Sine and Cosine ratios + SOH CAH TOA

Hi ! Hola ! marhaba ! >>> i have just greeted you in three languages :p

I hope you liked the song  in the previous post and found it helpful . Although it’s not my production but I thought it would be fun to hear trigonometry in a song , and a nice addition to the website.

If you heard the Song , you will notice that they were talking about sine ,cosine and tangent ratios. All of these three are called Primary trigonometric ratios . They are abbreviated as sin, cos and tan.  You know the rule for tan already. These are the new rules.

Cos = adjacent/hypotenuse                      sin= opposite / hypotenuse

If you have been wondering about the meaning of SOH CAH TOA , try looking carefully on the three rules you know . Haven’t figured it out ? its ok , you will feel smart in the next exercise ;p


soh cah toa


Read this now , just to have an idea then, return again  to this part after reading the post.

Summary :

*know what is needed

*Determine the suitable rule to solve the problem.

if its finding :

-Primary Trigonometric ratio/s: You will be given a right angle triangle , then according to the mentioned sides , determine the suitable rule to use remember SOH CAH TOA. (look at second picture’s diagrams) or you can be given all side lengths , and be asked to find all three Ratios.

– tan or cos or sin of an angle : You will be given the name of the Trigonometric Ratio and a number beside it ( example : cos 32°) , then just punch that in to your calculator, and  voila ! that is your answer.

-an angle using the suitable trigonometric ratio: You will look at the right angle triangle you have , and notice were your theta O lies . (Dont forget to label your triangle!) . After that, according to the mentioned sides , you will determine the suitable rule to use. Then you will apply the inverse of the rule you used( inverse sin , inverse cos , inverse tan)  to get the final answer – the angle.

-a side using the suitable trigonometric ratio :You will have to label your triangle , then according to the side you have and the side you need , you will determine which rule or formula to use (example : if you have adj and you want the hypotenuse and the angle O you will use the cosine formula – CAH ) . Now you have your formula so just sub in the stuff you know (adj and the angle O) and solve for h – the hypotenuse to get it.

I hope you got it !


This review has 4 parts :

1- Finding the primary Trigonometric Ratios

2-Finding the Sine and Cosine of an angle

3-Finding an angle using the sine and cosine ratios

4- Solve a Right Triangle ( find the six measures : three side lengths and three angles.)


1- Finding the primary Trigonometric Ratios :

Find the three primary trigonometric ratios for  . Express the ratios as decimals , rounded to four decimal places.

Answer : sin O = oopp/hyp 

= 13.5/15.8

=   0.8544

cos O = adj /hyp

=   8.2/ 15.8


tan O = opp/adj

=   13.5/8.2

=  1.6463

Done !


2- Finding the Sine and Cosine of an angle .

Evaluate the following to four decimal places .

a) sin 24 ° = 0.4067

b) cos 63 ° = 0.454


3- Finding an angle using the sine and cosine ratios

Example :a) A pilot is navigating an aircraft towards point C  which is directly north of the plane’ s location ( point B ). He was ordered to aim for point A  20 Km west of  point  C  because the first order was a mistake . Assuming that the point A is 35 Km from his position now , at what bearing must the pilot head his plane ?

Answer : We will use sine since we have opp and hyp  (that means we are using SOH) :).

sin O =opp/ hyp

= 20/35

= 4/7

= 0.5714

To find O  calculate the inverse sine of 0.5714

O =sin^-1 (0.5714)


Statement : the pilot must head his plane on a bearing of approximately N35°W

b) A captain of a ship is in communication with a submarine that is cruising at a depth of   550m  below sea level . If the captain’s radar tells him that the submarine is  650m  from him , due north of  his ship , at what angle is the submarine located , with respect to the captain’s ship , to the nearest degree ?

Answer : Since the adj and the hyp are known , we will use the cosine ratio.

cos O = adj / hyp

cos = 550/650

= 11/13

Now to find the angle theta we use inverse cosine .

O =cos^-1 (11/13)


statement :The submarine is approximately 32° north of the captain’s ship


4- Solving a Right Triangle

Example: Solve triangle  KJA . Round side lengths to the nearest unit and angles to the nearest degree .

Answer :First you will have to label the corresponding angles.

Since we have two known angles we can use them to find angle C.

Since the sum of the angles of any triangle is = 180°

Then : 180° -(24°+90°) = 66°

To find side j we use the cosine ratio ( because we have the adj and we need the hypotenuse )

cos J = adj/hyp

cos J =a /j

cos 24°=18/j

j(cos 24°)=18 >>>>> multiply sides by j

j= 18/cos 24° >>>>>>>divide both sides by cos 24°

j= 19.7034 >>>>>>>>>>> Rounded to the nearest unit :   j=20

To find k , we will have to use the tangent ratio .

tan A = opp/ adj

tan A = k/a

tan 24° =k/ 18

18(tan 24°) = k >>>>>>> multiply both sides by 18

k= 8.01412  >>>>>>>> Rounded to the nearest unit : k= 8

All done ! , Well done ! if you got the same answers 🙂

i wish you the best luck and  a happy new year ! ( maybe too late ;p)

Bye! Stay tuned for more !


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