chapter 5 trigonometry

FHMM1014 Mathematics I

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Centre For Foundation Studies

Department of Sciences and Engineering

Chapter 5

Trigonometry

FHMM1024 Calculus

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Trigonometric Functions of Angles

Evaluating Trigonometric Functions for all Angles

Trigonometric Identities

Area of Triangles

The Law of Sines

The Ambiguous Case

The Law of Cosines

Simplifying Trigonometric Expression

Proving Trigonometric Identities

Topics


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Addition and Subtraction Formulas

Double Angle Formulas

Half Angle Formulas

Sum to Product Formulas

Solving Trigonometric Equations

Equations involving Trigonometric functions of Multiple Angles

Expression of the form A sin x + B cos x

Topics


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Trigonometric Functions of angles


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Trigonometric Functions of Angles
Using the Pythagorean Theorem, we see that the
hypotenuse has length
Thus,The other trigonometric
ratios can be found
in the same way.

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Fundamental Identities


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Reciprocal Identities
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Trigonometric Functions
If
is the distance
from the origin
to the point P(x, y),
then


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Trigonometric Functions

We move in
a counterclockwise
direction if t is positive
and in a clockwise
direction if t is negative.


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Even-Odd Properties
Sine, cosecant, tangent, and cotangent are odd functions.Cosine and secant are even functions.

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Use the even-odd properties of the trigonometric functions to determine each value.

Example 1


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Cofunction Identities


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Cofunction Identities
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Evaluating Trigonometric Functions for All Angles


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From the definition, we see the values of
the trigonometric functions are all positive if
the angle has its terminal side in quadrant I.
This is because x and y are positive
in this quadrant.Of course, r is always positivesince it is simply the distance from the origin to the point P(x, y).

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However, if the terminal side of is in quadrant II, x is negative and y is positive. Thus, in quadrant II, the functions sin and csc are positive, and all the other trigonometric functions have negative values. You can check the other entries in the following table.


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Signs of the Trigonometric Functions


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QuadrantPositive FunctionsNegative FunctionsIallNoneIIsin, csccos, sec, tan, cotIIItan, cotsin, csc, cos, secIVcos, secsin, csc, tan, cot
FHMM1024 Calculus

This mnemonic device will help you
remember which trigonometric functions are

positive in each quadrant:
All of them, Sine, Tangent, or Cosine
You can remember this as:
All Students Take Calculus.
Signs of the Trigonometric Functions


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Find:

cos 135

tan 390

Example 2


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The trigonometric functions of angles are related to each other through
several important equations called trigonometric identities.


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Fundamental Identities


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Pythagorean Identities
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(a) Express sin in terms of cos .

(b) Express tan in terms of sin , where is in quadrant II.

Example 3


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If tan = and is in quadrant III,
find cos .

Example 4


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If sec = 2 and is in quadrant IV,
find the other five trigonometric functions of .

Example 5


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Area of Triangles


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Areas of Triangles

If is an acute angle, the height of the triangle in the figure is given by h = b sin .

Thus, the area is:
A = x base x height
= ab sin


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Areas of Triangles
If the angle is not acute, from the figure, we see that the height of the triangle is:
h = b sin(180 ) = b sin This is so because
the reference
angle of is the
angle 180 .

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Areas of Triangles

Thus, in this case also, the area of
the triangle is:
A = x base x height
= ab sin


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Area of a Triangle

The area A of a triangle with sides
of lengths a and b and with included
angle is:


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Find the area
of triangle ABC
shown here.

Example 6


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Herons Formula

The area A of triangle ABC is given by

where s = (a + b + c) is the semiperimeter of the trianglethat is, s is half the perimeter.


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Example 7

A businessman wishes to buy a triangular lot in a busy downtown location.

The lot frontages on
the three adjacent streets
are 125, 280, and 315 ft.

Find the area of the lot.


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The Law of Sines


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Introduction

The trigonometric functions can also be used
to solve oblique trianglestriangles with no right angles.

To do this, we first study the Law of Sines here and then the Law of Cosines in the next section.


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Introduction

To state these laws (or formulas) more easily, we follow the convention of labeling:

- The angles of a triangle as A, B, C.

- The lengths of the corresponding opposite
sides as a, b, c.


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Solving a Triangle

To solve a triangle, we need to know certain information about its sides and angles.
To decide whether we have enough information, its often helpful to make a sketch.

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Two Angles and Included Side

For instance, if we are given two angles and the included side, then its clear that one and only one triangle can be formed.


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Two Sides and Included Angle

Similarly, if two sides and the included angle are known, then a unique triangle
is determined.


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Three Angles and No Sides

However, if we know all three angles and no sides, we cannot uniquely determine the triangle.
Many triangles can have the same three angles.All these triangles would be similar, of course.So, we wont
consider this case.

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Determining a Triangle

In general, a triangle is determined by three of its six parts (angles and sides)
as long as at least one of these three parts is a side.
So, the possibilities are as follows.

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CaseAngles and SidesAbbrvn.1One side and two anglesASA/SAA2Two sides and the angle opposite one of those sidesSSA3Two sides and the included angleSAS4Three sidesSSS
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Cases 1 and 2 are solved using the Law
of Sines. Cases 3 and 4 require the Law of
Cosines.


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Law of Sines

The law says that, in any triangle, the
lengths of the sides are proportional to the sines of the corresponding opposite angles.
In triangle ABC, we have:

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A satellite orbiting the earth passes directly overhead at observation stations in Phoenix and Los Angeles, 340 mi apart.At an instant when it is
between these two stations,
its angle of elevation is
simultaneously observed
to be 60 at Phoenix and
75 at Los Angeles.
Example 8


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How far is the satellite from Los Angeles?

Example 9


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Solve the triangle in the figure.

Example 10


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Example 10(a)

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The Ambiguous Case


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The Ambiguous Case

In Examples 7 and 8, a unique
triangle was determined by the information given.
This is always true of Case 1 (ASA or SAA).

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The Ambiguous Case

However, in Case 2 (SSA), there may be
two triangles, one triangle, or no triangle with the given properties.
For this reason, Case 2 is sometimes
called the ambiguous case.

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The Ambiguous Case

To see why this is so, here we show
the possibilities when angle A and sides a and b are given.


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The Ambiguous Case
In part (a), no solution is possible.Side a is too short to complete the triangle. In part (b),
the solution is
a right triangle.

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The Ambiguous Case
In part (c), two solutions are possible.In part (d), there is a unique triangle
with the given properties.We illustrate
the possibilities
of Case 2 in
the following
examples.

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Solve triangle ABC, where:

Example 11


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The Ambiguous Case

In Example 9, there were two possibilities for angle B.
One of these was not compatible
with the rest of the information.

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The Ambiguous Case
In general, if sin A < 1, we must
check the angle and its supplementary value as possibilities.This is because any angle smaller than 180
can be in the triangle.To decide whether either possibility works,
we check to see whether the resulting sum
of the angles exceeds 180.

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The Ambiguous Case

It can happen that both possibilities are compatible with the given information.
In that case, two different
triangles are solutions to
the problem.

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Solve triangle ABC if:

Example 12


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The No-Solution Case

The next example presents a situation for which no triangle is compatible with the given data.


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Solve triangle ABC, where:

Example 13


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The Law of Cosines


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Introduction

The Law of Sines cannot be used directly to solve triangles if we know either:
Two sides and the angle between themAll three sides
In these two cases, the Law of Cosines
applies.


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Law of Cosines

In any triangle ABC, we have:


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Pythagorean Theorem as Law of Cosines

If one of the angles of a triangle, say C,
is a right angle, then cos C = 0.

Hence, the Law of Cosines reduces to
the Pythagorean Theorem, c2 = a2 + b2.
Thus, the Pythagorean Theorem is
a special case of the Law of Cosines.

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A tunnel is to be built through a mountain.

To estimate the length of the tunnel,
a surveyor makes the measurements shown.
Use the
surveyors data
to approximate
the length of
the tunnel.
Example 14


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The sides of a triangle are:

a = 5, b = 8, c = 12

Find the angles of the triangle.

Example 15


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Example 16

Solve triangle ABC, where

A = 46.5, b = 10.5, c = 18.0


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Example 16(a)

P

S

R

Q

13cm

15cm

14cm

10cm

Law of Cosines

We could have used the Law of Sines to find
B and C in Example 16 since we knew all three sides and an angle in the triangle.
However, knowing the sine of an angle does
not uniquely specify the angle.This is because an angle and its supplement 180 both have the same sine.

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Law of Cosines

Thus, we would need to decide which
of the two angles is the correct choice.
This ambiguity does not arise when we use
the Law of Cosines.This is because every angle between 0 and 180 has a unique cosine. So, using only the Law of Cosines is preferable
in problems like Example 16.

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Simplifying Trigonometric Expressions


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Simplifying Trigonometric Expressions

To simplify algebraic expressions,
we used:
FactoringCommon denominatorsSpecial Product Formulas

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Simplify the expression

Example 17


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Simplify the expression

Example 18


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Proving Trigonometric Identities


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Guidelines for Proving Trigonometric Identities

Start with one side.

Use known identities.

Convert to sines and cosines.


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Verify the identity

Example 19


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Verify the identity

Example 20


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Verify the identity

Example 21


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Verify the identity

Example 22


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Addition and Subtraction Formulas


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Formulas for sine


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Formulas for cosine


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Formulas for tangent


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Find the exact value of each expression.

cos 75

cos /12

Example 23


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Find the exact value of:

sin 20 cos 40 + cos 20 sin 40

Example 24


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Prove the cofunction identity

Example 25


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Double Angle Formulas


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Double-Angle Formulas


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Formula for sineFormulas for cosine
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Double-Angle Formulas


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Formula for tangent
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If cos x = (2/3) and x is in quadrant II, find cos 2x and sin 2x.

Example 26


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Example 27

Write cos 3x in terms of cos x.


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Example 28

Prove the identity


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Half Angle Formulas


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Half-angle Formulae


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Example 29

If is an acute angle, evaluate each of these without using calculator,

(i)

(ii)


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Sum-Product Formulas


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Sum-to-Product Formulas


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Sum-to-Product Formulas for Sine
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Sum-to-Product Formulas


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Sum-to-Product Formulas for Cosine
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Write sin 7x + sin 3x as a product.

Example 30


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Express sin 3x sin 5x as a sum of trigonometric functions.

Example 31


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Example 32

Verify the identity


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Example 33

Prove that

Deduce that


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To solve a trigonometric equation,
we use:

The rules of algebra to isolate the trigonometric function on one side of the equal sign.

Our knowledge of the values of the trigonometric functions to solve for the variable.


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Solve the equation

2 sin x 1 = 0 for

Example 34


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Solve the equation

tan2x 3 = 0 for

Example 35


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Find the values of x for which the graphs of

f(x) = sin x and g(x) = cos x

intersect. (Give answer for )

Example 36


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Solve the equation 3 sin 2 = 0 for

solutions in the interval [0, 2), correct to

five decimals.

Example 37


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Solving Trigonometric Equations by Factoring


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Solving Trigonometric Equations by Factoring

Factoring is one of the most useful techniques for solving equations, including trigonometric equations.
The idea is to:
Move all terms to one side of the equation.

Factor the equation.

Use the Zero-Product Property.


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Solve the equation

2 cos2x 7 cos x + 3 = 0

for

Example 38


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Solve the equation

1 + sin x = 2 cos2x

for

Example 39


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Solve the equation

sin 2x cos x = 0

for

Example 40


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Solve the equation

cos x + 1 = sin x

in the interval [0, 2).

Example 41


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Solve the equation tan2x tan x 2 = 0

for

Example 42


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Equations withTrigonometric Functions
of Multiple Angles


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Equations with Trig. Functions of Multiple Angles

When solving trigonometric equations that involve functions of multiples of angles, we:

Solve for the multiple of the angle.

Divide to solve for the angle.


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Consider the equation 2 sin 3x 1 = 0.

Find the solutions in the interval [0, 2).

Example 43


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Consider the equation .

Find the solutions in the interval [0, 4).

Example 44


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Expression of the form
a sin + b cos


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Expressions of the Form
a sin + b cos

We can write expressions of the form

a sin + b cos

in terms of a single trigonometric function using the addition formula for sine and cosine.


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Sums of Sines and Cosines

If a and b are real numbers, then

where


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Express 3sin x + 4cos x in the form

Example 45


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Write the function

in the form

Example 46


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Example 47

Express in the form

and hence find such that has a maximum value.


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Example 48

Express in the form

where R>0 and

State the maximum and minimum values of

Hence, solve the equation

for


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Example 49

Express in the form

where r is positive and is acute.

Hence, determine the ranges of x

such that


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Solution of Equation


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Example 50

Solve the equation

for


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Example 51

Find the values of r and , where r > 0 and

is acute, given that can be expressed in

the form

Hence, show that


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The End
Of
Chapter 5


FHMM1024 Calculus

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2

2

y

x

r

+

=

x

y

r

x

r

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q

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The ratio of three angles in a triangle

is 4:5:6,

given that its perimeter is 100cm, find

the

largest length of the triangle.

7

2

7

45

0

=

=

=

b

a

A

6

.

248

2

.

186

1

.

43

0

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122

70

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a

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cos

2

2

2

2

2

2

2

2

2

2

-

+

=

-

+

=

-

+

=

2

The diagram shows a quadrilateral PQRS.

The area of the triangle PQR is 72.45 cm

.

(a) Calculate PQR.

(b) Given triangle PQR is an acute trian

gle.

Calculate

(i) the length, in cm, of PR,

2

(ii) PSR,

(iii) the area, in cm, of quadrila

teral PQRS.

q

q

q

sin

tan

cos

+

sincos

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qq

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)

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x

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A

B

A

B

A

B

A

B

A

B

A

sin

cos

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sin(

sin

cos

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sin(

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B

A

B

A

B

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B

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sin

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A

B

A

B

A

B

A

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A

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tan

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x

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x

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2

2

2

2

2

2

2

2

2

2

2

2

tan

1

tan

2

tan

sin

2

1

1

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2

sin

cos

cos

cos

sin

2

sin

q

q

q

q

q

q

q

q

q

q

q

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=

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=

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=

=

q

q

and

tan

4

3

=

q

2

tan

2

tan

q

2

sin

2

cos

2

sin

sin

2

cos

2

sin

2

sin

sin

Q

P

Q

P

Q

P

Q

P

Q

P

Q

P

-

+

=

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-

+

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2

sin

2

sin

2

cos

cos

2

cos

2

cos

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Q

P

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Q

P

Q

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Q

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Q

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360

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360

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chapter 5 trigonometry

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