Trigonometry 2 Aims Solve oblique triangles using sin & cos laws Objectives Calculate angles and...
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Transcript of Trigonometry 2 Aims Solve oblique triangles using sin & cos laws Objectives Calculate angles and...
2
Unit 4:Mathematics
Aims• Solve oblique
triangles using sin & cos laws
Objectives• Calculate angles and
lengths of oblique triangles.
Write out the each of the trigonometric functions (sin, cos, and tan) of the following
1. 45º
2. 38º
3. 22º
4. 18º
5. 95º
6. 63º
7. 90º
8. 152º
9. 112º
10. 58º
When solving oblique triangles, simply using
trigonometric functions is not enough. You need…
The Law of Sines
C
c
B
b
A
a
sinsinsin
The Law of Cosines
a2=b2+c2-2bc cosA
b2=a2+c2-2ac cosB
c2=a2+b2-2ab cosC
a
c
bA
B
C
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
a2 = b2 + c2 - (2 × b × c × Cos A) b2 = a2 + c2 - (2 × a × c × Cos B) c2 = a2 + b2 - (2 × a × b × Cos C)
The relationship between the three sides of ageneral triangle is given by The Cosine Rule. There are three forms of this rule. All are equivalent.
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.
C
c
B
b
A
a
sinsinsin
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,
C
c
A
a
C
c
B
b
A
a
sinsin.........
sinsinsin
This can be rearranged to 42.3
326.4sin
sin..sin
xSinC
a
AxcC