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Trigonometric Functions

TRIGONOMETRIC FUNCTIONS

DEFINITION OF TRIGONOMETRIC FUNCTIONS FOR A RIGHT TRIANGLE

Triangle ABC has a right angle (90°) at C and sides of length a, b, c. The trigonometric functions of angle A are denned as follows

sine of A = sine A =

RELATIONSHIP BETWEEN DEGREES AND RADIANS

A radian is that angle e subtended at center O of a circle by an arc MN equal to the radius r.
Since 2a- radians = 360° we have

1 radian = 180°/π = 57.29577 95130 8232. . .°
1° = π/180 radians = 0.01745 32925 19943 29576 92. . .radians

RELATIONSHIPS AMONG TRIGONOMETRIC FUNCTIONS


SIGNS AND VARIATIONS OF TRIGONOMETRIC FUNCTIONS

EXACT VALUES FOR TRIGONOMETRIC FUNCTIONS OF VARIOUS ANGLES

 

GRAPHS OF TRIGONOMETRIC FUNCTIONS



FUNCTIONS OF NEGATIVE ANGLES

  • sin(-A) = - sin A
  • cos (-A) = cos A
  • tan (-A) = - tan A
  • csc(-A) = -esc A
  • sec(-A) = sec A
  • cot(-A)   = -cot A

ADDITION FORMULAS

FUNCTIONS OF ANGLES IN ALL QUADRANTS IN TERMS OF THOSE IN QUADRANT I

RELATIONSHIPS AMONG FUNCTIONS OF ANGLES IN QUADRANT I

DOUBLE ANGLE FORMULAS

HALF ANGLE FORMULAS

MULTIPLE ANGLE FORMULAS

POWERS OF TRIGONOMETRIC FUNCTIONS

  • sin² A = ½ - ½ cos 2A
  • cos² A = ½ + ½ cos 2A
  • sin³ A = ¾ sin A - ¼ sin 3A
  • cos² A = ¾ sin A + ¼ sin 3A

SUM, DIFFERENCE AND PRODUCT OF TRIGONOMETRIC FUNCTIONS

  • sin A + sin B = 2 sin ½(A + B) cos ½(A — B)
  • sin A - sin B = 2 cos ½(A + B) sin ½(A — B)
  • cos A + cos B = 2 cos ½(A + B) cos ½(A - B)
  • cos A - cos B = 2 sin ½(A + B) sin ½(B - A)
  • sin A sin B = ½{cos (A - B) - cos (A + B)}
  • cos A cos B =½{cos (A — B) + cos (A + B)}
  • sin A cos B = ½{sin (A - B) + sin (A + B)}

GENERAL FORMULAS

INVERSE TRIGONOMETRIC FUNCTIONS

If x = sin y then y = sin¹x, i.e. the angle whose sine is x or inverse sine of x, is a many-valued function of x which is a collection of single-valued functions called branches. Similarly the other inverse trigonometric functions are multiple-valued.
For many purposes a particular branch is required. This is called the principal branch and the values for this branch are called principal values.

PRINCIPAL VALUES FOR INVERSE TRIGONOMETRIC FUNCTIONS

Principal values for x ≥ 0 Principal values for x < 0
0 ≤ sin-1 x ≤  π/2 -π/2 ≤ sin-1 x ≤  0
0 ≤ cos-1 x ≤  π/2 π/2 ≤ cos-1 x ≤  π
0 ≤ tan-1 x ≤  π/2 -π/2 ≤ tan-1 x ≤  0
0 ≤ cot-1 x ≤  π/2 π/2 ≤ cot-1 x ≤  π
0 ≤ sec-1 x ≤  π/2 π/2 ≤ sec-1 x ≤  0
0 ≤ csc-1 x ≤  π/2 -π/2 ≤ csc-1 x ≤  π

Relations Between Inverse Trigonometric Functions


sin-1x + cos-1x = π/2 sin-1 (-x)= -sin -1 x;
tan-1x + cot-1x = π/2 cos-1 (-x)= π -cos -1 x
sec-1x + csc-1x = π/2 tan-1 (-x)= -tan-1 x
csc-1x = sin-1(1/x) cot-1 (-x)= π -cot -1 x
sec-1x = cos-1(1/x) sec-(-x)= π -sec -1 x
cot-1x = tan-1(1/x) csc-1 (-x)=-csc -1 x
 

Graphs Of Inverse Trogometric Functions

RELATIONSHIPS BETWEEN SIDES AND ANGLES OF A PLANE TRIANGLE


The following results hold for any plane triangle ABC with sides a, b, c and angles A, B, C.
Law of Sines
laws of sines

Law of Cosines
laws of cosines
with similar relations involving the other sides and angles.
Law of Tangents
tangents
with similar relations involving the other sides and angles.


Similar relations involving angles B and C can be obtained.

RELATIONSHIPS BETWEEN SIDES AND ANGLES OF A SPHERICAL TRIANGLE

Spherical triangle ABC is on the surface of a sphere as shown in Fig. above Sides a, b, c [which are arcs of great circles] are measured by their angles subtended at center O of the sphere. A,B,C are the angles opposite sides a, b, c respectively. Then the following results hold.

Law of Sines

Law of Cosines

with similar relations involving the other sides and angles.

Law of Tangents

with similar relations involving the other sides and angles.



Similar results hold for other sides and angles.

NAPIER'S RULES FOR RIGHT ANGLED SPHERICAL TRIANGLES

Except for right angle C, there are five parts of spherical triangle ABC which if arranged in the order as given in Fig. would be a, b, A, c, B.

Napier

Suppose these quantities are arranged in a circle as in Fig above where we attach the prefix co [indicating complement] to hypotenuse c and angles A and B.
Any one of the parts of this circle is called a middle part, the two neighboring parts are called adjacent parts and the two remaining parts are called opposite parts.  Then Napier's rules are

  • The sine of any middle part equals the product of the tangents of the adjacent parts.
  • The sine of any middle part equals the product of the cosines of the opposite parts.

Since co-A=90°-A, co-B = 90°-B,
we have

sin a = tan b tan (co-B) or sin a = tan b cot B

sin (co-A) = cos a cos (co-B) or cos A = cos a sin B