Transcript Trigonometry

Trigonometry
Unit 4:Mathematics
Aims
•
Solve oblique
triangles using
sin & cos laws
Objectives
• Calculate angles and
lengths of oblique
triangles.
2
B
c=5.2
a=2.4
A
b=3.5
C
The table shows
some of the values of
these functions for
various angles.
Sines increase from 0 to 1
Between 0o a 90o:
Cosines decrease from 1 to 0
Between 0o a 90o:
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. 45º
6. 63º
2. 38º
7. 90º
3. 22º
8. 152º
4. 18º
9. 112º
5. 95º
10. 58º
The Law of Sines
a
sin A

b
sin B

B
c
c
sin C
a
The Law of Cosines
a2=b2+c2-2bc
cosA
b2=a2+c2-2ac cosB
c2=a2+b2-2ab cosC
A
C
b
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 180o.
A + B + C = 180o
The area of this triangle is given by
one of the following three formulae
Area = (a × b × Sin C) = (a × c × Sin B) =
2
2
(b × c × Sin A)
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.
a2 = b2 + c2 - (2 × b × c × Cos A)
b2 = a2 + c2 - (2 × a × c × Cos B)
c2 = a2 + b2 - (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 90o = 0. The equation then
reduces to Pythagoras'
Theorem.
a2 = b2 + c2 - (2 × b × c × Cos 90o) = b2 + c2 - 0 = b2 + c2
The relationship between the sides
and angles of a general triangle is
given by The Sine Rule.
a
sin A

b
sin B

c
sin C
Find the missing length and the missing angles in the following triangle.
By the Cosine Rule,
a2 = b2 + c2 - (2 × b × c × Cos A)
Find the missing length and the missing angles in the following triangle.
Now, from the Sine Rule,
a
sin A

b
sin B

c
.........
sin C
This can be rearranged to
sin C 
a
c

sin A
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
B
22
25
12
A
14
B
C
31 º
a
28 º
A
b
C
B
B
c
c
168 º 5
A
8
C
15
35 º
A
24
C