Transcript Trigonometry
Trigonometry
Unit 4:Mathematics
•
Aims
Solve oblique triangles using sin & cos laws •
Objectives
Calculate angles and lengths of oblique triangles.
2
A c=5.2
b=3.5
B a=2.4
C
The table shows some of the values of these functions for various angles.
Sines increase from 0 to 1 Between
0 o a 90 o :
Cosines decrease from 1 to 0 Between
0 o a 90 o :
Tangents increase from 0 to infinity.
Cos(90 - X) = Sin(X) Sin(90 - X) = Cos(X)
Write out the each of the trigonometric functions (sin, cos, and tan) of the following 1. 45 º 2. 38 º 3. 22 º 4. 18 º 5. 95 º 6. 63 º 7. 90 º 8. 152 º 9. 112 º 10. 58 º
The Law of Sines sin
a A
sin
b B
c
sin
C
The Law of Cosines a 2 =b 2 +c 2 -2bc cosA b 2 =a 2 +c 2 -2ac cosB c 2 =a 2 +b 2 -2ab cosC B a C c b A
Whenever possible, the law of sines should be used. Remember that at least one angle measurement must be given in order to use the law of sines.
The law of cosines in much more difficult and time consuming method than the law of sines and is harder to memorize. This law, however, is the only way to solve a triangle in which all sides but no angles are given.
Only triangles with all sides, an angle and two sides, or a side and two angles given can be solved.
The triangle has three sides,
a
,
b
, and
c
. There are three angles,
A
,
B
,
C
(where angle A is opposite side a, etc). The height of the triangle is
h
. The sum of the three angles is always 180 o .
A + B + C = 180 o
The area of this triangle is given by one of the following three formulae
Area = (a × b × Sin C) = (a × c × Sin B) = 2 (b × c × Sin A) 2 2 = b × h 2
The relationship between the three sides of a general triangle is given by
The Cosine Rule
. There are three forms of this rule. All are equivalent.
a
2
b
2
c
2
= b
2
= a
2
= a
2
+ c
2
+ c
2
+ b
2
- (2 × b × c × Cos A) - (2 × a × c × Cos B) - (2 × a × b × Cos C)
Show that Pythagoras' Theorem is a special case of the Cosine Rule.
In the first version of the Cosine Rule, if angle A is a right angle,
Cos 90 o = 0
. The equation then reduces to Pythagoras' Theorem.
a 2 = b 2 + c 2 - (2 × b × c × Cos 90 o ) = b 2 + c 2 - 0 = b 2 + c 2
The relationship between the sides and angles of a general triangle is given by
The Sine Rule
.
sin
a A
sin
b B
c
sin
C
Find the missing length and the missing angles in the following triangle.
By the Cosine Rule,
a 2 = b 2 + c 2 - (2 × b × c × Cos A)
Find the missing length and the missing angles in the following triangle.
Now, from the Sine Rule , sin
a A
sin
b B
sin
c C
.........
sin
a A
c
sin
C
This can be rearranged to sin
C
c
.
x
.
sin
a A
sin
C
4 .
6
xSin
32 3 .
42
Side a is opposite angle A Side b is opposite angle B Side c is opposite angle C
Solve the following oblique triangles with the dimensions given A A c 22 14 8 C 12 A B B
168
º 5 C A 25
28
º b c
35
º 24 C
31
º a B B C 15