Trigonometry is the study of relationships between sides and angles of triangles. It was originally developed to solve geometric problems involving triangles. Today, trigonometry has many applications in fields like electrical engineering, physics, navigation, construction and more. The document discusses key concepts in trigonometry including defining angles using radians and degrees, trigonometric functions like sine, cosine and tangent, and important trigonometric identities.

2. INTRODUCTIOINTRODUCTIO
NNTHE WORD TRIGONOMETRY IS DERIVED FROM
THE GREEK WORDS TRIGON AND METRON AND
IT MEANS MEASURING THE SIDES OF A
TRIANGLE.
THE SUBJECT WAS ORIGINALLY DEVELOPED
TO SOLVE GEOMETRIC PROBLEMS INVOLVING
TRIANGLES.
CURRENTLY TRIGONOMETRY IS USED IN MANY
AREAS SUCH AS DESIGNING ELECTRICAL
CIRCUITS ,DESCRIBING THE STATE OF AN
ATOM,PREDICTING THE HEIGHTS OF TIDES IN
OCEAN AND IN MANY OTHER AREAS.
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3. KEY FOR TRIGONOMETRICKEY FOR TRIGONOMETRIC
FUNCTIONSFUNCTIONS
INTRODUCTIONINTRODUCTION
ANGLESANGLES
DEGREE MEASUREDEGREE MEASURE
RADIANRADIAN
FORMULASFORMULAS

4. AnAngglesles
Measuring angles
Units
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5. Measuring AngleMeasuring Angle
 The value ofThe value of θθ thus defined is independent of the sizethus defined is independent of the size
of the circle: if the length of the radius is changedof the circle: if the length of the radius is changed
then the arc length changes in the same proportion,then the arc length changes in the same proportion,
so the ratioso the ratio ss//rr is unaltered.is unaltered.
 In many geometrical situations, angles that differ byIn many geometrical situations, angles that differ by
an exact multiple of a full circle are effectivelyan exact multiple of a full circle are effectively
equivalent (it makes no difference how many times aequivalent (it makes no difference how many times a
line is rotated through a full circle because it alwaysline is rotated through a full circle because it always
ends up in the same place). However, this is notends up in the same place). However, this is not
always the case. For example, when tracing a curvealways the case. For example, when tracing a curve
such as asuch as a spiralspiral usingusing polar coordinatespolar coordinates, an extra full, an extra full
turn gives rise to a quite different point on the curve.turn gives rise to a quite different point on the curve.
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6. Degrees
30° 45° 60° 90°
Radians
Degrees
120° 150° 180° 360°
Radians

7. UnitsUnits
 Angles are considered dimensionless, since they areAngles are considered dimensionless, since they are
defined as the ratio of lengths.defined as the ratio of lengths.
 With the notable exception of the radian, most units of angularWith the notable exception of the radian, most units of angular
measurement are defined such that one full circle (i.e. onemeasurement are defined such that one full circle (i.e. one
revolution) is equal torevolution) is equal to nn units, for some whole numberunits, for some whole number nn (for(for
example, in the case of degrees,example, in the case of degrees, nn = 360). This is equivalent to= 360). This is equivalent to
settingsetting kk == nn/2/2ππ in the formula above. (To see why, note that onein the formula above. (To see why, note that one
full circle corresponds to an arc equal in length to the circle'sfull circle corresponds to an arc equal in length to the circle's
circumferencecircumference, which is 2, which is 2πrπr, so, so ss = 2= 2πrπr. Substituting, we get. Substituting, we get θθ ==
ksks//rr = 2= 2πkπk. But if one complete circle is to have a numerical. But if one complete circle is to have a numerical
angular value ofangular value of nn, then we need, then we need θθ == nn. This is achieved by. This is achieved by
settingsetting kk == nn/2/2ππ.).)
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Angles

8. Degree MeasureDegree Measure
• TheThe degreedegree, denoted by a small superscript, denoted by a small superscript
circle (°) is 1/360 of a full circle, so one full circlecircle (°) is 1/360 of a full circle, so one full circle
is 360°. One advantage of this oldis 360°. One advantage of this old sexagesimalsexagesimal
subunit is that many angles common in simplesubunit is that many angles common in simple
geometry are measured as a whole number ofgeometry are measured as a whole number of
degrees. (The problem of havingdegrees. (The problem of having allall
"interesting" angles measured as whole"interesting" angles measured as whole
numbers is of course insolvable.) Fractions of anumbers is of course insolvable.) Fractions of a
degree may be written in normal decimaldegree may be written in normal decimal
notation (e.g. 3.5° for three and a half degrees),notation (e.g. 3.5° for three and a half degrees),
but the following sexagesimal subunits of thebut the following sexagesimal subunits of the
"degree-minute-second" system are also in use,"degree-minute-second" system are also in use,
especially forespecially for geographical coordinatesgeographical coordinates and inand in
astronomyastronomy andand ballisticsballistics::
θθ == ss//rr rad = 1 rad.rad = 1 rad.
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9. Radian
• The radian is the angle subtended by an arc of a
circle that has the same length as the circle's
radius (k = 1 in the formula given earlier). One full
circle is 2π radians, and one radian is 180/π
degrees, or about 57.2958 degrees. The radian is
abbreviated rad, though this symbol is often
omitted in mathematical texts, where radians are
assumed unless specified otherwise. The radian is
used in virtually all mathematical work beyond
simple practical geometry, due, for example, to the
pleasing and "natural" properties that the
trigonometric functions display when their
arguments are in radians. The radian is the
(derived) unit of angular measurement in the SI
system

10. • In order to measure an angle θ, a
circular arc centered at the vertex of
the angle is drawn, e.g. with a pair of
compasses. The length of the arc s
is then divided by the radius of the
circle r, and possibly multiplied by a
scaling constant k (which depends
on the units of measurement that are
chosen):
• The value of thus defined is
independent of the size of the circle:
if the length of the radius is changed
then the arc length changes in the
same proportion, so the ratio s/r is
unaltered.
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11. Trigonometric functions
The tangent (tan) of an angle is the ratio of the sine to the
cosine:

12. Pythagorean identity
The basic relationship between the sine
and the cosine is the Pythagorean
trigonometric identity:

13. • sin(π/2–x) =cos xsin(π/2–x) =cos x
• sin(sin(ππ/2+x) =cos x/2+x) =cos x
• cos(π/2–x) =sin xcos(π/2–x) =sin x
• cos(cos(ππ/2+x) =-sin x/2+x) =-sin x
• cos(cos(ππ/2-x) =cos x/2-x) =cos x
• sin(sin(ππ/2-x) =-sin x/2-x) =-sin x

14. Sign of trigonometric
function

15. • sin (x + y) =sin x cos y + cos x sin ysin (x + y) =sin x cos y + cos x sin y
• sin (x – y) =sin x cos y – cos x sin ysin (x – y) =sin x cos y – cos x sin y
• cos (x + y) =cos x cos y – sin x sin ycos (x + y) =cos x cos y – sin x sin y
• cos (x – y) =cos x cos y + sin x sin ycos (x – y) =cos x cos y + sin x sin y

16. • Tan(x+y) = tanx +tany/1- tanx tany
• Tan (x-y) = tanx-tany/1+tanx tany
• Cot (x+y) = cotx+coty/1-cotx coty
• Cot (x-y) = cotx-coty/1+cotx coty
• cos2x= = cos

17. Multiple-angle formulae

19. • Made By:-Made By:- AnandAnand
YadavYadav Class-Class-
XIXI BB
• Made By:-Made By:- AnandAnand
YadavYadav Class-Class-
XIXI BB