Trigonometry And Its Short Tricks

Trigonometric Function

Trigonometric Functions (Right Triangle)

Special Angles

Trigonometric Function Values in Quadrants II, III, and IV

Examples:

Example2:

Example: 3:

Unit Circle

Addition Formulas:

• cos(X+Y) = cosXcoxY – sinXsinY
• cos(X-Y) = cosXcoxY + sinXsinY
• sin(X+Y) = sinXcoxY + cosXsin
• sin(X-Y) = sinXcoxY – cosXsinY
• tan(X+Y) = [tanX+tanY]/ [1– tanXtanY]
• tan(X-Y) = [tanX-tanY]/ [1+ tanXtanY]
• cot(X+Y) = [cotX+cotY-1]/ [cotX+cotY]
• cot(X-Y) = [cotX+cotY+1]/ [cotX-cotY]

Sum to Product Formulas:

• cosX + cosY = 2cos [(X+Y) / 2] cos[(X-Y)/2]
• sinX + sinY = 2sin [(X+Y) / 2] cos[(X-Y)/2]

Difference to Product Formulas

• cosX - cosY = - 2sin [(X+Y) / 2] sin[(X-Y)/2]
• sinX + sinY = 2cos [(X+Y) / 2] sin[(X-Y)/2]

Product to Sum/Difference Formulas

• cosXcosY = (1/2) [cos (x-Y) + cos (X+Y)]
• sinXcoxY = (1/2) [sin (x+Y) + sin (X-Y)]
• cosXsinY = (1/2) [sin (x+Y) + sin (X-Y)]
• sinXsinY = (1/2) [cos (x-Y) + cos (X+Y)]

Difference of Squares Formulas

• sin²X – sin²Y = sin (X+Y) sin (X-Y)
• cos²X – cos²Y = - sin (X+Y) sin (X-Y)
• cos²X – sin²Y = cos (X+Y) cos (X-Y)

Double Angle Formulas

• sin (2X) = 2 sin X cos X
• cos (2X) = 1 – 2sin²X= 2cos²X – 1
• tan(2X) = 2tanX/[1-tan²X]

Multiple Angle Formulas

More half-angle formulas

Key Angle Formulas (cont’d)Co-functions

Each trigonometric function has a co-function with symmetric properties in Quadrant I. The following identities express the relationships between co-functions.

• sinq= con(90⁰- q)
• Cosq = sin(90⁰- q)
• tanq = cot(90⁰- q)
• cotq = tan(90⁰- q)
• secq = cosec(90⁰- q)
• cosecq =sec(90⁰- q)

Law of Sines

a/sinA = b/sinB= c/sinC

Law of Cosines

a² = b² +c² – 2bcCosA

b² = a² + c² – 2ac CosB

c² = a² + b² – 2abCosC

Pythagorean Identities

a. sin² X + cos² X = 1

b. 1 + tan² X = cec² X

a. 1 + cot² X = csc² X

Given Three Sides and no Angles (SSS)

• Given three segment lengths and no angle measures, do the following:
• Use the Law of Cosines to determine the measure of one angle.
• Use the Law of Sines to determine the measure of one of the two remaining angles.

Subtract the sum of the measures of the two known angles from 180˚ to obtain the measure of the remaining angle.

Given Two Sides and the Angle between Them (SAS)

Given two segment lengths and the measure of the angle that is between them, do the following:

• Use the Law of Cosines to determine the length of the remaining leg.
• Use the Law of Sines to determine the measure of one of the two remaining angles.
• Subtract the sum of the measures of the two known angles from 180˚ to obtain the measure of the remaining angle.

Given One Side and Two Angles (ASA or AAS)

Given one segment length and the measures of two angles, do the following:

• Subtract the sum of the measures of the two known angles from 180˚ to obtain the measure of the remaining angle.
• Use the Law of Sines to determine the lengths of the two remaining legs.

Some Important Tricks

Remember Useful Point :

•  tan1. tan2. ……… tan89 = 1
• cot1. cot2 ……. Cot89⁰ = 1
• cos1⁰.cos2⁰…… cos90⁰ = 0
• cos1⁰.cos2⁰…… to (greater than cos90⁰) = 0
• sin1⁰.sin2⁰.sin3⁰ ……… sin180⁰ = 0
• sin1⁰. sin2⁰ sin3⁰ ….. to (greater than sin180⁰) = 0