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Similar figures

All functions from one function



PLANE TRIGONOMETRY is based on the fact of similar figures.  (Topic 1.)  We saw:

Figures are similar if they are equiangular
and the sides that make the equal angles
are proportional.

For triangles to be similar, however, it is sufficient that they be equiangular. (Theorem 15 of "Some Theorems of Plane Geometry.") From that it follows:

Right triangles will be similar if an acute angle of one
is equal to an acute angle of the other.

Similar right triangles

In the right triangles ABC, DEF, if the acute angle at B is equal to the acute angle at E, then those triangles will be similar.  Therefore the sides that make the equal angles will be proportional.  Whatever ratio CA has to AB,  FD will have to DE.  If CA were half of AB, for example, then FD would also be half of DE.

A trigonometric Table is a table of ratios of sides.  In the Table, each value of sin θ represents the ratio of the opposite side to the hypotenuse -- in every right triangle with that acute angle.

If angle θ is 28°, say, then in every right triangle with a 28° angle, its sides will be in the same ratio.  We read in the Table,

sin 28° = .469

This means that in a right triangle having an acute angle of 28°, its opposite side is 469 thousandths of the hypotenuse, which is to say, a little less than half.

It is in this sense that in a right triangle, the trigonometric ratios -- the sine, the cosine, and so on -- are "functions" of the acute angle.  They depend only on the acute angle.

Example.  Indirect measurement.   When we cannot measure things directly, we can use trigonometry.


For example, to measure the height h of a flagpole, we could measure a distance of, say, 100 feet from its base.  From that point P we could then measure the angle required to sight the top .  If that angle, called the angle of elevation, turned out to be 37°, then

so that       h  
 = tan 37°
so that       
so that     h  = 100 × tan 37°.

From the Table,

tan 37° = .754

Therefore, on multiplying by 100,

h = 75.4 feet.

(Lesson 4 of Arithmetic..)

All functions from one function

If we know the value of any one trigonometric function, then -- with the aid of the Pythagorean theorem -- we can find the rest.

  Example 1.   In a right triangle, sin θ  5 
.  Sketch the triangle, place

the ratio numbers, and evaluate the remaining functions of θ.


5-12-? triangle

To find the unknown side x, we have

x² + 5² = 13²

x² = 169 − 25 = 144.


x = = 12.

(Lesson 26 of Algebra.)

We can now evaluate all six functions of θ:

sin θ   =   5 
        csc θ   =  13
cos θ   =  12
        sec θ   =  13
tan θ   =   5 
        cot θ   =  12

Example 2.   In a right triangle, sec θ = 4.  Sketch the triangle, place the ratio numbers, and evaluate the remaining functions of θ.

4-1-? triangle

To say that sec θ = 4, is to say that the hypotenuse is

to the adjacent side in the ratio 4 : 1.  (4 =  4

To find the unknown side x, we have

x² + 1² =
x² = 16 − 1 = 15.


x = .

We can now evaluate all six functions of θ:

sin θ   = 
        csc θ   =     4   
cos θ   =  1
        sec θ   =  4
tan θ   =          cot θ   =     1   
  Problem 1.    In a right triangle, cos θ 2
.  Sketch the triangle and

evaluate sin θ.

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

Problem 2.   cot θ = .  Sketch the triangle and evaluate csc θ.


Two angles are called complements of one another if together they equal a right angle.  Thus the complement of 60° is 30°.  This is the degree system of measurement in which a full circle, made up of four right angles at the center, is called 360°.  (But see Topic 12:  Radian Measure.)

Problem 3.   Name the complement of each angle.

 a)  70°   20°        b)    20°   70°        c)  45°   45°        d)  θ   90° − θ

A right triangle

The point about complements is that, in a right triangle, the two acute angles are complementary.  For, the three angles of the right triangle are together equal to two right angles (Theorem 9); therefore, the two acute angles together will equal one right angle.

(When we come to radian measure, we will see that 90° =  π
, and
  therefore the complement of θ is  π
 − θ.)


There are three pairs of cofunctions:

The sine and the cosine

The secant and the cosecant

The tangent and the cotangent

Here is the significance of a cofunction:

A function of any angle is equal to the cofunction
of its complement.

This means, for example, that

sin 80° = cos 10°.

The cofunction of the sine is the cosine.  And 10° is the complement of 80°.

Problem 4.   Answer in terms of cofunctions.

a)   cos 5° = sin 85°       b)   tan 60° = cot 30°       c)   csc 12° = sec 78°

d)   sin (90° − θ) = cos θ        e)   cot θ = tan (90° − θ)

In the figure:

  sin θ =  a
    cos φ =  a

Thus the sine of θ is equal to the cosine of its complement.

  sec θ =  c
    csc φ =  c

The secant of θ is equal to the cosecant of its complement.

  tan θ =  a
    cot φ =  a

The tangent of θ is equal to the cotangent of its complement.

Next Topic:  The Isosceles Right Triangle

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