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Trigonometry

Trigonometric Ratios

If one of the angles of a triangle is 90º (a right angle), the
triangle is called a right angled triangle.
We indicate the 90º (right) angle by placing a box in its corner.) Because the
three (internal) angles of a triangle add up to 180º, the other two angles are
each less than 90º; that is they are acute.

In the above triangle, the side H opposite the right angle is called
the hypotenuse. Relative to the angle θ, the side O
opposite the angle θ is called the opposite
side
. The remaining side A is called the adjacent
side
.

Warning:
This assignment of the opposite and adjacent sides is relative to θ. If the angle
of interest (in this case θ) is located in the upper right hand corner of the
above triangle the assignment of sides is then:

Pythagoras
Theorem states that a triangle is right angled if and only if

.

This means that given any two sides of a right angled triangle, the
third side is completely determined.

For example, if O = 1, A = 2, then .

If H = 5, and O = 3, then .

Trigonometric ratios provide relationships between the
sides and angles of a right angle triangle. The three most commonly used ratios are

 sine cosine tangent

Note that .

Other ratios are defined by using the above three:

 cosecant secant cotangent

These six ratios define what are known as the trigonometric
(trig in short) functions. They
are independent of the unit used.

Exercise 1

Given the lengths of the three sides
of a right angled triangle find the values of the trig functions, corresponding to
the angle θ.

 H= O = A =

 sin θ = 2 d.p. cos θ = 2 d.p. tan θ= 2 d.p. csc θ = 2 d.p. sec θ = 2 d.p. cot θ= 2 d.p.

Exercise 2.

Given the lengths of two
sides of a right angled triangle find the length of the third side (use Pythagoras
Theorem). Then find the values of the given trig functions corresponding to the angle
θ.

 H= O = A =

 sin θ= 2 d.p. cos θ= 2 d.p. tan θ= 2 d.p.

These ratios are independent of the unit used to measure the
sides as long as the same unit is used for all the sides.

In particular, if we take H = 1, then

O = sin θ  and  A = cos θ.

Special Angles

The trigonometric ratios of the angles 30º, 45º and 60º
are often used in mechanics and other branches of mathematics. So it is useful to
calculate them and know their values by heart.

45º

In this case, the triangle is isosceles. Hence the opposite side and
adjacent sides are equal, say 1 unit.
The hypotenuse is therefore of length units
(by Pythagoras Theorem ).

We have

60º & 30º

Let us draw an equilateral triangle, ABC, of sides 2 units
in length. Next draw a line AD from A perpendicular to BC
AD bisects BC giving BD = CD = 1.

From this we can determine the following trig ratios for the special
angles 30º and 60º:

General Angles

For any other angle θ, you can calculate approximately the values
of sin θ, cos θ, tan θ by using a scientific calculator.

Make sure you set the mode on your calculator to DEG if the angle is
measured in degrees or RAD if the angle is measured in radians.

Exercise 3

Use
a calculator to compute the values of the trig functions at the given θ, where
θ is measured in degrees.

to 2 decimal places)

 θ= º sin θ = 2 d.p. cos θ = 2 d.p. tan θ= 2 d.p. csc θ = 2 d.p. sec θ = 2 d.p. cot θ= 2 d.p.

<<  Pythagoras
Theorem  |  Trigonometry Index  |  Inverse
Trig Functions  >>

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Trigonometric Ratios

“Trigon” is Greek for

triangle

, and “metric” is Greek for measurement. The

trigonometric ratios

are special measurements of a

right triangle

(a triangle with one

angle

measuring



90

°

). Remember that the two sides of a right triangle which form the right angle are called the

legs

, and the third side (opposite the right angle) is called the

hypotenuse

.

There are three basic trigonometric ratios:

sine

,

cosine

, and

tangent

. Given a right triangle, you can find the sine (or cosine, or tangent) of either of the non-



90

°

angles.

$\begin{array}{}\end{array}$

sine

=

length of the leg opposite to the angle

length of hypotenuse

abbreviated “sin”

cosine

=

length of the leg adjacent to the angle

length of hypotenuse

abbreviated “cos”

tangent

=

length of the leg opposite to the angle

length of the leg adjacent to the angle

abbreviated “tan”

Example:

Write expressions for the sine, cosine, and tangent of



A

.

The length of the leg opposite



A

is

. The length of the leg adjacent to



A

is

, and the length of the hypotenuse is

.

The sine of the angle is given by the ratio “opposite over hypotenuse.” So,



sin

A

=

a

c

The cosine is given by the ratio “adjacent over hypotenuse.”



cos

A

=

b

c

The tangent is given by the ratio “opposite over adjacent.”



tan

A

=

a

b

Generations of students have used the mnemonic ”

SOHCAHTOA

” to remember which ratio is which. (

S

ine:

O

pposite over

H

ypotenuse,

C

osine:

A

djacent over

H

ypotenuse,

T

angent:

O

pposite over

A

djacent.)

Other Trigonometric Ratios

The other common trigonometric ratios are:

$\begin{array}{}\end{array}$

secant

=

length of hypotenuse

length of the leg adjacent to the angle

abbreviated “sec”

sec

(

x

)

=

1

cos

(

x

)

cosecant

=

length of hypotenuse

length of the leg opposite to the angle

abbreviated “csc”

csc

(

x

)

=

1

sin

(

x

)

secant

=

length of the leg adjacent to the angle

length of the leg opposite to the angle

abbreviated “cot”

cot

(

x

)

=

1

tan

(

x

)

Example:

Write expressions for the secant, cosecant, and cotangent of



A

.

The length of the leg opposite



A

is

. The length of the leg adjacent to



A

is

, and the length of the hypotenuse is

.

The secant of the angle is given by the ratio “hypotenuse over adjacent”. So,



sec

A

=

c

b

The cosecant is given by the ratio “hypotenuse over opposite”.



csc

A

=

c

a

The cotangent is given by the ratio “adjacent over opposite”.



cot

A

=

b

a

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