# 5.2 RIGHT TRIANGLE TRIGONOMETRY by happo7

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```									      5.2 RIGHT TRIANGLE TRIGONOMETRY
Trigonometry is study involving triangle measurement. Right triangles are used to
study functions that involve trigonometry (trigonometric functions)

Trigonometric function is a ratio of any two sides of a right angle triangle.

Hypotenuse(r)

Opposite(y)

θ

x2 + y2 = r 2    (Pythagoras theorem)

The 6 ratios defined for a right triangle are:

y                                 x                              y                r
sin θ =   ,                  cos θ =        ,                  tan θ =     ,    csc θ =     ,
r                                 r                              x                y
r                         x
sec θ =                 cot θ =
x                         y

Example 1
5
Let θ be an acute angle such that sin θ =                find:
3
a) secθ + tan θ      b) ( cot θ ) + ( csc θ )
2              2

Solution
y
( )
2
a) sin θ =   ⇒ 32 = 5 + x 2 ⇒ x = 2
r
r 3                 y  5
⇒ sec θ = = and ⇒ tan θ = =
x 2                 x 2
3+ 5
⇒ sec θ + tan θ =
2
x     2                4
b) cot θ = =       ⇒ ( cot θ ) =
2

y      5               5
2

r   3                9
csc θ =          ⇒ ( csc θ ) =
2
=
y    5               5

4 9 13
⇒ ( cot θ ) + ( csc θ ) =
2                2
+ =
5 5 5

Trigonometric ratios of special acute angles

The equilateral and isosceles triangles shown below illustrates how trigonometric
ratios for angles 30° 45° and 60° are derived.

30°30°

x            h                x

x                         x
60°               2                         2         60°

Note:
h = x 2 − ( 2 )2 =
x         3x
2

x                        3x
1            h       3
sin 30° =   2
=      , sin 60° = = 2 =
x        2            x  x   2

x
1                         h    3
cos 60° =      2
=     ,           cos 30° =     =
x       2                         x   2

r
y

45°
y

y         y            y        1     2
sin 45° = cos 45° =         =               =         =      =
r       y2 + y2       y 2        2   2
3

Using the above two triangles, the following are obtained;

1. sin 0° = cos 90° = 0
1
2. sin 30° = cos 60° =
2
1     2
3. sin 45° = cos 45° =     =
2   2
3
4. sin 60° = cos 30° =
2
5. sin 90° = cos 0° = 1
6. tan 0° = cot 90° = 0
3
7. tan 30° = cot 60° =
3
8. tan 45° = cot 45° = 1
9. tan 60° = cot 30° = 3

Example 2

Find the exact values of;

a)    3 sin 45° cos 30° − cot 30°

π             π         π
b)      2 tan          + cos         csc
4             3         6

π         π                     π
c)     2 cos           cot       + 2 3sec
3         4                     6

Solution
2 3      3 6 −4 3
a) 3sin 45° cos 30° − cot 30° = 3.                       .  − 3=
2 2           4

π             π           π        1
b) 2 tan       + cos         csc       = 2.1 + .2 = 3
4             3           6        2

π         π                     π     1          2 3     2 +8
c)    2 cos        cot       + 2 3sec            = 2. .1 + 2 3.     =
3         4                     6     2           3       4
4

Application of Trigonometric Functions of Acute Angles

Angle of Elevation is an angle measured upwards from the horizontal as shown
in the figure below

θ

Angle of depression is an angle measured downwards from the horizontal as
shown below

θ
5

Example 3

A 5 meters ladder is resting against a wall and makes an angle of 30° with the
ground. Find the height to which the ladder will reach the wall.

Solution

5               h

30°

h            1
= sin 30° = ⇒ h = 2.5m
5            2

Example 4

Find the height of a building if the angle of elevation from the top of the building
changes from 60° to 30° as the observer moves 30m further from the building.

Solution

h
30°             60°
30                x

h
= tan 60° = 3 ⇒ h = x 3
x

h                 3          30
x 3 30 3
= tan 30° =    ⇒h=    +
x + 30             3       3     3

x 3 30 3
⇒x 3=       +
3    3
x 3 30 3    ⎛      3 ⎞ 30 3
⇒ x 3−      =     ⇒ x⎜ 3 −
⎜        ⎟=
3    3     ⎝     3 ⎟⎠   3
6

⎛       3 ⎞ 30 3
⇒ x⎜ 3 −
⎜         ⎟=
⎝      3 ⎟⎠   3
2 3 30 3
⇒ x.     =
3       3
⇒ x = 15 3

⇒ h = 15 3. 3 = 45

Example 5

Find the exact height h of the triangle below

hm

30°                  60°
5 3 m

Solution

hm

30°                  60°
5 3 m                  x

h
= tan 60° = 3 ⇒ h = x 3
x
h                  3     x 3
= tan 30° =    ⇒h=     +5
x+5 3                3       3
7

x 3
⇒x 3=       +5
3
⎛      3⎞
⇒ x⎜⎜ 3−    ⎟=5
⎝    3 ⎟⎠
2 3
⇒ x.     =5
3

5 3
⇒x=
2
5 3      15
⇒h=     . 3 = = 7.5m
2        2

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