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 ?

Here is the answer

From the above figures, we can derive formulas for the three trigonometric ratios sin, cos and tan as given below.

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

**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,

AB^{2} = BC^{2} + CA^{2}

AB^{2} = 7^{2} + 24^{2}

AB^{2} = 49 + 576

AB^{2} = 625

AB^{2} = 25^{2}

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,

AC^{2} = AB^{2} + BC^{2}

15^{2} = AB^{2} + 12^{2}

225 = AB^{2} + 144

Subtract 144 from each side.

81 = AB^{2}

9^{2} = AB^{2}

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,

AB^{2} = AD^{2} + BD^{2}

13^{2} = AD^{2} + 5^{2}

169 = AD^{2} + 25

Subtract 25 from each side.

144 = AD^{2}

12^{2} = AD^{2}

12 = AD

In the right ΔACD, by Pythagorean Theorem,

AC^{2} = AD^{2} + CD^{2}

AC^{2} = 12^{2} + 16^{2}

AC^{2} = 144 + 256

AC^{2} = 400

AC^{2} = 20^{2}

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.

HTML Comment Box is loading comments...

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 **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**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**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**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 **

**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 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**