# SOHCAHTOA

This is a way to remember how values the trigonometric ratios sin, cosine and tangent of an angle can be computed.

Let us see, how this shortcut works to remember the above mentioned trigonometric ratios.

Before we discuss this shortcut, let us know the name of each side of a right triangle from the figure given below. To understand the shortcut, first we have to divide SOHCAHTOA in to three parts as given below. What do SOH, CAH and TOA stand for ?   From the above figures, we can derive formulas for the three trigonometric ratios sin, cos and tan as given below.   ## Reciprocal Relations

The trigonometric ratios csc θ, sec θ and cot θ are the reciprocals of sin θ, cos θ and tan θ respectively. ## Practice Problems

Problem 1 :

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

In the above right angled triangle, note that for the given angle θ, PQ is the ‘opposite’ side and PR is the ‘adjacent’ side.

Then,

sin θ  =  opposite side / hypotenuse  =   PQ/QR  =  5/13

cos θ  =  adjacent side / hypotenuse  =  PR/QR  =  12/13

tan θ  =  opposite side / adjacent side  =  PQ/PR  =  5/12

csc θ  =  1/sin θ  =  13/5

sec θ  =  1/cos θ  =  13/12

cot θ  =  1/tan θ  =  12/5

Problem 2 :

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

In the right angled triangle ABC shown above,

AC  =  24

BC  =  7

By Pythagorean theorem,

AB2  =  BC2 + CA2

AB2  =  72 + 242

AB2  =  49 + 576

AB2  =  625

AB2  =  252

AB  =  25

Now, we can use the three sides to find the six trigonometric ratios of angle θ.

sin θ  =  opposite side / hypotenuse  =   BC/AB  =  7/25

cos θ  =  adjacent side / hypotenuse  =  AC/AB  =  24/25

tan θ  =  opposite side / adjacent side  =  BC/AC  =  7/24

csc θ  =  1/sin θ  =  25/7

sec θ  =  1/cos θ  =  25/24

cot θ  =  1/tan θ  =  24/7

Problem 3 :

In triangle ABC, right angled at B, 15sin A = 12. Find the other five trigonometric ratios of the angle A.

Solution :

15sin A = 12

sin A  =  12/15

sin A  =  opposite side / hypotenuse  =  12 / 15 By Pythagorean theorem,

AC2  =  AB2 + BC2

152  =  AB2 + 122

225  =  AB2 + 144

Subtract 144 from each side.

81  =  AB2

92  =  AB2

9  =  AB

Now, we can use the three sides to find the five trigonometric ratios of angle A and six trigonometric ratios of angle C.

cos A :

=  adjacent side / hypotenuse  =  AB/AC  =  9/15  =  3/5

tan A :

=  opposite side / adjacent side  =  BC/AB  =  12/9  =  4/3

csc A  =  1/sin A  =  15/12  =  5/4

sec A  =  1/cos A  =  5/3

cot A  =  1/tan A  =  4/3

Problem 4 :

In the figure shown below, find the values of

sin B, sec B, cot B, cos C, tan C and csc C In the right ΔABD, by Pythagorean Theorem,

Subtract 25 from each side.

In the right ΔACD, by Pythagorean Theorem,

AC2  =  122 + 162

AC2  =  144 + 256

AC2  =  400

AC2  =  202

AC  =  20

Then,

sin B  =  opposite side / hypotenuse  =  AD/AB  =  12/13

sec B  =  hypotenuse / adjacent side  =  AB/BD  =  13/5

cot B  =  adjacent side / opposite side  =  BD/AD  =  5/12

cos C :

=  adjacent side / hypotenuse  =  CD/AC  =  16/20  =  4/5

tan C :

=  opposite side / adjacent side  = AD/CD  =  12/16  =  3/4

csc C :

=  hypotenuse/ opposite side  =  AC/AD  =  20/12  =  5/3 Apart from the stuff given in this section if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

v4formath@gmail.com

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

Featured Categories

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test 