Right Triangle Trigonometry

1
Right Triangle Trigonometry

• Section 6.5

2
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

3
Similar Triangles

• Triangles are similar if two conditions are met
• The corresponding angle measures are equal.
• 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
• E d F
• B a C

4
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

5
Example

• Find the missing side lengths for the similar
  triangles.
• 3.2 3.8
• y
• 
  54.4 x
• 42.5

6
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.

7
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

8
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

9
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 v3 2
• 2 1 60

10
30 60 90

• Now we can define the sine cosine and tangent of
  30 and 60.
• sin(60)v3 / 2 cos(60) ½ tan(60) v3
• sin(30) ½ cos(30) v3 / 2 tan(30)
  1/v3

11
45 45 90

• Consider a right triangle in which the lengths of
  each leg are 1. This implies the hypotenuse is
  v2.
• 45 sin(45) 1/v2
• v2 cos(45) 1/v2
• 1 tan(45) 1
• 1 45

12
Example

• Find the missing side lengths and angles.
• 60 A 180-90-6030
• sin(60)y/10
• 10 x thus y10sin(60)
• A y

13
Inverse Trig Functions

• What if you know all the sides of a right
  triangle but you dont 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.

14
Notation

• Asin-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)
• Acos-1(z) is read as the inverse cosine of A.
• Atan-1(z) is read as the inverse tangent of A.

15
Inverse Trig definitions

• Referring to the right triangle from the
  introduction slide. The inverse trig functions
  are defined as follows
• Asin-1(opp/hyp)
• Acos-1(adj/hyp)
• Atan-1(opp/adj)

16
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 Asin-1(6/10)36.9.
• Then B 180 - 90 - 36.9 53.1.
• B
• 6 10
• 8 A