1.6 Trig Functions

1.6 Trig Functions

The Mean Streak, Cedar Point Amusement Park, Sandusky, OH

Trigonometry Review

(I) Introduction

By convention, angles are measured from the initial line or the x-axis with respect to the origin.

If OP is rotated counter-clockwisefrom the x-axis, the angle so formed is positive.

But if OP is rotated clockwisefrom the x-axis, the angle so formed is negative.

O

P

xnegative angle

P

O xpositive angle

(II) Degrees & Radians

Angles are measured in degrees or radians.

rr

r1c

Given a circle with radius r, the angle subtended by an arc of length r measures 1 radian.

Care with calculator! Make sure your calculator is set to radians when you are making radian calculations.

180rad

(III) Definition of trigonometric ratios

r

y

hyp

oppsin

r

x

hyp

adjcos

x

y

adj

opptan

cos

sin

sin

1 cosec

cos

1sec

sin

cos

tan

1cot

x

y P(x, y)

r y

x

Note:

1sin

sin

1

Do not write cos1tan1

From the above definitions, the signs of sin , cos & tan in different quadrants can be obtained. These are represented in the following diagram:

All +ve sin +ve

tan +ve

1st2nd

3rd 4th

cos +ve

What are special angles?

(IV) Trigonometrical ratios of special angles

Trigonometrical ratios of these angles are worth exploring

,...,2

,3

,4

30o, 45o, 60o, 90o, …

1

00sin

0sin

12

sin

02sin

12

3sin

sin 0° 0

sin 360° 0sin 180° 0

sin 90° 1 sin 270° 1

xy sin

0 2

23

1

10cos 1cos

02

cos

12cos

02

3cos

cos 0° 1

cos 360° 1

cos 180° 1

cos 90° 0cos 270°

1xy cos

0 2

23

1

00tan 0tan

undefined. is 2

tan

02tan

undefined. is 2

3tan

tan 180° 0

tan 0° 0

tan 90° is undefined tan 270° is undefined

tan 360° 0

xy tan

0 2

23

Using the equilateral triangle (of side length 2 units) shown on the right, the following exact values can be found.

2

3

3sin60sin

2

3

6cos30cos

2

1

6sin30sin

2

1

3cos60cos

33

tan60tan

3

1

6tan30tan

2

2

2

1

4sin45sin

2

2

2

1

4cos45cos

14

tan45tan

Complete the table. What do you observe?

2nd quadrant sin)sin(

cos)cos(

tan)tan(

Important properties:Important properties:

3rd quadrant sin)sin(

cos)cos(

tan)tan(

1st quadrant sin)2sin(

cos)2cos(

tan)2tan(

or 2

Important properties:Important properties:

4th quadrant

sin)2sin( cos)2cos(

tan)2tan(or

or 2

sin)sin( cos)cos(

tan)tan(In the diagram, is acute. However, these relationships are true for all sizes of

Complementary angles

E.g.: 30° & 60° are complementary angles.

Two angles that sum up to 90° or radians are called complementary angles.

2

2

and are complementary angles.

Recall:

2

160cos30sin

2

3

6cos

3sin

3

160cot30tan 330cot60tan

Principal Angle & Principal Range

Example: sinθ = 0.5

2

2

Principal range

Restricting y= sinθ inside the principal range makes it a one-one function, i.e. so that a unique θ= sin-1y exists

Example: sin . Solve for θ if 2

1)

2

3( 0

4

Basic angle, α =

Since sin is positive, it is in the 1st or 2nd quadrant )2

3(

42

3

42

3 or

Therefore

4

3)(

4

5 orleinadmissib

Hence, 4

3

ry

xA

O

P(x, y)By Pythagoras’ Theorem,

222 ryx

122

r

y

r

x

(VI) 3 Important Identities

sin2 A cos2 A 1

r

xA cos

r

yA sinSince and ,

1cossin 22 AA Note:

sin 2 A (sin A)2 cos 2 A (cos A)2

A2cos

1

(1) sin2 A + cos2 A 1

(2) tan2 A +1 sec2 A

(3) 1 + cot2 A csc2 A

tan 2 x = (tan x)2

(VI) 3 Important Identities

Dividing (1) throughout by cos2 A,

Dividing (1) throughout by sin2 A,

2)(sec A

2

cos

1

A

A2sec

(VII) Important Formulae

(1) Compound Angle Formulae

BABABA sincoscossin)sin( BABABA sincoscossin)sin(

BABABA sinsincoscos)cos( BABABA sinsincoscos)cos(

BA

BABA

tantan1

tantan)tan(

BA

BABA

tantan1

tantan)tan(

E.g. 4: It is given that tan A = 3. Find, without using calculator,(i) the exact value of tan , given that tan ( + A) = 5;(ii) the exact value of tan , given that sin ( + A) = 2 cos ( – A)

Solution:

(i) Given tan ( + A) 5 and tan A 3,

tan31

3tan5

3tantan155

8

1tan

A

AA

tantan1

tantan)tan(

(2) Double Angle Formulae

(i) sin 2A = 2 sin A cos A

(ii) cos 2A = cos2 A – sin2 A

= 2 cos2 A – 1

= 1 – 2 sin2 A

(iii)

A

AA

2tan1

tan22tan

Proof:

)sin(

2sin

AA

A

AAAA sincoscossin

AAcossin2

)cos(2cos AAA

AA 22 sincos

)cos1(cos 22 AA

1cos2 2 A

Trigonometric functions are used extensively in calculus.

When you use trig functions in calculus, you must use radian measure for the angles. The best plan is to set the calculator mode to radians and use when you need to use degrees.

2nd o

If you want to brush up on trig functions, they are graphed on page 41.

Even and Odd Trig Functions:

“Even” functions behave like polynomials with even exponents, in that when you change the sign of x, the y value doesn’t change.

Cosine is an even function because: cos cos

Secant is also an even function, because it is the reciprocal of cosine.

Even functions are symmetric about the y - axis.

Even and Odd Trig Functions:

“Odd” functions behave like polynomials with odd exponents, in that when you change the sign of x, the

sign of the y value also changes.

Sine is an odd function because: sin sin

Cosecant, tangent and cotangent are also odd, because their formulas contain the sine function.

Odd functions have origin symmetry.

The rules for shifting, stretching, shrinking, and reflecting the graph of a function apply to trigonometric functions.

y a f b x c d

Vertical stretch or shrink;reflection about x-axis

Horizontal stretch or shrink;reflection about y-axis

Horizontal shift

Vertical shift

Positive c moves left.

Positive d moves up.

The horizontal changes happen in the opposite direction to what you might expect.

is a stretch.1a

is a shrink.1b

When we apply these rules to sine and cosine, we use some different terms.

2sinf x A x C D

B

Horizontal shift

Vertical shift

is the amplitude.A

is the period.B

A

B

C

D 21.5sin 1 2

4y x

The sine equation is built into the TI-89 as a sinusoidal regression equation.

For practice, we will find the sinusoidal equation for the tuning fork data on page 45. To save time, we will use only five points instead of all the data.

Time: .00108 .00198 .00289 .00379 .00471 Pressure: .200 .771 -.309 .480 .581

.00108,.00198,.00289,.00379,.00471 L1 ENTER

2nd { .00108,.00198,.00289,.00379,.00471 2nd }

STO alpha L 1 ENTER

.2,.771, .309,.48,.581 L2 ENTER

SinReg L1, L2 ENTER

2nd MATH 6 3

Statistics Regressions

9

SinReg

alpha L 1 alpha L 2 ENTER

DoneThe calculator should return:

,

Tuning Fork Data

ShowStat ENTER

2nd MATH 6 8

Statistics ShowStat

ENTER

The calculator gives you an equation and constants: siny a b x c d

.608

2480

2.779

.268

a

b

c

d

2nd MATH 6 3

Statistics Regressions

9

SinReg

alpha L 1 alpha L 2 ENTER

DoneThe calculator should return:

,

ExpReg L1, L2 ENTER

We can use the calculator to plot the new curve along with the original points:

Y= y1=regeq(x)

2nd VAR-LINK regeq

x )

Plot 1 ENTER

ENTER

WINDOW

Plot 1 ENTER

ENTER

WINDOW

GRAPH

WINDOW

GRAPH

You could use the “trace” function to investigate the pressure at any given time.

2 3

2

2

2

3

2

2

Trig functions are not one-to-one.

However, the domain can be restricted for trig functions to make them one-to-one.

These restricted trig functions have inverses.

Inverse trig functions and their restricted domains and ranges are defined on page 47.

siny x