Download The Right Triangle

Right Triangle
Trigonometry
Obea Rizzi B. Omboy
Pythagorean Theorem
• Recall that a right triangle has a 90° angle as one of its
angles.
• The side that is opposite the 90° angle is called the
hypotenuse.
• The theorem due to Pythagoras says that the square of
the hypotenuse is equal to the sum of the squares of the
legs.
a
c
b
c2 = a2 + b2
Similar Triangles
Triangles are similar if two conditions are met:
1. The corresponding angle measures are equal.
2. Corresponding sides must be proportional. (That is, their
ratios must be equal.)
The triangles below are similar. They have the same shape, but
their size is different.
A
D
c
b
f
E
B
a
C
e
d
F
Corresponding Angles
and Sides
As you can see from the previous page we can see
that angle A is equal to angle D, angle B equals angle E,
and angle C equals angle F.
The lengths of the sides are different but there is a
correspondence. Side a is in correspondence with side d.
Side b corresponds to side e. Side c corresponds to side f.
What we do have is a set of proportions
a/d = b/e = c/f
Example
Find the missing side lengths for the similar
triangles.
3.2
3.8
y
54.4
x
42.5
ANSWER
• Notice that the 54.4 length side corresponds to the
3.2 length side. This will form are complete ratio.
• To find x, we notice side x corresponds to the side
of length 3.8.
• Thus we have 3.2/54.4 = 3.8/x. Solve for x.
• Thus x = (54.4)(3.8)/3.2 = 64.6
• Same thing for y we see that 3.2/54.4 =
y/42.5. Solving for y gives y = (42.5)(3.2)/54.4
= 2.5.
Introduction to Trigonometry
In this section we define the three basic
trigonometric ratios, sine, cosine and tangent.
• opp is the side opposite angle A
• adj is the side adjacent to angle A
• hyp is the hypotenuse of the right triangle
hyp
opp
adj
A
Definitions
Sine is abbreviated sin, cosine is
abbreviated cos and tangent is abbreviated
tan.
•
•
•
•
•
The sin(A) = opp/hyp
The cos(A) = adj/hyp
The tan(A) = opp/adj
Just remember sohcahtoa!
Sin Opp Hyp Cos Adj Hyp Tan Opp Adj
Special Triangles
Special triangle is a triangle with 30 – 60 – 90
degree measurement in its angles.
Consider an equilateral triangle with side lengths 2.
Recall the measure of each angle is 60°. Chopping the
triangle in half gives the 30 – 60 – 90 degree triangle.
30°
2
2
2
√3
1
60°
30° – 60° – 90°
Now we can define the sine cosine
and tangent of 30° and 60°.
• sin(60°)=√3 / 2; cos(60°) = ½; tan(60°) = √3
• sin(30°) = ½ ; cos(30°) = √3 / 2; tan(30°) = 1/√3
45° – 45° – 90°
Consider a right triangle in which the
lengths of each leg are 1. This implies the
hypotenuse is √2.
sin(45°) = 1/√2
45°
cos(45°) = 1/√2
tan(45°) = 1
1
√2
1
45°
Example
Find the missing side lengths and angles.
60°
A = 180°-90°-60°=30°
sin(60°)=y/10
thus y = 10sin(60°)
10
x
y
10 3
5 3
2
x 2  y 2  10 2
x 2  100  (5 3 ) 2
A
y
x 2  100  75
x 2  25
x5