Right Triangle Trigonometry

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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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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:
• 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
• 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

• 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
• hyp is the hypotenuse of the right triangle

hyp

opp

Definitions
• Sine is abbreviated sin, cosine is abbreviated cos and tangent is abbreviated tan.
• The sin(A) = opp/hyp
• Just remember sohcahtoa!
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°
• 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°)

A y

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)