Trigonometrically Equivalent

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The name trigonometry deals with the subject who provides the relationships between the measurements of sides and the angles of a triangle. Trigonometrically equivalent is the both sides of equal for trigonometric identities and trigonometric ratios.

Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle.
Trigonometric Equivalent Ratios:

Some trigonometric equivalent ratios as follows

Sintheta =1/cosectheta ;

Costheta =1/sectheta

tan theta =1/cottheta

cosec theta =1/sintheta

sectheta =1/costheta

cot theta =1/ tantheta
Examples of Trigonometric Equivalent Ratio:

Example 1:

Find the trigonometrically equivalent ratios by using Pythagoras theorem

Solution:

We note that for the angle theta , length of opposite side = 6; length of adjacent side = 8. By,

(Length of hypotenuse side)2 = 82 + 62 = 64 + 36 = 100.

Length of hypotenuse side = v100 = 10.

Right triangle ...
... (1)

Example 2:

In ?ABC, m B= 90°, AB = 8cm, AC = 17cm. Find all the trigonometrically ratios for the angles A and C by using Pythagoras theorem

Solution:

Here A = m? BAC and C = m? BCA

By Pythagoras formula, AC2 = AB2 + BC2

BC2 = AC2 – AB2 = 172 – 82 = 289 – 64 = 225.

BC =v 225 = 15. Hence we have

Right triangle (2)

MATHEMATICS

Example 3:

Prove that (tan A+ tan b) / (1-tanA+tanB) = (sinAcosB + cosAsinB) / (cos Acos B-sinA sinB)

(tanA+tanB) / (1-tanA+tanB) = (sinA/cosA)+(sinB /cos B)/(1-[(sinA/cosA)+(sinB/cosB))

=[(sinAcosB+sinBcosA)/cosAcosB]/

[(cosAcosB-sinAsinB)/cosAcosB]

=(sinAcosB+cosAsinB) / (cosAcosB-sinAsinB)

Show that 1+tan0/1+cot0=(sin0+tan0)/(1+cos0)

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Example3

Evaluate 2cos2 30 tan260 - sec245 sin260.

Solution: cos 30 =v 3 / 2 , tan 60 =v 3, sec 45 = v2, sin 60° =v 3/2.

= 2cos230 tan260 - sec2 45 sin2 60

= 2(v3/2)2 32 - 2(v3/2)2 v22=(v3/2)4-(v3/2) *2

= 2 × (3 /4)× 3 – 2 × 3/4 = 9/2-3/2 = 3.

Practice problem:

1. If cosecA=2, find (i) sin A + cos A (ii) tan A + cot A.

Answer: (1) sin A + cos A =2

(2) tan A + cot A = 2

2.prove that ( tanA+cotB) / ( cotA + tanB ) = tan A / tan B

Measurement is defined as a magnitude and unit of a physical quantity, which is used for measuring different things which are used in our daily life.

The magnitude represents numerical quantity (i.e. 5 or 7or 44 ...etc) while the unit is used (i.e. meter, liter, gram....) to represent any quantity in different form of measurement.

Examples:

1) When we go to a shop and want to buy a cloth we usually measure it in a Meter.

2) When we buy any liquid we mostly use Liter.

When we change one unit to another then there is some change in magnitude but quantity does not change is known as Equivalent Measurement.

Example:

1 kilometer (km) = 1000 meters (m)

In the above example when we change unit from kilometer to meter there is 1000 times change in magnitude.

Equivalent measure
Different Unit of Measurement for Different Items

Mass

The standard unit to measure the mass in metric system is’ Gram’. Other units of mass and its equivalents are as follows

1 kilogram (kg) = 1000 grams (g)

1 decigram (dg) = 0.1 gram (g)

1 centigram (cg) = 0.01 gram (g)
1 milligram (mg) = 0.001 gram (g)

Length

The standard unit to measure the length in metric system is ‘Meter’. Other units of length and its equivalents are as follows

1 kilometer (km) = 1000 meters (m)

1 decimeter (dm) = 0.1 meter (m)

1 centimeter (cm) = 0.01 meter (m)

1 millimeter (mm) = 0.001 meter (m)

Time

The different standard units used for time measurement are second,minute,hour etc.. The equivalent conversion of these units are as follows

60 seconds =1 minute
60 minutes = 1 hour = 3600 seconds
1 day = 24 hours
1 week = 7 days
Area

The standard form to measure area of any section is ‘Square Meter’. Other units of area and its equivalents are as follows

1 Square Meter (m2) = 10000 Square Centimeter (cm2)

Volume

The standard unit to measure the volume in metric system is ‘Liter’. Other units of volume and its equivalents are as follows

1 kiloliter (kl) = 1000 liters (l)

1 deciliter (dl) = 0.1 liter (l)

1 centiliter (cl) = 0.01 liter (l)

1 milliliter (ml) = 0.001 liter (l)

Temperature

The different standard units used for temperature measurement are Celsius and fahrenheit. In metric system it is expressed in 'degrees Celsius'. The boiling point of water is 100°Celsius. The freezing point of water is 0° Celsius. It is denoted by 'C'.

The temperature of body is measured in fahrenheit. It is denoted by 'F'.

Relation between degree Celsius(C) and fahrenheit(F).

(F-32)/(9) = (C)/(5)

Learn more on about the pythagorean theorem and its Examples. Between, if you have problem on these topics Data Sets for Graphing, Please share your comments.