Download Right Triangle Trig Powerpoint

6.1 Right Triangle
Trigonometry
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
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