Following are some useful results about angles:
1. Interior angles formed when two straight lines intersect are equal. For exam- ple, in Figure 0.8, the two angles and are equal.
2. When two parallel lines are intersected by a diagonal straight line, the alter- nate interior angles are equal. For example, in Figure 0.9, the two angles and
are equal.
3. When the sides of one angle are each perpendicular to the corresponding sides of a second angle, then the two angles are equal. For example, in Figure 0.10, the two angles and are equal.
4. The sum of the angles on one side of a straight line is 180°. In Figure 0.11, 5. The sum of the angles in any triangle is 180°.
u 1 f 5180°.
u f
f u
u f
V5 pr2h.
V543pr3. A54pr2
A5 pr25 pd2/4. d52r, C52pr5 pd,
0.7 Plane Geometry and Trigonometry 0-11
Interior angles formed when two straight lines intersect are equal:
u 5 f
u f
䊱FIGURE 0.8
When two parallel lines are intersected by a diagonal straight line, the alternate
interior angles are equal:
u 5 f u f
䊱FIGURE 0.9
90° 90°
When the sides of one angle are each perpen- dicular to the corresponding sides of a second angle, then the two angles are equal:
u 5 f u
f
䊱FIGURE 0.10 Area A 5 ab
Area A 5 pr2 Radius Circumference C 5 2pr 5 pd
Diameter d 5 2r
Surface area A 5 4pr2
Volume V 5 abc
Volume V 5 pr2h
Volume V 5 pr3
Radius
h
r b
r
Radius r a
b c
a
4 3
䊱FIGURE 0.7
The sum of the angles on one side of a straight line is 180°:
u 1 f 5 180°
u f
䊱FIGURE 0.11 b1
b2 a1
a2
c1
c2 f1
u1
u2
g1
g2
f2
Two similar triangles: Same shape but not necessarily the same size.
䊱FIGURE 0.12
Similar Triangles
Triangles are similarif they have the same shape, but different sizes or orientations.
Similar triangles have equal corresponding angles and equal ratios of corresponding sides. If the two triangles in Figure 0.12 are similar, then
and a1
a2 5b1
b2
5 c1
c2.
g15 g2, f15 f2,
u15 u2,
If two similar triangles have the same size, they are said to be congruent.If triangles are congruent, one can be flipped and rotated so that it can be placed precisely on top of the other.
Right Triangles and Trig Functions
In a right triangle,one angle is 90°. Therefore, the other two acute angles (acute means less than 90°) have a sum of 90°. In Figure 0.13, The side opposite the right angle is called the hypotenuse(side cin the figure). In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For the triangle in Figure 0.13,
This formula is called the Pythagorean Theorem.
If two right triangles have the same value for one acute angle, then the two triangles are similar and have the same ratio of corresponding sides. This true statement allows us to define the functions sine, cosine,and tangent that are ratios of a pair of sides. These functions, called trigonometric functions or trig functions,depend only on one of the angles in the right triangle. For an angle , these functions are written and
In terms of the triangle in Figure 0.13, the sine, cosine, and tangent of the angle are as follows:
Note that For angle , and
In physics, angles are expressed in either degrees or radians, where radians (For more on radians, see Section 9.1.) Most calculators have a key for switching between degrees and radians. Always be sure that your calcula- tor is set to the appropriate angular measure.
Inverse trig functions, denoted, for example, by (or arcsin x) have a value equal to the angle that has the value xfor the trig function. For example, Note that does notmean Also, note that when you determine an angle using inverse trigono- metric functions, the calculator will always give you the smallest correct angle, which may or may not be the right answer. Use the knowledge of which quadrant you are working in to determine the correct angle in the situation.
1 sinx.
sin21x sin30°50.500, so sin2110.5002 5arcsin10.5002 530°.
sin21x 5180°.
p tanf 5b
a. cosf 5a
c, sinf 5b
c, f
tanu 5 sinu cosu.
tanu 5 opposite side adjacent side 5a
b. cosu 5adjacent side
hypotenuse 5 b c, and sinu 5opposite side
hypotenuse 5 a c, u
tanu. cosu,
sinu,
u c25a21b2. u 1 f 590°.
EXAMPLE 0.13 Using trigonometry I
A right triangle has one angle of 30° and one side with length 8.0 cm, as shown in Figure 0.14. What is the angle and what are the lengths xf and yof the other two sides of the triangle?
S O L U T I O N
S E T U P A N D S O LV E 30 90 , so 60 .
so
To find y,we use the Pythagorean Theorem:
so y516.0 cm.
113.9 cm22,
y25 18.0 cm221 x5 8.0 cm
tan30o513.9 cm.
tan30°58.0 cm x ,
5 ° f 5 °
1 ° f
Or we can say so
which agrees with the previous result.
REFLECT Notice how we used the Pythagorean Theorem in
combination with a trig function. You will use these tools con- stantly in physics, so make sure that you can employ them with confidence.
16 cm,
y58.0 cm/sin30o5
sin30°58.0 cm/y,
Hypotenuse c
90°
a
b u
f
For a right triangle:
u 1 f 5 90°
c2 5 a2 1 b2 (Pythagorean theorem)
䊱FIGURE 0.13
8.0 cm
y
x
90° 30°
f
䊱FIGURE 0.14
u
x
3.0 m 5.0 m
90°
䊱FIGURE 0.15
In a right triangle, all angles are in the range from 0° to 90°, and the sine, cosine, and tangent of the angles are all positive. This must be the case, since the trig functions are ratios of lengths. But for other applications, such as finding the components of vectors, calculating the oscillatory motion of a mass on a spring, or describing wave motion, it is useful to define the sine, cosine, and tangent for angles outside that range. Graphs of and are given in Figure 0.16. The values of and vary between and Each function is periodic, with a period of 360°. Note the range of angles between 0° and 360° for which each function is positive and negative. The two functions and are 90° out of phase (that is, out of step). When one is zero, the other has its maximum magni- tude (i.e., its maximum or minimum value).
For any triangle (see Figure 0.17)—in other words, not necessarily a right triangle—the following two relations apply:
1. (law of sines).
2. (law of cosines).
Some of the relations among trig functions are called trig identities. The follow- ing table lists only a few, those most useful in introductory physics:
Useful trigonometric identities
even
sin (180 ) sin cos (180 ) –cos
sin (90 ) cos cos (90 ) sin For small angle (in radians),
cos 1 1
sinu<u u2<
2 2
<
u u
u 5 u 2
°
u 5 u 2
°
u 5 u 2
°
u 5 u 2
°
cos1u 6 f2 5cosu cosf 7sinu sinf sin1u 6 f2 5sinu cosf 6cosu sinf cos 2u 5cos2u 2sin2u 52 cos2u 215122sin2u
sin 2u 52 sinu cosu
function2 1cosu is an
cos12u2 5cos1u2
1sinu is an odd function2 sin12u2 5 2sin1u2
c25a21b222abcosg sina
a 5 sinb
b 5 sing c
cosu sinu
21.
11 cosu
sinu sinu cosu
0.7 Plane Geometry and Trigonometry 0-13
EXAMPLE 0.14 Using trigonometry II
A right triangle has two sides with lengths as specified in Figure 0.15. What is the length xof the third side of the triangle, and what is the angle , in degrees?u
S O L U T I O N
S E T U P A N D S O LV E The Pythagorean Theorem applied to
this right triangle gives so
(Since xis a length, we take the positive root of the equation.) We also have
so 53 .u 5cos2110.602 5 ° cosu 53.0 m
5.0 m50.60,
" 15.0 m222 13.0 m2254.0 m.
x5 13.0 m221x25 15.0 m22,
R E F L E C T In this case, we knew the lengths of two sides, but none of the acute angles, so we used the Pythagorean Theorem first and then an appropriate trig function.
cos u
90° 0 –1
1
180°270°360° u sin u
90° 0 –1
1
180°270°360° u
䊱FIGURE 0.16
a
b g b
a c
䊱FIGURE 0.17
Problems
0.1 Exponents
Use the exponent rules to simplify the following expressions:
1. ( 3x4y2)2 2.
3. 4.
0.2 Scientific Notation and Powers of 10 Express the following expressions in scientific notation:
5. 475000 6. 0.00000472
7. 123 10–6 8.
0.3 Algebra
Solve the following equations using any method:
9. 4x 6 9x 14
10.F 9 5 C 32 (solve for C) 11. 4x2 6 3x2 18
12. 196 9.8t2 13.x2 5x 6 0 14.x2 x 1 0 15.
16. 5x 4y 1, 6y 10x 4
17. 2, 2x y 1