Transcript Right Triangle Trigonometry

Right Triangle Trigonometry
Section 6.5
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.
c2 = a2 + b2
a
c
b
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
• 30 – 60 – 90 degree triangle.
• 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 traingle.
30°
2
2
2
√3
2
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.
45°
sin(45°) = 1/√2
√2
cos(45°) = 1/√2
1
tan(45°) = 1
1
45°
Example
• Find the missing side lengths and angles.
60°
A = 180°-90°-60°=30°
sin(60°)=y/10
10
x
thus y=10sin(60°)
y 
10 3
5 3
2
x 2  y 2  10 2
A
y
x 2  100  (5 3 ) 2
x 2  100  75
x 2  25
x5
Inverse Trig Functions
• What if you know all the sides of a right triangle
but you don’t know the other 2 angle measures.
How could you find these angle measures?
• What you need is the inverse trigonometric
functions.
• Think of the angle measure as a present. When
you take the sine, cosine, or tangent of that
angle, it is similar to wrapping your present.
• The inverse trig functions give you the ability to
unwrap your present and to find the value of the
angle in question.
Notation
• A=sin-1(z) is read as the inverse sine of A.
• Never ever think of the -1 as an exponent.
It may look like an exponent and thus you
might think it is 1/sin(z), this is not true.
• (We refer to 1/sin(z) as the cosecant of z)
• A=cos-1(z) is read as the inverse cosine of
A.
• A=tan-1(z) is read as the inverse tangent of
A.
Inverse Trig definitions
• Referring to the right triangle from the
introduction slide. The inverse trig
functions are defined as follows:
• A=sin-1(opp/hyp)
• A=cos-1(adj/hyp)
• A=tan-1(opp/adj)
Example using inverse trig
functions
• Find the angles A and B given the following right
triangle.
• Find angle A. Use an inverse trig function to find
A. For instance A=sin-1(6/10)=36.9°.
• Then B = 180° - 90° - 36.9° = 53.1°.
B
6
10
8
A