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