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Right Triangle Trigonometry

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Title: 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
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