## Wednesday, August 8, 2007

### Geometry Chapter 9.5 Trigonometric Ratios

Chapter 9.5 Trigonometric Ratios:

I) Vocabulary:
A) A Trigonometric Ratio - is a ratio of the lengths of two sides of a right triangle.
1) Sine - abbreviated sin
2) Cosine - abbreviated cos
3) Tangent - abbreviated tan

check out: http://www.tpub.com/math1/20b.htm

Let ΔABC be a right triangle. The sine, the cosine, and the tangent of the acute angle ∠A are defined as follows:

The side opposite angle A = side a
The side opposite angle B = side b
The side opposite angle C = side c = hypotenuse

Sin A = (side opposite angle A)/hypotenuse = a/c

Cos A = (side adjacent angle A)/hypotenuse = b/c

Tan A = (side opposite angle A)/(side adjacent angle A) = a/b

Example: Given ΔABC with AB = 13, AC = 5 and BC = 12 with right ∠C, find the 3 different ratios for angle A and angle B:
Sin A = 12/13
Cos A = 5/13
Tan A = 12/5

Sin B = 5/13
Cos B = 12/13
Tan B = 5/12

Do you notice a pattern?
The Sin A = Cos B and Cos A = Sin B plus the tangents are just reciprocals of each other. This is because the Sine and Cosine are cofunctions of each other.

Example: Find the sine, cosine and tangent of 30°, 45° and 60°.
Sin 30° = 1/2
Cos 30° = √(3)/2
Tan 30° = √(3)/3

Sin 45° = √(2)/2
Cos 45° = √(2)/2
Tan 45° = 1

Sin 60° = √(3)/2
Cos 60° = 1/2
Tan 60° = √3

B)Angle of Elevation - the angle that your line of sight makes with a line drawn horizontally.
Example: The angle of elevation from the base of a water slide to the top of a waterslide is about 13°. The slide extends horizontally about 58.2 meters. Find the vertical height of the slide.

1) Draw a diagram. Place the appropriate values where they belong.
2) Using the trigonometric functions, find the one you need.
a) We need to know the side opposite the angle and we know the side adjacent to the angle so this would be tangent.
b) Tan 13° = h/58.2
h = 58.2 (tan 13°)
h = 13.43652872 meters

Chapter 9.6 Solving Right Triangles:

I) Vocabulary:
A) Solve a right triangle - to determine the measures of all six parts, 3 sides and 3 angles.
Example: Given ΔABC, angle C is the right angle, angle A = 26°, AB = 4.5 inches, find AC, BC and angle B.
measure of angle A = 26°
measure of angle B = ?
measure of angle C = 90°
AB = c = 4.5 inches
BC = a = ?
AC = b = ?

A triangle has 180° so 26°+ 90° = 116°, 180° - 116° = 64° = measure of angle B

From the angle A, we know the hypotenuse or side "c" = 4.5, to find side "a", this is opposite to angle A so we would use sine function.
sin 26° = a/4.5
a = 1.972670161 inches

From the angle A, we know the hypotenuse and need to find side "b", this is adjacent to angle A so we would use the cosine function.
cos 26° = b/4.5
b = 4.044573208 inches

Therefore we have now solved the triangle:
measure of angle A = 26°
measure of angle B = 64°
measure of angle C =90°
a = 1.972670161 inches
b = 4.044573208 inches
c = 4.5 inches