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Trigonometric Functions (including applications of trigonometric ratios)

(13.1 – 13.7)

13.1 Circular measure of angles. Angle, arc, sector.

13.2 The functions sin x, cos x, tan x, cosec x, sec x, cot x and their graphs.

13.3 Periodicity and other simple properties of the functions sin x, cos x and tan x.

13.4 Approximations to sin x, cos x, tan x when x is small. The result = 1

13.5 Differentiation of cos x, sin x, tan x.

13.6 Primitive functions of sin x, cos x, sec²x.

13.7 Extension of 13.2 – 13.6 to functions of the form a sin(bx + c), etc.” (syllabus)

For a more detailed description of the requirements for this topic, see the mathematics syllabus on the Board of Studies website click here

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RADIANS

Just as people speak different languages, so too there are different languages when it comes to angular measurement.

The radian is related to the radius of a circle.

The radian is defined as the angle subtended at the centre of a circle of an arc of length 1 radius.

You are aware that the circumference of a circle has the formula C = 2pr

From this it can be seen that there are 2p radians in a complete revolution.

The conversion from radians to degrees and vice-versa is best understood by considering 1 revolution.

1 revolution = 2p radians = 360^o

To convert radians to degrees multiply by or

To convert degrees to radians multiply by or

Example 1: Convert 140^o to radians.

Answer 1: 140^o = revolution = x 2p radians = 2.44 radians.

Example 2: Convert 5 radians to degrees.

Answer 2: 5 radians = revolution = x 360 degrees = 286.5 degrees.

Exercise 1: Convert the following angles from radians to degrees.

(i) p radians (ii) radians (iii) 3p radians (iv) radians (v) radians

(vi) 1 radian (vii) 2.5 radians (viii) 8 radians (ix) 3.2 radians (x) 7.6 radians

Answers 1:

(i) 180^o (ii) 30^o (iii) 540^o (iv) 45^o (v) 154.3^o

(vi) 57.3^o (vii) 143.2^o (viii) 458.4^o (ix) 183.3^o (x) 435.4^o

Exercise 2: Convert the following angles from degrees to radians.

(i) 90^o (ii) 60^o (iii) 300^o (iv) 315⁰ (v) 450^o

(vi) 74^o (vii) 157^o (viii) 258^o (ix) 380^o (x) 215^o

Answers 2:

(i) = 1.57 rad. (ii) = 1.05 rad. (iii) = 5.24 rad. (iv) = 5.5 rad. (v) = 7.85 rad (vi) 1.29 rad. (vii) 2.74 rad. (viii) 4.5 rad. (ix) 6.63 rad. (x) 3.75 rad.

Arcs & Sectors

An arc is a part of the circumference of a circle.

The length of an arc is given by: arc = rq where r is the radius of the circle and q is the angle in radians subtended by the arc.

Example 1: What is the length of an arc that subtends an angle of 2 radians at the centre of a circle of radius 20 cm?

Answer 1: arc = rq = 20 x 2 = 40 cm.

Example 2: An arc 30 cm long subtends an angle of radians at the centre of a circle. What is the radius of the circle?

Answer 2: = =38.2 cm.

A sector is a portion of a circle subtended by two radii.

The area of a circle is given by A = pr² and the fraction of a circle occupied by a sector is so the area of a sector is given by x pr² = r²q A = r²q

Example 3: Find the area of a sector that subtends an angle of radians at the centre of a circle of radius 20 cm.

Answer 3: A = r²q = 20² x = = 100p cm²

Exercise:

1. What is the length of an arc that subtends an angle of 2.5 radians at the centre of a circle of radius 12 cm?

2. An arc 30 mm long subtends an angle of radians at the centre of a circle. What is the radius of the circle?

3. Find the area of a sector that subtends an angle of radians at the centre of a circle of radius 15 cm.

Answers: 1. 30 cm. 2.= 57.3 mm. 3. 75p = 235.6 cm²

TABLES OF TRIGONOMETRIC VALUES

You will use these values in the exercises that follow.

1. Complete the following table. Give all values to two decimal places:

q^o	q radians in terms of p	Sinq	Cosq	Tanq
0
30
45
60
90
120
135
150
180
210
225
240
270
300
315
330
360

3. Complete the following table. Give values to the nearest whole number:

q^o	Tan q
89^o59’
90^o01’
269^o59’
270^o01’

Table 1.

q^o	q radians	Sinq	Cosq	Tanq
0	0	0.00	1.00	0.00
30	p/6	0.50	0.87	0.58
45	p/4	0.71	0.71	1.00
60	p/3	0.87	0.50	1.73
90	p/2	1.00	0.00	N/A
120	2p/3	0.87	-0.50	-1.73
135	3p/4	0.71	-0.71	-1.00
150	5p/6	0.50	-0.87	-0.58
180	p	0.00	-1.00	0.00
210	7p/6	-0.50	-0.87	0.58
225	5p/4	-0.71	-0.71	1.00
240	4p/3	-0.87	-0.50	1.73
270	3p/2	-1.00	0.00	N/A
300	5p/3	-0.87	0.50	-1.73
315	7p/4	-0.71	0.71	-1.00
330	11p/6	-0.50	0.87	-0.58
360	2p	0.00	1.00	0.00

Table 2.

q^o	Tan q
89^o59’	3438
90^o01’	-3438
269^o59’	3438
270^o01’	-3438

TRIGONOMETRY: GRAPHS OF SINE FUNCTIONS

Q.1. On the same axes, draw the graphs of y = sinq and y = -sinq.

Describe the effect of the minus sign in front of the sinq.

Q.2. On the same axes, draw the graphs of y = sinq and y = sin2q.

Describe the effect of the coefficient of q.

Q.3. On the same axes, draw the graphs of y = sinq, y = 2sinq and y = 3sinq.

Describe the effect of the coefficient of sinq.

Answers:

1.

The – sign causes the y = sinq graph to be reflected about the q axis. The graphs y = sinq and y = -sinq are reflections of each other.

2.

The coefficient of q reduces the period of the graph. In this case, the coefficient 2 reduces the period to ½ of its original value. A graph of y = sin aq would have a period of 1/a that of y = sinq.

3.

The coefficient of sinq changes the amplitude (height) of the graph. A graph of y = asinq will have an amplitude of a times that of y = sinq.

TRIGONOMETRY: GRAPHS OF COSINE FUNCTIONS

Q.1. On the same axes, draw the graphs of y = cosq and y = -cosq.

Describe the effect of the minus sign in front of the cosq.

Q.2. On the same axes, draw the graphs of y = cosq and y = cos2q.

Describe the effect of the coefficient of q.

Q.3. On the same axes, draw the graphs of y = cosq, y = 2cosq and y = 3cosq.

Describe the effect of the coefficient of cosq.

Answers:

1.

The – sign causes the y = cosq graph to be reflected about the q axis. The graphs y = cosq and y = -cosq are reflections of each other.

2.

The coefficient of q reduces the period of the graph. In this case, the coefficient 2 reduces the period to ½ of its original value. A graph of y = cos aq would have a period of 1/a that of y = cosq.

3.

The coefficient of cosq changes the amplitude (height) of the graph. A graph of y = acosq will have an amplitude of a times that of y = cosq.

Differentiating Trigonometric Functions

If y = a sin bx then = ab cos bx

If y = a cos bx then = - ab sin bx

If y = a tan bx then = ab sec² bx

Examples: Differentiate the following

(i) 7sin 5x (ii) 8cos 3x (iii) 5tan 4x

Answers: (i) 35cos5x (ii) -24sin 3x (iii) 20sec²4x

Exercise 1: Differentiate the following functions:

(i) y = sin x (ii) y = 2 cos x (iii) y = 4 cos 2x (iv) y = tan 3x (v) y = 3 sin 5x (vi) y = 4 tan 6x (vii) y = 8 sin 3x

(viii) y = (ix) y =

(x) y = (xi) y = (xii) y =

Answers 1:

(i) cos x (ii) -2 sin x (iii) -8 cos 2x (iv) 3 sec² 3x (v) 15 cos 5x (vi) 24 sec²6x (vii) 24 cos 3x (viii) – ½ sin x

(ix) 4 sec²3x (x) 5 cos 2x (xi) -9 sin 6x (xii) –sin 3x

Example: Find the equations of the tangent and the normal to the curve y = sin 2x at the point where x =

Answer: When x = y = sin = sin = 1 tangent at the point (, 1)

y = sin 2x y’ = 2cos 2x

when x = , y’ = 2cos = 2cos= 0 (since cos = 0)

Hence equation of tangent is y = 0x + b and substituting (, 1) gives b = 1

So the equation of the tangent is y = 1.

Gradient of normal is -1

Equation of normal is y = -x + b

Substituting (, 1) gives 1 = -+ b b = 1 +

So the equation of the normal is y = -x + 1 + and when y = 1 x = .

Tangent y = 1, Normal x =

Exercise: Find the equations of the tangent and the normal to the curve y = 2cos 4x at the point where x =

Answer: Tangent y = -2, Normal x =