 180).
4th quadratic, 270º < θ<360º .
Sin θ = - Sin (360 - θ)
Cos θ = + Cos (360- θ)
Tan θ = - Tan (360 - θ)
https://youtu.be/Vn4TnlrFpsw
Examples:
Use tables to find the values of the sine, cosine and tangent of the following :
(a) 153º (b)- 120.
(a) Since 153º is in the 2nd quadrant, then
Sin 153º = + Sin (180º - 153º)
= + Sin 270º
= + 0.4540.
Cos 153º = -cos (180º-153º)
Cos 27º
- 0.8910.
Tan 153º = - tan (180º-153º)
- tan 27º
= - 0.5095.
(b) - 210º = 360º + ( -210º)
= 360º - 210º
= 150º
Sin (-210) = sin 150º
= + sin (180º-150º)
= + sin 30º
- + 0.5000. or - 0.5.
cos (-260º) = cos 150º
tan -210º = tan 150º
= tan (180º -159º)
= - tan 30º,
= - 5.774.
2. Find the values of θ lying between 0º and 360º for each of the following:
(a) Cos θ = - 0.3453. (b) Sin θ = 0.4939 (c) Tan θ = - 1.402.
Solution.
(a) Cos θ = 0.3453.
First find the acute angle whose Cosine is 0.3453,
From tables,
0.3453 = cos 69.8º
Since Cos θ is negative θ is in the 2nd or 3rd quadrant.
i.e θ = 180º - 69.8º
θ = 110.2
or
θ-180 = 69.8
θ = 69.8 + 180
θ = 249.8º
(b) Sin θ = - 0.4939
First find the acute angle whose Sinθ is 0.4939.
From tables, 0.4939 = sin 29.6º
Since Sin θ is negative in the 3rd or 4th quadrant,
then θ - 180º = 29.6º
θ = 29. 6º + 180º
θ = 209.6º
or
360 -θ = 29.6º
360 -29.6º = θ
330.4º = θ or θ = 330.4º.
(c ) tan θ = 1.402.
First find the acute angle whose tangent is 1.402. from tables, 1.402 = tan 54.5º
Since tan θ is positive in the 1st or 3rd quadrant, then, θ = 54.5º ( for 1st quad) or
(θ -180º) = 54.5º (for 3rd quad)
θ = 54.5º + 180º
θ = 234.5º
https://youtu.be/tfGeXCFV6tY
EVALUATION.
1 .Use tables to find the values of the following:
(a) tan (-220º)
( b) Sin 239º
( c) cos (-120º)
2. Find the values of θ lying between 0º and 360º for each of the following:
(a) tan θ - 0.5555
(b) cos θ = 0.9703
( c) Sin θ = -0.7314,
ASSIGNMENT
Use table to find the value of Cos ( - 230º)
(a) -0.5 (b) -0.4830 ( c ) -.0.6570 ( d) -0.2700 (e ) -0.6428.
2. Use tables to find tan 273º
(a) 19.08 (b) 14.30 (c) 28.54 (d) -19.08 (e) 57.29.
3. Use tables to solve the following equation correct to the nearest 0.10 0 = 4 - 9 Sin θ.
(a) 26.2º (b) 25.9º ( c) 30.1º (d) 26.4º (e) 30.5º
4. Use tables to find the angles whose Cosine is -0.75.
(a) 130º, 230º (b) 138º,222º (c) 148º, 212º (d) 168º, 192º (e) 200º,160º
5. Find the value of Sin 252º using tables to 2 decimal places:
(a) -0.75 (b) -0.85 (c )- 0.15 9d) 0.68 (e) -.095.
THEORY.
Copy and complete the table below giving the trigonometry ratios correct to 2 decimal places. The first two rows have been done as examples:
Reading Assignment.
NGM SS Bk 1 pg 187 - 192 , Ex 17a Nos 4b, 4d, 1a and 1r pg 189.
GRAPHS OF TRIGONOMETRICAL FUNCTIONS I.E GRAPHS OF SIN θ, COS θ AND TAN θ
The figures above show the development of
(a) the Sine curve (b) the Cosine curve from a unit circle.
Each circle in the figure above has a radius of I unit. The angle θ that the radius OP makes with OX changes as
P moves on the circumference of the circles. Since P is the general point (x,y) and OP = I unit, then:
Sin θ = y
Cos θ = x
Hence the values of x and y gives cos θ and sin θ respectively. These values are used to draw the corresponding sine and cosine curves.
The following points should be noted on the graphs of sin θ and cos θ :
I .AII values of sin θ and cos θ lie between +1 and -1 .
2. The sine and cosine curves have the same wave shape but they start from different points. Sine θ starts from 0 while Cos θ starts from 1 .
3.Each curve is symmetrical about its crest (high point) and trough (low point). Hence, for the values of Sin θ and Cos θ there are usually two corresponding values of θ between 0º and 360º for each of them. The only exceptions to this are at the quarter turns, where sin θ and Cos θ have values as given in the table below:
https://youtu.be/h53862-27lQ
Graph of Tan θ
Values can be taken from a unit circle to draw a tangent curve. In the figure below, a tangent is drawn to the unit circle Ox. A typical radius is drawn and extended to meet the tangent at T. The y- coordinate of T give a measure of Tan θ, where θ is the angle that the radius makes Ox
Note that Tan θ is not defined when θ equals 90º and 270º
https://youtu.be/hIE3NBdkDwE
Evaluation
I .(a) Copy and complete the table below giving values of Sin θ correct to 2 decimal places corresponding to θ = 0º, 12º, 24º...... in intervals of 12º up to 360º. Use tables to find sin θ.
(b)Using scales of2cm to 60º on the θ axis and 10cm to 1 unit on the Sin θ axis, draw the graph of Sin θ.
2(a) Copy and complete the table below giving values of θ correct to 1 decimal place corresponding to θ = 0º, 12º, 24º... ........ in intervals Of 12º up to 360º . Use tables to find tan θ
(b) Using scales of 2cm to 60º on the θ axis and 1cm to 1 unit on the tan θ axis draw the graph of tan θ.
ASSIGNMENT
Draw the graph of y = Sin θ from 0º to 360º with interval of 30º, Use 2cm to represent 0.5 unit on the Sin θ axis and 2cm to represent 60º on the θ axis. Use your graph to find the values of the following:
(a) Sin 294º (b) Sin 78º
( c) Sin 198º (d) sin 326º (e) Sin 162º
THEORY.
(a) Copy and complete the table below giving corresponding values of θ and Cos θ from 0º to 360º.
(b) Draw the graph of cos θ using 2cm to represent 0.5 unit on the cos θ axis and 2cm to represent 60º on the θ axis.
2. Use your graph in question 1 above to find the angle whose cosines are : (a) -.0.15' (b) 0.35.
Reading Assignment
NGM SS Bk I pg 195, Ex 17C Nos 6a, 6c, 2e and 2f pg 194-195.
HARDER PROBLEMS ON TRIGONOMETRIC GRAPHS
Example:
1 (a) Copy and complete the table below giving values of y = I + Cos 2x, correct to one decimal places
(b) Using a scale of 2cm to 30º on the horizontal axis and 2cm to 1 unit on the vertical axis, draw the graph of y = I + cos 2x for θº ≤ x ≤ 360º
(c ) use your graph to solve the following equations. Give your answers to the nearest degree.
(i) I + 2x = 0
(ii) I + cos 0.8.
Solutions.
The table below is the table of values,
(b) the figure below is the graph of y = I + Cos 2x
(c) from the graph:
(i) 90º, 270º
(ii) 51º, 129º, 231º, 309º
2(a) Copy and complete the table below to give values of y = sin 2 θ cos θ
(b) Using a scale of 2cm to 30º on the horizontal axis and 5cm to 1 unit on the vertical axis, draw the graph of y
= sin 2 θ - cos θ for 0º ≤ θº ≤ 180º
( c) Use your graph to find the
(i) Solution of the equation :
Sin 2θ — Cos θ = 0, correct to the nearest degree.
(ii) maximum value of y, correct to one decimal place.
Solution.
(a) The table below is the complete table of values :
( c) (i) From the graph Sin 2 θ — Cos θ when θ = 30º, 90º 150º. (ii) The maximum value of y is 1.
https://youtu.be/WiRVl6bzvF8
EVALUATION.
(a) Copy and complete the table below to give values of y = 7 cos x + 3 Sin x correct to one decimal place.
(b) using a scale of 2cm to 30º on the horizontal axis and 1cm to 1 unit on the vertical axis)
(i) draw the graph of y = 7 Cos x + 3 sin x for θ ≤ x ≤ 210º
( c) use your graph to solve the equation :
7 cos x + 3 Sin x = 0
Correct to the nearest degree
(d) Find the maximum value of y, correct to I decimal place.
ASSIGNMENT.
1. Tan θ is positive and sin θ is negative . In which quadrant does θ lies?
(a) second (b) third only ( c) Third only (d) first and fourth only.
2, Which of the graphs in the figure below represents y = cos x?
I)
II & III)
(a) I only (b) II only (c) III only (d) I and II only . (e) I and III only.
3. Which of these is a graph of sin x.
I&II)
III)
(a) I only (b) 11 only ( c) 111 only (d) l and 111 only (e) and 11 only.
4. Use tables to solve the equation, 1 +2 Sin θ = 2 given that 0º ≤ θ ≤ 360º
(a) 60º and 300º (b) 30º and 150º (c) 30º and 210º (d) 60º and 240º (e) 150º and 330º
5. Use tables to solve the equation: 2.5 —3 Cos θ = 1
Given that 0º ≤ θ ≤ 360
(a) 60º and 300º (b) 150º and 330º (c) 60º and 120º (d) 30º and 150º (e) 60º and 300º
THEORY.
Sketch the following curves for values of θ from 0º to 360º
(a) cos ½h θ (b) 2 Sin θ (c ) - sin θ -1.
2.(a) Copy and complete the tables below to give values of I + 2 Sin θ for 0º ≤ 0 ≤ 360º in intervals of 30º
(b)using scales of 1cm to 30º on the horizontal axis and 2cm to 1 unit on the vertical axis, draw the graph of
1+2 Sine θ
Reading Assignment
NGM SS Bk 3 page 45-51 and Exam Focus pgs 158 -161 ex 6.5 No 12 pgs 160-161
The following summary shows how to find the sign and magnitude of the Sine, Cosine, and Tangent of angles in the four quadrants of the Cartesian plane.
First Quadratic 0 < θ < 90º
In the figure above.
Sin θ = + sin θ
Cos θ = + cos θ
Tan θ = +tan θ
2nd quadrant, 90º < θ < 180º
In the figure above.
Sin θ = + sin (180-θ)
Cos θ = - cos (180-θ)
Tan θ = -tan (180-θ)
3rd quadrant, 180º < θ < 270º
In the figure above
Sin θ = - sin (θ - 180)
Cos θ = - cos (θ -180)
Tan θ = + tan (θ - 180).
4th quadratic, 270º < θ<360º .
Sin θ = - Sin (360 - θ)
Cos θ = + Cos (360- θ)
Tan θ = - Tan (360 - θ)
https://youtu.be/Vn4TnlrFpsw
Examples:
Use tables to find the values of the sine, cosine and tangent of the following :
(a) 153º (b)- 120.
(a) Since 153º is in the 2nd quadrant, then
Sin 153º = + Sin (180º - 153º)
= + Sin 270º
= + 0.4540.
Cos 153º = -cos (180º-153º)
Cos 27º
- 0.8910.
Tan 153º = - tan (180º-153º)
- tan 27º
= - 0.5095.
(b) - 210º = 360º + ( -210º)
= 360º - 210º
= 150º
Sin (-210) = sin 150º
= + sin (180º-150º)
= + sin 30º
- + 0.5000. or - 0.5.
cos (-260º) = cos 150º
tan -210º = tan 150º
= tan (180º -159º)
= - tan 30º,
= - 5.774.
2. Find the values of θ lying between 0º and 360º for each of the following:
(a) Cos θ = - 0.3453. (b) Sin θ = 0.4939 (c) Tan θ = - 1.402.
Solution.
(a) Cos θ = 0.3453.
First find the acute angle whose Cosine is 0.3453,
From tables,
0.3453 = cos 69.8º
Since Cos θ is negative θ is in the 2nd or 3rd quadrant.
i.e θ = 180º - 69.8º
θ = 110.2
or
θ-180 = 69.8
θ = 69.8 + 180
θ = 249.8º
(b) Sin θ = - 0.4939
First find the acute angle whose Sinθ is 0.4939.
From tables, 0.4939 = sin 29.6º
Since Sin θ is negative in the 3rd or 4th quadrant,
then θ - 180º = 29.6º
θ = 29. 6º + 180º
θ = 209.6º
or
360 -θ = 29.6º
360 -29.6º = θ
330.4º = θ or θ = 330.4º.
(c ) tan θ = 1.402.
First find the acute angle whose tangent is 1.402. from tables, 1.402 = tan 54.5º
Since tan θ is positive in the 1st or 3rd quadrant, then, θ = 54.5º ( for 1st quad) or
(θ -180º) = 54.5º (for 3rd quad)
θ = 54.5º + 180º
θ = 234.5º
https://youtu.be/tfGeXCFV6tY
EVALUATION.
1 .Use tables to find the values of the following:
(a) tan (-220º)
( b) Sin 239º
( c) cos (-120º)
2. Find the values of θ lying between 0º and 360º for each of the following:
(a) tan θ - 0.5555
(b) cos θ = 0.9703
( c) Sin θ = -0.7314,
ASSIGNMENT
Use table to find the value of Cos ( - 230º)
(a) -0.5 (b) -0.4830 ( c ) -.0.6570 ( d) -0.2700 (e ) -0.6428.
2. Use tables to find tan 273º
(a) 19.08 (b) 14.30 (c) 28.54 (d) -19.08 (e) 57.29.
3. Use tables to solve the following equation correct to the nearest 0.10 0 = 4 - 9 Sin θ.
(a) 26.2º (b) 25.9º ( c) 30.1º (d) 26.4º (e) 30.5º
4. Use tables to find the angles whose Cosine is -0.75.
(a) 130º, 230º (b) 138º,222º (c) 148º, 212º (d) 168º, 192º (e) 200º,160º
5. Find the value of Sin 252º using tables to 2 decimal places:
(a) -0.75 (b) -0.85 (c )- 0.15 9d) 0.68 (e) -.095.
THEORY.
Copy and complete the table below giving the trigonometry ratios correct to 2 decimal places. The first two rows have been done as examples:
Reading Assignment.
NGM SS Bk 1 pg 187 - 192 , Ex 17a Nos 4b, 4d, 1a and 1r pg 189.
GRAPHS OF TRIGONOMETRICAL FUNCTIONS I.E GRAPHS OF SIN θ, COS θ AND TAN θ
The figures above show the development of
(a) the Sine curve (b) the Cosine curve from a unit circle.
Each circle in the figure above has a radius of I unit. The angle θ that the radius OP makes with OX changes as
P moves on the circumference of the circles. Since P is the general point (x,y) and OP = I unit, then:
Sin θ = y
Cos θ = x
Hence the values of x and y gives cos θ and sin θ respectively. These values are used to draw the corresponding sine and cosine curves.
The following points should be noted on the graphs of sin θ and cos θ :
I .AII values of sin θ and cos θ lie between +1 and -1 .
2. The sine and cosine curves have the same wave shape but they start from different points. Sine θ starts from 0 while Cos θ starts from 1 .
3.Each curve is symmetrical about its crest (high point) and trough (low point). Hence, for the values of Sin θ and Cos θ there are usually two corresponding values of θ between 0º and 360º for each of them. The only exceptions to this are at the quarter turns, where sin θ and Cos θ have values as given in the table below:
https://youtu.be/h53862-27lQ
Graph of Tan θ
Values can be taken from a unit circle to draw a tangent curve. In the figure below, a tangent is drawn to the unit circle Ox. A typical radius is drawn and extended to meet the tangent at T. The y- coordinate of T give a measure of Tan θ, where θ is the angle that the radius makes Ox
Note that Tan θ is not defined when θ equals 90º and 270º
https://youtu.be/hIE3NBdkDwE
Evaluation
I .(a) Copy and complete the table below giving values of Sin θ correct to 2 decimal places corresponding to θ = 0º, 12º, 24º...... in intervals of 12º up to 360º. Use tables to find sin θ.
(b)Using scales of2cm to 60º on the θ axis and 10cm to 1 unit on the Sin θ axis, draw the graph of Sin θ.
2(a) Copy and complete the table below giving values of θ correct to 1 decimal place corresponding to θ = 0º, 12º, 24º... ........ in intervals Of 12º up to 360º . Use tables to find tan θ
(b) Using scales of 2cm to 60º on the θ axis and 1cm to 1 unit on the tan θ axis draw the graph of tan θ.
ASSIGNMENT
Draw the graph of y = Sin θ from 0º to 360º with interval of 30º, Use 2cm to represent 0.5 unit on the Sin θ axis and 2cm to represent 60º on the θ axis. Use your graph to find the values of the following:
(a) Sin 294º (b) Sin 78º
( c) Sin 198º (d) sin 326º (e) Sin 162º
THEORY.
(a) Copy and complete the table below giving corresponding values of θ and Cos θ from 0º to 360º.
(b) Draw the graph of cos θ using 2cm to represent 0.5 unit on the cos θ axis and 2cm to represent 60º on the θ axis.
2. Use your graph in question 1 above to find the angle whose cosines are : (a) -.0.15' (b) 0.35.
Reading Assignment
NGM SS Bk I pg 195, Ex 17C Nos 6a, 6c, 2e and 2f pg 194-195.
HARDER PROBLEMS ON TRIGONOMETRIC GRAPHS
Example:
1 (a) Copy and complete the table below giving values of y = I + Cos 2x, correct to one decimal places
(b) Using a scale of 2cm to 30º on the horizontal axis and 2cm to 1 unit on the vertical axis, draw the graph of y = I + cos 2x for θº ≤ x ≤ 360º
(c ) use your graph to solve the following equations. Give your answers to the nearest degree.
(i) I + 2x = 0
(ii) I + cos 0.8.
Solutions.
The table below is the table of values,
(b) the figure below is the graph of y = I + Cos 2x
(c) from the graph:
(i) 90º, 270º
(ii) 51º, 129º, 231º, 309º
2(a) Copy and complete the table below to give values of y = sin 2 θ cos θ
(b) Using a scale of 2cm to 30º on the horizontal axis and 5cm to 1 unit on the vertical axis, draw the graph of y
= sin 2 θ - cos θ for 0º ≤ θº ≤ 180º
( c) Use your graph to find the
(i) Solution of the equation :
Sin 2θ — Cos θ = 0, correct to the nearest degree.
(ii) maximum value of y, correct to one decimal place.
Solution.
(a) The table below is the complete table of values :
( c) (i) From the graph Sin 2 θ — Cos θ when θ = 30º, 90º 150º. (ii) The maximum value of y is 1.
https://youtu.be/WiRVl6bzvF8
EVALUATION.
(a) Copy and complete the table below to give values of y = 7 cos x + 3 Sin x correct to one decimal place.
(b) using a scale of 2cm to 30º on the horizontal axis and 1cm to 1 unit on the vertical axis)
(i) draw the graph of y = 7 Cos x + 3 sin x for θ ≤ x ≤ 210º
( c) use your graph to solve the equation :
7 cos x + 3 Sin x = 0
Correct to the nearest degree
(d) Find the maximum value of y, correct to I decimal place.
ASSIGNMENT.
1. Tan θ is positive and sin θ is negative . In which quadrant does θ lies?
(a) second (b) third only ( c) Third only (d) first and fourth only.
2, Which of the graphs in the figure below represents y = cos x?
I)
II & III)
(a) I only (b) II only (c) III only (d) I and II only . (e) I and III only.
3. Which of these is a graph of sin x.
I&II)
III)
(a) I only (b) 11 only ( c) 111 only (d) l and 111 only (e) and 11 only.
4. Use tables to solve the equation, 1 +2 Sin θ = 2 given that 0º ≤ θ ≤ 360º
(a) 60º and 300º (b) 30º and 150º (c) 30º and 210º (d) 60º and 240º (e) 150º and 330º
5. Use tables to solve the equation: 2.5 —3 Cos θ = 1
Given that 0º ≤ θ ≤ 360
(a) 60º and 300º (b) 150º and 330º (c) 60º and 120º (d) 30º and 150º (e) 60º and 300º
THEORY.
Sketch the following curves for values of θ from 0º to 360º
(a) cos ½h θ (b) 2 Sin θ (c ) - sin θ -1.
2.(a) Copy and complete the tables below to give values of I + 2 Sin θ for 0º ≤ 0 ≤ 360º in intervals of 30º
(b)using scales of 1cm to 30º on the horizontal axis and 2cm to 1 unit on the vertical axis, draw the graph of
1+2 Sine θ
Reading Assignment
NGM SS Bk 3 page 45-51 and Exam Focus pgs 158 -161 ex 6.5 No 12 pgs 160-161