Download 10.1B Right Triangle Trigonometry

10.1B Right-Angle Trigonometry
Objectives:
F.TF.3: Use special triangles to determine geometrically the values of sine, cosine, and tangent
for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and
tangent for π – x, π + x, and 2π – x in terms of their values for x, where x is any real number.
F.TF.5: Choose trigonometric functions to model periodic phenomena with specified amplitude,
frequency, and midline.
For the board: You will be able to understand and use trigonometric relationships of acute angles in a
triangle and determine side lengths of right triangles by using trigonometric functions.
Anticipatory Set:
Geometry Review:
A 45°-45°-90° triangle is a special right triangle because the
short leg and the long leg have the same measure.
45
hypotenuse
The triangle is an isosceles right triangle.
leg
Since the legs are equal, instead of labeling the sides a, b,
(a)
and c, we can label them 1, 1, and c.
45
leg
The Pythagorean Theorem becomes 12 + 12 = c2 or 2 = c2.
45
2
1
45
1
Finally c = 2 .
sin 45° =
1
2

2
2
cos 45° =
1
2

2
2
A 30°-60°-90° triangle is a special right triangle because it is
½ of an equilateral triangle.
Recall: An equilateral triangle has all 3 angles equal.
2
Since the 3 angles of a triangle must add to 180°,
this requires that each angle be 180°/3 = 60°.
Recall: An equilateral triangle has all 3 sides equal.
60
Use 2 for each of the 3 sides.
tan 45° = 1
60
30 30
2
2
2
b
60
2
60
60
1
1
2
To get ½ of an equilateral triangle, draw the bisector of the top angle.
This splits this 60° angle into two 30 angles and the equilateral triangle into 2 right triangles.
The base measure of 2 now becomes 1 and 1. Note: the hypotenuse = 2 ∙ short leg.
Label the bisector b. Concentrate on one of the 30-60-90 triangles.
The Pythagorean Theorem can now be written
12 + b2 = (2)2
sin 30° =
sin 60° =
1
2
3
2
1 + b2 = 4
cos 30° =
cos 60° =
b2 = 3
3
2
1
2
tan 30° =
tan 60° =
b=
1
3

3°
3
3
3
30°
2
3
1
60°
Instruction:
Open the book to page 694 and read example 2.
Example: Use a trigonometric function to find the value of x.
x
sin 30  
74
1 x

x  37
2 74
White Board Activity:
Practice: Use a trigonometric function to find the value of x.
x
sin 45 
20
2
x

x  10 2
2
20
Assessment:
Question student pairs.
Independent Practice:
Text: pg. 697 prob. 5 – 7, 16 – 18.
For a Grade:
Text: pg. 697 prob. 16, 18.
74
x
30°
x
20
45°