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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 .

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pythagorean theorem
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
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

corresponding angles and sides
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
Example
  • Find the missing side lengths for the similar triangles.

3.2 3.8

y

54.4 x

42.5

answer
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
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
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 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 √3 2

2 1 60°

30 60 90
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
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°

example12
Example
  • Find the missing side lengths and angles.

60° A = 180°-90°-60°=30°

sin(60°)=y/10

10 x thus y=10sin(60°)

A y

inverse trig functions
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
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
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
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