Transcript document

A Review for Ms. Ma’s
Physics 11
Trigonometry is the study and solution of
triangles. Solving a triangle means finding the
value of each of its sides and angles. The
following terminology and tactics will be
important in the solving of triangles.
Pythagorean theorem (a2 + b2 = c2).
True ONLY for right angle triangles
Sine (sin), and its inverse, Cosecant (csc or sin-1)
Cosine (cos), and its inverse, Secant (sec or cos-1)
Tangent (tan), and its inverse, Cotangent (cot or tan-1)
Right or oblique triangle
A trig function is a ratio of certain parts of a triangle. Again, the names of
these ratios are: the sine, cosine, tangent, cosecant, secant, and cotangent.
Let us look at this triangle …
B
Given the assigned letters to the sides and angles, we can
determine the following trigonometric functions.
Remember this: SOH CAH TOA
The cosecant is the inverse of
the sine, the secant is the
inverse of the cosine, & the
A cotangent is the inverse of the
tangent.
c
a
ө
C
b
Side Opposite
Hypothenuse
Side Adjacent
Cos θ= Hypothenuse
Side Opposite
Tan θ= Side Adjacent
Sin θ=
=
a
c
b
c
=
a
b
=
With this, we can find the sine of
the value of angle A by dividing
side a by side c. In order to find
the angle itself, we must take the
sine of the angle and invert it (in
other words, find the cosecant of
the sine of the angle).
Try finding the angles of the following triangle from the
side lengths using the trigonometric ratios from the
previous slide.
B
6
A first step is to use the trig
functions on ∠A.
α
10
β
C
Sin θ = 6/10
θ
8
Sin θ = 0.6
A Csc0.6 ~ 36.9
Angle A ~ 36.9°
Note: SIG FIGS are
WRONG – we’ll review
these soon
Because all ∠’s add up to 180°
in a triangle, ∠B = 90° - 36.9°
= 53.1°
The measurements have changed. Find side BA and side AC.
Sin34 = 2/BA
B
0.559 = 2/BA
2
α
0.559BA = 2
BA = 2/0.559
β
34º
C
A BA ~ 3.578 (wrong SIG FIGS again)
The Pythagorean theorem when used on this triangle states that …
BC2 + AC2 = AB2
AC2 = AB2 - BC2
AC2 = 12.802 - 4 = 8.802
AC = 8.8020.5 ~ 3
When solving oblique triangles, using trig functions
is not enough. You need …
The Law of Sines
a
b
c


sin A sin B sin C
B
c
a
The Law of Cosines
a2 =b2+c2-2bc cosA
b2=a2+c2-2ac cosB
c2=a2+b2-2ab cosC
C
b
It is useful to memorize these
laws. They can be used to
solve any triangle if enough
measurements are given.
A
When solving a triangle, you must remember to choose the
correct law to solve it with.
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 timeconsuming 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. You need 3 pieces
of information.
Solve this triangle.
B
c=6
a=4
28º
C
b
A
Because this triangle has an angle given, we can use the law of sines to
solve it.
a/sin A = b/sin B = c/sin C and substitute: 4/sin28º = b/sin B = 6/sin C.
Because we know nothing about b/sin B, let’s start with 4/sin28º and use it
to solve 6/sin C.
B
c=6
a=4
28º
C
b
Cross-multiply those ratios: 4*sin C = 6*sin 28, then divide by 4:
sin C = (6*sin28)/4.
A
6*sin28=2.817. Divide that by four: 0.704. This means that
sin C = 0.704. Find the Csc (sin-1) of 0.704 º.
Csc0.704º = 44.749. Angle C is about 44.749º. Angle B is
about 180º - 44.749º – 28º = 17.251º.
The last unknown is side is b.
a/sinA = b/sinB
4/sin28º = b/sin17.251º
4*sin17.251 = sin28*b
(4*sin17.251) / sin28 = b  b ~ 2.53
Solve this triangle.
Hint: use the law of cosines (as you only have three side
lengths, and no angles)
B
c=5.2
a=2.4
A
b=3.5
C
a2 = b2 + c2 -2bc cosA. Substitute values.
2.42 = 3.52 + 5.22 -2(3.5)(5.2) cosA
5.76 - 12.25 - 27.04 = -2(3.5)(5.2) cos A
33.53 = 36.4 cosA
33.53 / 36.4 = cos A  0.921 = cos A, take the secant  A = 67.1
Now for B.
b2 = a2 + c2 -2ac cosB
(3.5)2 = (2.4)2 + (5.2)2 -2(2.4)(5.2) cosB
12.25 = 5.76 + 27.04 -24.96 cos B
12.25 - 5.76 - 27.04 = -24.96 cos B
20.54 / 24.96 = cos B
0.823 = cos B. Take the secant ...
B = 34.61º
C = 180º - 34.61º - 67.07º = 78.32º
Whew!
Trigonometric identities are ratios
and relationships between certain
trigonometric functions.
In the following few slides, you
will learn about different
trigonometric identities that take
place in each trigonometric
function.
What is the sine of 60º? 0.866. What is the cosine of 30º?
0.866. If you look at the name of cosine, you can actually
see that it is the cofunction of the sine (co-sine). The
cotangent is the cofunction of the tangent (co-tangent), and
the cosecant is the cofunction of the secant (co-secant).
Sine60º=Cosine30º
Secant60º=Cosecant30º
tangent30º=cotangent60º
The following trigonometric identities are useful to remember.
Sin θ=1/csc θ
Cos θ=1/sec θ
Tan θ=1/cot θ
Csc θ=1/sin θ
Sec θ=1/cos θ
Tan θ=1/cot θ
(sin θ)2 + (cos θ)2=1
1+(tan θ)2=(sec θ)2
1+(cot θ)2=(csc θ)2
Degrees and pi radians are two methods of
showing trigonometric info. To convert
between them, use the following equation.
2π radians = 360 degrees
1π radians= 180 degrees
Convert 500 degrees into radians.
2π radians = 360 degrees, 1 degree = 1π radians/180,
500 degrees = π radians/180 * 500
500 degrees = 25π radians/9
Write out the each of the trigonometric functions (sin, cos, and tan) of the following
degrees to the hundredth place.
(In degrees mode). Note: you do not have to do all of them 
1. 45º
7. 90º
13. 47º
19. 75º
2. 38º
8. 152º
14. 442º
20. 34º
3. 22º
9. 112º
15. 123º
21. 53º
4. 18º
10. 58º
16. 53º
22. 92º
5. 95º
11. 345º
17. 41º
23. 153º
6. 63º
12. 221º
18. 22º
24. 1000º
Solve the following right triangles with the dimensions given
B
B
c
52 º
5
C
B
c
a
22
A
C
B
20
A
13
c
9
C
18
A
8º
C
12
A
Solve the following oblique triangles with the dimensions given
B
22
25
12
A
14
B
C
A
28 º
b
31 º
a
C
B
B
c
c
A
8
168 º 5
A
C
35 º
24
15
C
Find each sine, cosecant, secant, and cotangent using different
trigonometric identities to the hundredth place
(don’t just use a few identities, try all of them.).
1. 45º
7. 90º
13. 47º
19. 75º
2. 38º
8. 152º
14. 442º
20. 34º
3. 22º
9. 112º
15. 123º
21. 53º
4. 18º
10. 58º
16. 53º
22. 92º
5. 95º
11. 345º
17. 41º
23. 153º
6. 63º
12. 221º
18. 22º
24. 1000º
Convert to radians
34º
15º
156º
272º
994º
52º
36º
174º
532º
732º
35º
37º
376º
631º
897º
46º
94º
324º
856º
1768º
74º
53º
163º
428º
2000º
Convert to degrees
3.2π rad
6.7π rad
3.14π rad
72.45π rad
52.652π rad
3.1π rad
7.9 rad
6.48π rad
93.16π rad
435.96π rad
1.3π rad
5.4π rad
8.23π rad
25.73π rad
14.995π rad
7.4π rad
9.6π rad
5.25π rad
79.23π rad
745.153π rad
Producer
Director
Creator
Author
Basic Mathematics
Second
edition
MathPower
Nine,
chapter
6
Eric Zhao
By Haym Kruglak, John T. Moore, Ramon Mata-Toledo