PLANE TRIGONOMETRY
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1. TRIGONOMETRY
• Derived from Greek means “ measurement of triangles “ The subject is principally concerned with the
measurement of triangles (i.e., their sides and angles), or ,more specifically, with the indirect measurement of line
segments and angles. For example, it is possible, by trigonometry, to measure the with of a river without crossing it,
or the height of a pole or cliff without climbing to the top.
• Uses of trigonometry are many. Sciences of physics, mechanics, and astronomy could hardly have developed without
it; practical art such as engineering find it indispensable. It’s a valuable aid in the study of periodic phenomena such
as tides, or even economic data which seem to be cyclic in the nature. Various specific uses will be illustrated
through the book, particularly in examples and exercise.
2 TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE.
• Consider the right triangle A B C, with the right angle at C. The side opposite the angles A, B, C will be denoted by
the corresponding small letters a b c respectively. Then, by taking ratios of the sides of the triangle, we define three
trigonometric functions of the acute angle A as follows:
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B
c
a
A
b
C
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• sine of A (abbreviated sin A)
side opposite A
a
hypotenuse
c
side opposite A
b
• cosine of A (abbreviated cos A)
hypotenuse
c
side opposite A
a
• tangent of A (abbreviated tan A)
hypotenuse
b
Thus, for example, in a right triangle in which a = 3, b = 4, c = 5
3
sin A 5
4
cos A= 5
3
tan A= 4
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The values of these functions are completely determined by the angle A. Thus, if we had an other right triangle with the same acute
angle A, it would be similar to the above triangle and its sides would be in the same proportion. Example, they might all be twice as
long, namely a = 6, b= 8, c=10. Then we should have sin A= 6/10 =3/5, as before, and similary for the other functions. On the other
hand, if the size of angle A were changed, the values of these functions would be changed.
B
cosecant of A (abbreviated csc A)
secant of A (abbreviated sec A)
hypotenuse
c
side opposite A
a
hypotenuse
c
side adjacent to A
b
coatngent of A (abbreviated cot A)
hypotenuse
side adjacent to A
c
b
a=3
Three, and only three, other ratios may also be formed from sides of triangle ABC. They are
A
b=4
C
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It will noted that these three functions are the reciprocals * of the other three,a and we may write
csc A =
1
𝑠𝑖𝑛 𝐴
sin A =
1
csc 𝐴
sec A =
1
cos 𝐴
cos A =
1
𝑠𝑒𝑐 𝐴
1
tan 𝐴
tan A =
1
cot 𝐴
cot A =
ASSIGNMENT 1. A
DRAW THE RIGHT TRIANGLES WHOSE SIDES HAVE THE FOLLOWING VALUES, AND FIND THE SIX TRIGONOMETRIC FUNCTIONS OF
THE ANGLE A:
1. a = 4, b= 3, c= 5
2. a = 2, b= 3, c= √13
3. a = 1, b= 1, c= √2
4. a = 2, b= √5, c= 3
5. a = 7, c= 25
6. a = 5, b= 3
ASSIGNMENT 1. A2
15. A guy wire 15 feet long is fastened to a point 13 feet above the foot of vertical pole, which stands on level
ground. Find the sine of the angle that the wire makes with the horizontal. * the reciprocal of a number is 1 divided by
the number.
16. A yardstick, held vertically on a level surface, cast a shadow 1 foot 8 inches long. Find the tangent of the angle
that the rays of sun make with horizontal.
17. A roadway rises 55 feet in a horizontal distance of ½ mile. Find the tangent of the angle that it makes with
the horizontal.
18. An airplane is descending 225 feet per 1000 ft of horizontal distance covered. What is the cosine of the angle
that its path of descent makes with the horizontal?
19. One end of a foot ruler is placed against a vertical wall; the other end of the ruler reaches a point on the
floor 9 inches from the base of the wall. Find the sine, cosine, and tangent of the angle that the ruler makes
(a)with the wall, (b) with the floor.
20. A box is 3 inches by 4 inches by 1 foot. Find the sine of the angle that a diagonal of the box makes with its
longest edge.
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3. FUNCTIONS OF COMPLEMENTARY ANGLES
By referring to the definitions of the trigonometric functions (section 2) and to Fig. 1, we see that, for the acute angle B,
sin B =
𝑏
cos B =
𝑎
tan B =
𝑏
𝑐
𝑐
𝑎
csc B =
sec B =
cot B =
𝑐
𝑏
𝑐
𝑎
𝑎
𝑏
Comparing these formulas with formulas (1)-(6) of section 2, and making use of the fact that A and B are complementary angles (i.e., A
+ B= 90°), we have
sin B = sin(90
A) = cos A,
cos B = sin(90
A) = sin A,
tan B = sin(90
A) = cot A,
csc B = sin(90
A) = sec A,
sec B = sin(90
A) = tan A,
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It is convenient to arrange the functions in pairs as follows : sine and cosine, tangent and cotangent, secant and
cosecant. In any pair, either function may be called the confunction of the other. Relations (2) may then be ex-pressed
by the single statement: Any functions of the complement of an angle is equal to the confunction of the angle.
4. FINDING THE OTHER FUNCTIONS OF AN ACUTE ANGLE WHEN ONE FUCTION IS GIVEN
The following examples will illustrate how the remaining functions of an acute angle can be found if the value of one
function is given.
Example 1.
Given sin A =
5
13
, A acute; find the other functions of A.
Example 2.
If a tan A = , what are the other functions of A, it being under stood that A is acute?
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ASSIGNMENT 1.C
Find the other five functions of the acute angle A , given that
4
1. cos A = 5
2
2. tan A = 3
1
3. cot A = 5
2
4. sin A = 5
5. sec A =√2
6. csc A =
4.1
9