Year 10 Circular Functions 1
CIRCULAR FUNCTIONS
Contents:
2. Special Angles and exercises
3. Plotting y = sin q
4. Plotting y = cos q
5. Plotting y = tan q
6 Transformations
7,8 Exercises in sketching circular functions with transformations,
amplitude and period
9,10 Determining equations, amplitude and period from graphs
11 Unit Circle – converting angles to first quadrant
12. Solving for all angles
13. Exercises in solving trig functions
Areas not covered:
Radian measure
Transformations
- Horizontal Reflection
- Horizontal Translation
Solving functions of the type y = sin 3q
CIRCULAR FUNCTIONS
Special Angles
These angles are special because 45o is exactly in the middle of 90o
and 30o and 60o are at one-third and two-thirds of 90o.
Although we can look up all of the values for sin q, cos q and tan q on the calculator,
in mathematics it is always preferable to use exact values. Because the special angles are so important we often use a table giving their trigonometric ratios in exact value.
1. Remember, the calculator normally gives the answer as a decimal,
which is an approximation of the actual answer.
2. Whole numbers, fractions and surds are examples of exact values.
degrees / radians / dec / exact / degrees / radians / dec / exact / degrees / radians / dec / exactsin 0o / sin 0 / 0 / 0 / cos 0o / cos 0 / 1 / 1 / tan 0o / tan 0 / 0 / 0
sin 30o / sin / 0.5 / / cos 30o / cos / 0.8660 / / tan 30o / tan / 0.5771 /
sin 45o / sin / 0.7071 / / cos 45o / cos / 0.7071 / / tan 45o / tan / 1 / 1
sin 60o / sin / 0.8660 / / cos 60o / cos / 0.5 / / tan 60o / tan / 1.732 /
sin 90o / sin / 1 / 1 / cos 90o / cos / 0 / 0 / tan 90o / tan / und / und
Give the exact value of the following:
sin 30o = tan 45o =
cos 45o = tan 60o =
sin 60o cos 90o =
tan 90o = sin 0o =
sin 45o = cos 30o =
tan 30o = cos 60o =
CIRCULAR FUNCTIONS
Circular functions include the equations y = sin q, y = cos q and y = tan q.
These are called circular functions because they can complete a full circle (360o)
1. Use your calculator to look up the value of sin q for each of the following angles and hence complete the table below.
y = sin q
q / 0o / 30o / 45o / 60o / 90o / 120o / 135o / 150o / 180oy
210o / 225o / 240o / 270o / 300o / 315o / 330o / 360o / 390o / 405o / 420o / 450o
2. Using the information from the table, plot the graph of y = sin q for {q: 0o £ q £ 360o}
on the set of axes below.
y = sin q
What happens to the graph after q = 360o ?
What is the maximum value for sin q ?
At what value of q does the maximum occur ?
What is the minimum value for sin q ?
At what value of q does the minimum occur ?
For which values of q does sin q equal zero ?
These are the 6 special points for sketching the graph of y = sin q.
CIRCULAR FUNCTIONS
3. Use your calculator to look up the value of cos q for each of the following angles and hence complete the table below.
y = cos q
q / 0o / 30o / 45o / 60o / 90o / 120o / 135o / 150o / 180oy
210o / 225o / 240o / 270o / 300o / 315o / 330o / 360o / 390o / 405o / 420o / 450o
2. Using the information from the table, plot the graph of y = cos q for {q: 0o £ q £ 360o}
on the set of axes below.
y = cos q
What happens to the graph after q = 360o ?
What is the maximum value for cos q ?
At what value of q does the maximum occur ?
What is the minimum value for cos q ?
At what value of q does the minimum occur ?
For which values of q does cos q equal zero ?
These are the 5 special points for sketching the graph of y = cos q.
CIRCULAR FUNCTIONS
4. Use your calculator to look up the value of cos q for each of the following angles and hence complete the table below.
y = tan q
q / 0o / 30o / 45o / 60o / 90o / 120o / 135o / 150o / 180oy
210o / 225o / 240o / 270o / 300o / 315o / 330o / 360o / 390o / 405o / 420o / 450o
2. Using the information from the table, plot the graph of y = tan q for {q: 0o £ q £ 360o}
on the set of axes below.
y = tan q
What happens to the graph after q = 180o ?
At what values of q does y = 1 ?
At what values of q does y = -1?
For which values of q does tan q equal zero ?
What is the value of tan q at q = 90o and q = 270o
What is the name of feature of the graph at y = 90o and 270o ?
These are the special features for sketching the graph of y = tan q.
Transformations
1. Vertical Reflection
- a negative sign before the function inverts the graph (top for bottom).
eg y = - sin q y
1
q
-1
2. Vertical Translation
- the number added on (or subtracted) slides the graph up (or down)
eg y = sin q - 3
y
q
180o 360o
-3
3. Vertical Dilation
- the number before the function stretches (or compresses) the graph vertically.
eg y = 3 tan q y
15
180o 3600 q
-15
4. Horizontal Dilation
- the number between the function and the angle (q) compresses (or stretches) the graph horizontally
eg y = sin 3q
y1
/ q
/ -1
1. Sketch the graphs of each of the following for{q : 0o q 360o}. Give the amplitude and period in
each case:
a. y = sin q b. y = cos q
c. y = tan q d. y = 3 sin q
e. y = - cos q f. y = -4 tan q
g. y = sin 3q h. y = - cos 2q
i. y = 4 cos q j. y = sin q - 2
k. y = 3 sin q + 2 l. y = -2 cos 3q + 1
3. By first finding the amplitude and period, give the equation for each of the following circular
functions:
y
3
0 q
360o
-3 y =
y
5
0 q
120o
-5 y =
y
2
0 q
180o
-2 y =
-4
UNIT CIRCLE
In the Second Quadrant
sin q = sin (180o - q) - still positive because sine is positive
in the second quadrant.
cos q = - cos (180o - q) - negative because cosine is negative
in the second quadrant.
tan q = - tan (180o - q) - negative because tangent is negative
in the second quadrant.
For example:
sin 150o = sin (180o – 150o) cos 150o = - cos (180o – 150o) tan 150o = tan (180o - 150o)
= sin 30o = - cos 30o = - tan 30o
= 0.5 = - 0.866 = - 0.577
Converting to First Quadrant Angles
180o - q Sin All q
2 1
3 4
q - 180o Tan Cos 360o - q
Similarly, in the Third Quadrant:
sin q = - sin (q - 180o) - negative because sine is negative
in the third quadrant.
cos q = - cos (q - 180o) - negative because cosine is negative
in the third quadrant.
tan q = tan (q - 180o) - still positive because tangent is positive
in the third quadrant.
and, in the Fourth Quadrant
sin q = - sin (360o - q) - negative because sine is negative
in the fourth quadrant.
cos q = cos (360o - q) - still positive because cosine is positive
in the fourth quadrant.
tan q = - tan (360o - q) - negative because tangent is negative
in the fourth quadrant.
Converting Angles
180o - q Sin All q
2 1
3 4
(to 1st) q - 180o Tan Cos 360o - q
(from 1st) 180o + q
1. Evaluating circular functions between 0o and 360o
Evaluate the following by first converting to a first quadrant angle:
(a) sin 330o (b) cos 330o (c) tan 330o
(d) sin 225o (e) cos 135o (f) tan 160o
(g) sin 115o (h) cos 325o (i) tan 330o
Solving for all angles between 0o and 360o - ie for {q : 0o q 360o}
eg1.
cos q = 0.866 (2 soln, 1st and 4th) identify correct quadrants
q = 30o look up cos-1 (0.866) on calculator to find 1st quad angle
q = 30o, (360o – 30o) use the rule to find 4th quadrant angle
\q = 30o, 330o
eg2.
tan q = - 3.732 (2 soln, 2,4) tan-1 (3.732) ignore the negative sign
[q = 75o] first quadrant angle
q = (180o – 75o), (360o – 75o) use the rule to find 2nd & 4th quad angles
q = 105o, 285o
Year 10 CIRCULAR FUNCTIONS
1. Will each of the following have a negative or positive (- ve or +ve) value?
a. sin 50o b. sin 150o c. sin 250o
d. cos 330o e. cos 230o f. cos 130o
g. tan 120o h. tan 220o i. tan 320o
2. Give the exact value of each of the following:
a. sin 30o b. sin 150o c. sin 225o
d. cos 330o e. cos 210o f. cos 150o
g. tan 120o h. tan 225o i. tan 330o
3. Find all of the values for q for {q : 0o q 360o} in each of the following
(Draw a small sketch diagram in each case to demonstrate the solutions):
(a) sin q = 0.5 (b) sin q = - 0.5 (c) cos q = 0.5
(d) sin q = 0.866 (e) cos q = - 0.5 (f) tan q = 1
(g) sin q = - 0.75 (h) cos q = 1 (i) tan q = - 0.5771
4. Find all of the values for q for {q : 0o q 360o} in each of the following
(a) 2 sin q = 1 (b) sin q = - 0.5 (c) 2 cos q =
(d) 2 sin q = (e) 2 cos q + = 0 (f) tan q + 1 = 0
(g) sin 2q = (h) 2 cos 2q =