TRIGONOMETRY
- Key -
Material AICLE. 4º de ESO: Trigonometry (Solucionario)
3
1.
2.
trigonometric
angle measure
chord
sine function
triangulation
trigonometric series
3.
4
Material AICLE. 4º de ESO: Trigonometry
(Solucionario)
4.
SINE
COSINE
TANGENT
COSECANT
SECANT
COTANGENT
Opposite / Adjacent
Opposite / Hypotenuse
Hypotenuse / Opposite
Adjacent / Hypotenuse
Adjacent / Opposite
Hypotenuse / Adjacent
5.
SINE RULE
a
b
c
=
=
sin A sin B sin C
side a divided by the sine of angle A equals side b divided by the sine
of angle B equals side c divided by the sine of angle C
COSINE RULE
a 2 = b 2 + c 2 − 2bc cos A side a squared equals side b squared plus side c squared minus twice
b times c times the cosine of angle A
Material AICLE. 4º de ESO: Trigonometry (Solucionario)
5
7.
A. For the figure given on the left, the value of sin C is
Answer c / b
The sine of an angle is defined as Opposite Side / Hypotenuse. Now for angle C, the opposite
side is c and the hypotenuse is b. Hence the correct answer is c/b.
B. From the figure given on the right, the value of sin A + cos A is
Answer (a + c)/b
We know sin A = a/b, and cos A = c/b. Hence sin A + cos A = (a + c)/b.
C. From the figure given on the left, the value of cos C is
Answer a / b
We know that cos of any angle = Base/Hypotenuse. Now for the angle C, the base is a and
hypotenuse is b. So cos C = a/b.
D. For the figure given on the right, which of the following relationships is true :
Answer cot A = c / a
By definition, cot A = 1 / tan A = c / a.
E. From the figure given on the left, the value of cos C + sin A is
Answer 2a/b
The value of cos C = a/b. Similarly the value of sin A = a/b. Hence cos C + sin A = 2a/b.
F. Which of the following relationships is true:
Answer sin A / cos A = tan A
The expression sin A / cos A = tan A is a useful one to remember in trigonometry.
G. tan A / sin A =
Answer sec A
tan A = sin A / cos A. Therefore tan A / sin A = 1 / cos A = sec A.
H. (sin A / tan A) + cos A =
Answer 2 cos A
We know tan A = sin A / cos A. Therefore (sin A / tan A) + cos A = cos A + cos A = 2 cos A.
I. cot A tan A =
Answer 1
cot A = 1 / tan A. Hence cot A tan A = 1.
Alternatively cot A = cos A/sin A and tan A= sin A/cos A. So cot A tan A = (cos A/sin A) (sin A/cos A) = 1.
J. From the figure, the value of cosec A + cot A is:
Answer (b + c)/a
We know cosec A = b/a and cot A = c/a. Hence cosec A + cot A = (b + c)/a.
6
Material AICLE. 4º de ESO: Trigonometry
(Solucionario)
K. Which of the following relationships is true:
Answer cos A sec A = 1
By definition, sec A = 1 / cos A. So cos A sec A = 1 is true.
L. From the figure, the value of sin2 A + cos2 A is
Answer 1
This question is a bit tricky. We know sin A = a/b and cos A = c/b. So sin2 A + cos2 A = (a2 + c2)
/ b2. By Pythagoras Theorem, a2 + c2 = b2 for a right-angled triangle. Hence sin2 A + cos2 A = 1,
which is a famous identity.
M. From the figure, the value of cot C + cosec C is
Answer (a + b)/c
cot C is Base/Opposite Side and cosec C is Hypotenuse/Opposite Side. From these
definitions, the values of cot C and cosec C are given by a/c and b/c respectively. Hence
the answer is (a + b)/c.
N. cosec A / sec A =
Answer cot A
By definition, cosec A = 1 / sin A and sec A = 1 / cos A. So cosec A / sec A = cos A / sin A =
cot A.
O. For the figure given on the right, the value of cot A is
Answer tan C
The value of cot A is c/a. Similarly the value of tan C is c/a. Hence cot A = tan C.
8.
Example 1
Example 2 Example 3
Material AICLE. 4º de ESO: Trigonometry (Solucionario)
7
Example 4
Example 5
9.
A.
C.
(a) a = 2, A = 30°, B = 40°
b = 2.571, c = 3.75
(a) a = 1, b = 2, C = 30°
c = 1.239, A = 23.8°, B = 126.2°
(b) b = 5, B = 45°, C = 60°
a = 7.044, c = 6.124
(b) a = 3, c = 4, B = 50°
b = 3.094, A = 48°, C = 82°
(c) c = 3, A = 37°, B = 54°
a = 1.806, b = 2.427
(c) b = 5, c = 10, A = 30°
a = 6.197, B = 23.8°, C = 126.2°
B.
D.
(a) a = 3, b = 5, A = 32°
Two possible triangles:
B = 62°, C = 86°, c = 5.647
and B = 118°, C = 30°, c = 2.833
(a) a = 2, b = 3, c = 4
A = 29.0°, B = 46.6°, C = 104.5°
(b) b = 2, c = 4, C = 63°
B = 27°, A = 88°, a = 4.487
(b) a = 1, b = 1, c = 1.5
A = 41.4°, B = 41.4°, C = 97.2°
(c) c = 2, a = 1, B = 108°
b = 2.457, A = 23.2°, C = 51.9°
8
Material AICLE. 4º de ESO: Trigonometry
(Solucionario)
11.
a) The distance of the man from the tower is 20.21 m
b) The length of the string used by the little boy is l = 2 h = 2 (15) = 30 m
c) The height of the second tower is 46.19 m
d) The distance of the ship from the lighthouse is 35.49 m
e) The velocity of the plane is given by V = distance covered / time taken
V= DE / 60 = 19.25 m/s
15.
sinA
cosA
tanA
secA
cosecA
cotA
0º
0
1
0
1
none
none
30º
0.5
0.8660
0.5773
1.1547
2
1.7320
45º
0.7071
0.7071
1
1.4142
1,4142
1
60º
0.8660
0.5
1.7320
2
1.1547
0.5773
90º
1
0
none
none
1
0
16.
tan A = 1.23
cos F = 0.725
A = 50.9 °
F = 43.5 °
tan B = 2.56
tan G = 0.786
B = 68.7 °
G = 38.2 °
tan H = 1.275
sin C = 0.78
H = 51.9 °
C = 51.3 °
sin I = 0.468
sin D = 0.527
I = 27.9 °
D = 31.8 °
sin J = 0.867
cos E = 0.352
J = 60.1 °
E = 69.4 °
Material AICLE. 4º de ESO: Trigonometry (Solucionario)
9
17.
Score
Requirement
1
2
3
4
No Story or logical sequence
of thoughts. One sentence.
No real story.
Lacks imagination
and thought. No
real application of
a trigonometry
problem. Spelling
and grammar
mistakes.
Story lacks one of
the following:
imagination,
complete
sentences,
complete thought,
but still involves a
trigonometry
problem. Some
grammar and
spelling mistakes.
Imaginative story
comprised of
mostly complete
sentences that
involves a
trigonometry
problem. Some
grammar mistakes
or a few
misspellings.
Imaginative story
comprised of
complete
sentences that
involves a
trigonometry
problem. No
grammar mistakes
or misspellings.
Nothing clearly
definable, or
understandable
Picture lacks at
least two of the
following: illustrates
the story, clarity,
appropriate size,
color, appropriate
subject matter.
The picture is not
very visually
appealing.
Picture lacks at
least one of the
following: illustrates
the story, clarity,
appropriate size,
and appropriate
subject matter.
The picture is still
visually appealing
and has color.
Creative picture or
drawing that
somewhat
illustrates the story
and the
trigonometry
problem. Mainly
clear, colorful and
visually appealing.
Appropriate size
and topic.
Creative picture or
drawing that
clearly illustrates
the story and the
trigonometry
problem. Clear,
colorful and
visually appealing.
Appropriate size
and topic.
Diagram of
triangle
No serious attempt to make
diagram.
Lacks labels, units and
accuracy to story.
Not drawn with a
straight edge or
computer, and
missing ONE of the
following: units,
labels. Correctly
drawn to match
story, picture and
solution.
Clearly drawn
using a straightedge
or computer.
Missing TWO of the
following: units,
labels. Correctly
drawn to match
story, picture and
solution.
Clearly drawn
using a straightedge
or computer.
Missing ONE of the
following: units,
labels. Correctly
drawn to match
story, picture and
solution.
Clearly drawn
using a straightedge
or computer.
Units included,
labeled with right
angle. Correctly
drawn to match
story, picture and
solution.
Calculations
Not neatly typed or written
and
missing TWO of the
following: units, formulas,
each
step of the problem, correct
answer.
Not neatly typed or
written and missing
ONE of the
following: units,
formulas, each
step of the
problem, correct
answer.
Neatly typed or
written. Missing
TWO of the
following: units,
formulas, each
step of the
problem, correct
answer.
Neatly typed or
written. Missing
One of the
following: units,
formulas, each
step of the
problem, correct
answer
Neatly typed or
written. Formulas
listed. Each step
clearly outlined
and included.
Units included in
answer. Correct
answer to problem
Neat and
Professional but
has ONE of the
following: Does not
use 3 colors, title
not included, not a
standard size.
Professional look
(no tape showing,
neatly constructed
or drawn)
Story should be
typed or neatly
printed
At least 3 Colors
Title should be
obvious and neat
Size limit: 8½ x 11
inches or a
standard sheet of
construction paper.
Story
Picture or
drawing
Presentation
10
0
Put together on lined paper
or notebook paper.
Rushed,
incomplete, not
professional.
Material AICLE. 4º de ESO: Trigonometry
Not quite
professional. Looks
rushed or quickly
put together.
Lacks colors, title or
does not fit
standard size
requirements
(Solucionario)