# FIND TRIGONOMETRIC RATIOS USING RIGHT TRIANGLES

To find all trigonometric ratios from the given right triangle, first we have to name the sides as hypotenuse side, opposite side and adjacent side.

Hypotenuse Side :

The side which is opposite to 90 degree is known as hypotenuse side.

Opposite Side :

The side which is opposite to θ is known as opposite side

Apart from hypotenuse and opposite side, the remaining third side of the triangle is known as adjacent side.

Generally we have 6 trigonometric ratios, those are sin θ, cos θ, tan θ, csc θ, sec θ and cot θ.

Formulas to find the values of the above six trigonometric ratios.

sin θ  =  Opposite side/hypotenuse side

cos θ  =  Adjacent side/hypotenuse side

tan θ  =  Opposite side/Adjacent side

csc θ  =  Hypotenuse side/Opposite side

sec θ  =  Hypotenuse side/Adjacent side

cot θ  =  Adjacent side/Opposite side

From the above formulas, we ca get the following results.

sin θ and csc θ are reciprocal to each other

cos θ and sec θ are reciprocal to each other

tan θ and cot θ are reciprocal to each other

## Examples

Example 1 :

In the right triangle shown below, find the six trigonometric ratios of the angle θ. Solution :

From the above triangle, we come to know that the right angled at B.

AC - hypotenuse side  =  10

AB - opposite side  =  6

BC - Adjacent side  =  8

Finding the vale of sin θ :

sin θ  =  Opposite side/Hypotenuse side

sin θ  =  AB/AC

sin θ  =  6/10

sin θ  =  3/5

Finding the vale of cos θ :

cos θ  =  Adjacent side/Hypotenuse side

cos θ  =  BC/AC

cos θ  =  8/10

cos θ  =  4/5

Finding the vale of tan θ :

tan θ  =  Opposite side/Adjacent side

tan θ  =  AB/BC

tan θ  =  6/8

tan θ  =  3/4

Finding the vale of csc θ :

csc θ = Hypotenuse side/Opposite side

csc θ  =  AC/AB

csc θ  =  10/6

csc θ  =  5/3

Finding the vale of sec θ :

sec θ = Hypotenuse side/Adjacent side

sec θ  =  AC/BC

sec θ  =  10/8

sec θ  =  5/4

Finding the vale of cot θ :

cot θ = Adjacent side/opposite side

cot θ  =  BC/AB

cot θ  =  8/6

cot θ  =  4/3

Example 2 :

In the right triangle shown below, find the six trigonometric ratios of the angle θ. Solution :

In the triangle above, right angle is at C.

AB - hypotenuse side = 25

AC - opposite side = 7

BC - Adjacent side = 24

Finding the vale of sin θ :

sin θ  =  Opposite side/Hypotenuse side

sin θ  =  AC/AB

sin θ  =  7/25

Finding the vale of cos θ :

cos θ  =  Adjacent side/Hypotenuse side

cos θ  =  BC/AB

cos θ  =  24/25

Finding the vale of tan θ :

tan θ  =  Opposite side/Adjacent side

tan θ  =  AC/BC

tan θ  =  7/24

Finding the vale of csc θ :

csc θ  =  Hypotenuse side/Opposite side

csc θ  AB/AC

csc θ  =  25/7

Finding the vale of sec θ :

sec θ  =  Hypotenuse side/Adjacent side

sec θ  =  AB/BC

sec θ   =  25/24

Finding the vale of cot θ :

cot θ  =  Adjacent side/opposite side

cot θ  =  BC/AC

Example 3 :

In the right triangle shown below, find the six trigonometric ratios of the angle θ. Solution :

From the above triangle right angled at C.

AB - hypotenuse side = 37

AC - opposite side = 35

BC - Adjacent side = 12

Finding the vale of sin θ :

sin θ  =  Opposite side/Hypotenuse side

sin θ  =  AC/AB

sin θ  = 35/37

Finding the vale of cos θ :

cos θ  =  Adjacent side/Hypotenuse side

cos θ  =  BC/AB

cos θ  =  12/37

Finding the vale of tan θ :

tan θ  =  Opposite side/Adjacent side

tan θ  =  AC/BC

tan θ  =  35/12

Finding the value of csc θ :

csc θ  =  Hypotenuse side/Opposite side

csc θ  =  AB/AC

csc θ  =  37/35

Finding the value of sec θ :

sec θ  =  Hypotenuse side/Adjacent side

sec θ  =  AB/BC

sec θ  =  37/12

Finding the value of cot θ :

cot θ  =  Adjacent side/opposite side

cot θ  =  BC/AC

cot θ  =  12/35 Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6 