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| ## Trigonometry## Trigonometric RatiosIf one of the angles of a triangle is 90º (a right angle), the 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 opposite the angle θ is called the opposite. The remaining side A is called the side adjacent.side
Pythagoras
This means that given any two sides of a right angled triangle, the For example, if If Trigonometric ratios provide relationships between the
Note that . Other ratios are defined by using the above three:
These six ratios define what are known as the . Theyfunctionsare independent of the unit used. ## Exercise 1
## Exercise 2.These ratios are independent of the unit used to measure the In particular, if we take
## Special AnglesThe trigonometric ratios of the angles 30º, 45º and 60º ## 45ºIn this case, the triangle is isosceles. Hence the opposite side and We have
## 60º & 30ºLet us draw an equilateral triangle,
From this we can determine the following trig ratios for the special ## General AnglesFor any other angle θ, you can calculate approximately the values Make sure you set the mode on your calculator to DEG if the angle is ## Exercise 3<< Pythagoras |

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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

∠

A

* *

is

. The length of the leg adjacent to

$$∠

A

is

$b$, and the length of the hypotenuse is

$c$.

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

$a$. The length of the leg adjacent to

$$∠

A

is

$b$, and the length of the hypotenuse is

$c$.

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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