1. Section 1.7
Applications

2. Objectives:
1. To review the slope of a line.
2. To find the inclination of a line.
3. To find the area of a triangle
using trigonometry or Heron’s
formula.

3. Definition
The slope of a line is the ratio of vertical change to horizontal change when moving from one point on the line to another point on the line. Dy
x1 – x2
y1 – y2
m = = Dx

4. Slope of a Line Dy x1 – x2 y1 – y2 m = = Dx P2(x2, y2) y P1(x1, y1)

5. EXAMPLE 1 Find the slope of the line that passes through the points (5, 9) and (-2, 4).
x1 – x2
y1 – y2 x y
m = =
m =
9 – 4
5 – (-2)
m = 5 7

6. Practice: Find the slope of the line that passes through the points (-3, -5) and (1, -3).
x1 – x2
y1 – y2 x y
m = =
m =
-5 + 3
-3 – 1
m = 1 2

7. Definition
The inclination of a line is the angle that the line makes with a positively directed ray on the x-axis.

8. Angle of inclination x y
tan  = Dy Dx y
tan  = m  x
 = tan-1 m

9. EXAMPLE 2 Find the angle of inclination of the line that passes through (2, 9) and (1, 2).
x1 – x2
y1 – y2 x y
m = =
m = = 7 7 1
m =
9 - 2
2 – 1
tan α = 7
α = 81.87

10. Caution: If the angle is obtuse the calculator will respond with a negative angle. To bring your answer in line with the definition of the angle of inclination, add 180 to the answer the calculator gives to produce the correct obtuse angle.
tan α = -2
α = -63.4
α = 116.6 (-63.4 + 180)

11. Practice: Find the angle of inclination of the line that passes through (2, 3) and (6, -2).
m = = -1.25 5 -4
m =
3 + 2
2 – 6
x1 – x2
y1 – y2 x y
m = =
tan α = -1.25
α = -51.3
α = 128.7

12. Area of a Triangle C B A a b c h a h sin C = h = a sin C A = ½bh
A = ½ab sin C

13. EXAMPLE 3 Find the area of ABC.8 12
82
A = ½ ab sin C
A = ½·8·12 sin 82
A = 48 sin 82
A = 47.5 sq. units

14. Practice: Find the area of XYZ.10 7
58
A = ½ yz sin X
A = ½·7·10 sin 58
A = 35 sin 58
A = 29.7 sq. units

15. EXAMPLE 4 Find the area of PQR, assuming that PQR is acute.10 9
48 9
sin 48 10
sin Q =
10 sin 48 9
sin Q =
mQ = 55.7
mP = 180 - (48 )
mP = 76.3

16. EXAMPLE 4 Find the area of PQR, assuming that PQR is acute.10 9
48
A = ½ qr sin P
A = ½·10·9 sin 76.3
A = 45 sin 76.3
A = 43.7 sq. units

17. Heron’s Formula
The area of a triangle having sides of length a, b, and c is given by:
A = s(s - a)(s - b)(s - c)
where s = ½(a + b + c)

18. Homework pp

19. ►A. Exercises x1 – x2 y1 – y2 x y m = = m = 2 - 1 -7 - 4 m = = -11 1
Find the slope of the line that passes through the given points.
1. (2, -7), (1, 4).
x1 – x2
y1 – y2 x y
m = =
m =
2 - 1
-7 - 4
m =
= -11 1
-11

20. ►A. Exercises m = 2 – (-5) 4 - 1 x1 – x2 y1 – y2 x y m = = m = 7 3
Find the angle of inclination of the line that passes through the given points. Round angle measure to the nearest minute.
7. (-5, 1), (2, 4).
m =
2 – (-5)
4 - 1
x1 – x2
y1 – y2 x y
m = =
m = 7 3

21. ►A. Exercises 7 3 tan  = tan  = m ÷ ø ö ç è æ 7 3 m = tan-1
Find the angle of inclination of the line that passes through the given points. Round angle measure to the nearest minute.
7. (-5, 1), (2, 4). 7 3
tan  =
tan  = m ÷ ø ö ç è æ 7 3
m = tan-1
m = 23.2

22. ►B. Exercises S = ½(a + b + c) S = ½(2 + 7 + 7) S = ½(16) S = 8
Find the area of triangle ABC.
15. a = 2, b = 7, c = 7
S = ½(a + b + c)
S = ½( )
S = ½(16)
S = 8

23. ►B. Exercises A = s(s - a)(s - b)(s - c) A = 8(8 - 2)(8 - 7)(8 - 7)
Find the area of triangle ABC.
15. a = 2, b = 7, c = 7
A = s(s - a)(s - b)(s - c)
A = 8(8 - 2)(8 - 7)(8 - 7)
A = 8(6)(1)(1)
A = = 4 3

24. ►B. Exercises
Find the slope of the line, given the angle of inclination.
19. 30°

25. ►B. Exercises
Find the slope of the line, given the angle of inclination.
21. 5 6

26. ■ Cumulative Review
32. Under what conditions may the law of cosines be applied to a triangle?

27. ■ Cumulative Review
33. In using the law of sines, which condition is the ambiguous case?

28. ■ Cumulative Review
34. When can you solve a triangle given two sides without using the law of sines or the law of cosines?

29. ■ Cumulative Review
35. Find the length of an arc of a circle if the radius is 6 ft. 3 in. and the central angle is 81.7°.

30. ■ Cumulative Review
36. List all angles θ, such that 0° ≤ θ < °, that have a reference angle of 30°.