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

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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 !

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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.)

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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

=0.519

tan O = opp/adj

=   13.5/8.2

=  1.6463

Done !

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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

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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)

=34.8479°

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)

=32.2042°

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

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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 !

The Tangent Ratio (part 2)

hi again ! this post is a continuing post for part 1 ,

So lets start right away!

2- Finding the Tangent of an angle

Example : Record each of the following with a calculator .Record your answer rounded to four decimal places .

a) Tan 50º  = 1.1918

b) Tan 20º  = 0.364

This is what it’s all about 🙂 just use your calculator

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Now , to the next part ,

3- Finding an angle using the tangent ratio

ramp

Example : An aircraft engineer is designing a ramp to help move passengers’  bags to an airplane . The point where he is attaching the ramp to the airplane is 2 meters high from the ground and he wants the ramp to take 4 meters in horizontal distance . At what acute angles should he cut the wooden main base piece , (rounded four decimal places ) ?

Answer :  First , find angle A ( Now since the needed one is angle A so the  side opposing it will be the opposite and the side sticking to it – other than the hypotenuse – would be the adjacent ).

                                                   
write the rule : tan  A =opposite / adjacent
                                                              
                       
tan A =    4/2
                     
        =   2

wait ! we are not done yet … now , to find angle calculate the inverse tangent of 2 (inverse tangent is just the opposite of tangent like subtraction and addition.)

angle A = tan-¹(2)

= 63.4349

statement : One of the acute angles  (rounded to four decimal places ) = 63.4349º

To find angle B  we do the same steps .(You can also do this part by using the rule: The sum of the three angles in a triangle =  180º)

But i will use the way involving tangent for the sake of the exercise.

Since angle B is the needed one so the side opposing it AC will be the opposite , and the side sticking to it BC is the adjacent.

Tan = opp/adj

Tan =    2/4
                   
         = 1/2
                    
        = 0.5

Remember the second step always the first step is just to get the tangent ratio Not the angle . This step is to get the angle.

Find inverse tangent of 0.5 using your calculator.

angle B =  tan-¹(0.5)

             = 26.5651º

Statement : The second acute angle = 26.5651

Done ! Good job if you have the same answers 😉 if you dont , keep trying !

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4-Finding a side length using the Tangent ratio

Example : Adam is a bridge engineer. His task is to rebuild a  bridge  which was in a really bad state. He has to know the width of the river to get the right amount and  sizes of  the materials he  need. Another bridge is placed 22.5 meters away from the one he is working on.   Adam is standing with his body towards pole H  and with his head turned 67 º looking at pole J. Help Adam get the width of the river.

Answer : Find angle J first by using geometric reasoning . I will work on getting angle J so that the adjacent side will be the know and the opposite will be the unknown.

since the sum of the angles of any triangle is 180º ,

*we already have two angles  H= 90º and A = 67º

so ,      J= 180º-(90º+67º)

J= 23º

Now , our opposite side is AH and the adjacent is JH which is 22.5 m.

Tan 23º =  opp/adj

tan 23º=   x/22.5                                                                                                                                                                                                                                                                          
                        
22.5 (tan 23º) = x  >>>>>>> multiply both sides by 22.5

9.5507=x                   >>>>>> Rounded to four decimal places

Statement : The river is about 9.5507 m wide.

Any questions ? I f you have a question then please don’t hesitate to post it in a comment , and I will answer within 24 hours !

Good luck , 🙂

Israa

 

The Tangent Ratio (part 1)

This review exercise consists of four parts :

1- Finding the tangent ratio from given sides 

2-Finding  the tangent of an angle .    

3- Find an angle using the Tangent ratio 

4- Finding a side length using the Tangent Ratio

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Now , lets start ! 

1-Finding the tangent ratio from given sides

Example :Find tan 0 for this triangle ,expressed as a decimal correct to four decimal places.

 Image

First , write the rule you think you are using to minimize your “goofy mistakes” as my teacher says. Then label your triangle . start with labeling the hypotenuse (its always the longest side infront of the 90 degrees angle) . After that look at were theta lies (the circle with a line in it) , the side which is opposing theta is the opposite and the last side left would be the adjacent.

Image

see? really simple 🙂 now finally write the final statement : so tan 0 = 1.2 for this triangle

to be continued …the next question on the second topic will be on my next post 

sincerely ,

israa

 

 

 

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