SOHCAHTOA rule and word problems.

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SOHCAHTOA rule and word problems.

Trigonometry is all about triangles and it has numerous applications in science and engineering.

Lets learn about some terms related with right triangle.

 

Opposite side: The side which is opposite to angle θ

Adjacent side: The side on which angle θ is based.

Hypotenuse:  The longest side and opposite to 90°  angle.

Basic three functions of trigonometry are Sine, Cosine and Tangent. These are just the ratios of sides of right triangle.  That’s why these are also called trigonometric  ratios and in short they are abbreviated as sin, cos and tan.

Fundamental trigonometric identities:

  • sinθ =1/cscθ     or    cosecθ  = 1/sinθ
  • cosθ=1/secθ      or    sec  = 1/cosθ
  • tanθ = 1/cotθ    or   cotθ  =1/tanθ
  • tanθ =sinθ/cosθ    or   cot  =cosθ/sinθ

Pythagorean identities:

  • \dpi{120} \mathbf{sin^{2}\theta +cos^{2}\theta =1}
  • \dpi{120} \mathbf{1+tan^{2}\theta =sec^{2}\theta }
  • \dpi{120} \mathbf{1+cot^{2}\theta =csc^{2}\theta }

 

Example: Using given right triangle find all six trigonometric  function of θ.

 

Solution:  First, we find hypotenuse using Pythagorean identity.

\dpi{120} \mathbf{\left ( hyp \right )^{2}= \left ( adj \right )^{2}+\left ( opp \right )^{2}}

\dpi{120} \left ( hyp \right )^{2}= \left ( 4 \right )^{2}+\left ( 3 \right )^{2}

\dpi{120} \left ( hyp \right )^{2}= 16+9=25

hyp = 5

Using  SOHCAHTOA  rule, we get

Sinθ = O/H= 3/5                                 Cosecθ =H/O= 5/3

Cosθ = A/H= 4/5                                 Secθ = H/A = 5/4

Tanθ =O/A = 3/4                                 Cotθ = A/O =4/3

 

Example: Find each angle measure to nearest degree.

   a)tanA = 0.6249                        b) sinX = 0.4540

Solution:  To  solve these type of problems we need calculator. Since these use  inverse functions so we need to be familiar with inverse trigonometric buttons on our calculator.

a)             tanA = 0.6249

\dpi{120} A= tan^{-1}(0.6249)

Enter 0.6249 in your calculator  and press  arctan \dpi{120} \left ( tan^{-1} \right )  button

We get,          A = 32.001

Which when rounded to nearest degree become 32°

If your calculator doesn’t have  inverse trigonometric functions  then press inverse(INV) button after entering the number and inverse functions will appear .

b)            sinx = 0.4540

\dpi{120} x=sin^{-1}(0.4540)

x = 27.0006  ≈  27°

 

Example: Find the missing side of given right triangle round to nearest tenth.

 

Solution:  Given Adjacent side(A) =x

Hypotenuse (H)= 17

Angle(θ ) = 30

We know that ,            Cosθ = A/H

Cos(30)  =  x/17

17 cos(30) = x

14.7   = x

 

Example: A wheelchair ramp is 4.2 m long. It rises 0.7 m. What is its angle of inclination to the nearest degree?

Solution: Lets  make a picture to understand  this problem better. Ramp makes the hypotenuse and rise of 0.7 make opposite side for right triangle.

 

Here we need to find angle θ.

We know that,  sinθ   = O/H

sinθ   = 0.7 /4.2

\dpi{120} \theta =sin^{-1}\left ( \frac{0.7}{4.2} \right )

θ  = 9.59 ≈ 10°

 

Example: The angle of elevation of the sun is 68° when a tree casts a shadow 14.3 m long. How tall is the tree, to the nearest tenth of a meter?

 

Solution:  Here opposite(vertical) side form the height of tree and adjacent(horizontal) side form the shadow of tree and angle  of elevation is given 68 degrees.

 

We know that                    tanθ= O/A

tan(68) = x/14.3

14.3*tan(68) =x

35.393  =   x

35.4 m  =  x

 

 

 

 

 

Practice problems:

  • Find all the six trigonometric functions of  θ for a given right triangle with hypotenuse as 41 and opposite side as 9.
  • An airplane is flying at an altitude of 6000 m over the ocean directly toward a coastline. At a certain time, the angle of depression to the coastline from the airplane is 14°. How much farther (to the nearest kilometer) does the airplane have to fly before it is directly above the coastline?
  • A 9.0 m ladder rests against the side of a wall. The bottom of the ladder is 1.5 m from the base of the wall. Determine the measure of the angle between the ladder and the ground, to the nearest degree.
  • A person flying a kite has released 176 m of string. The string makes an angle of 27° with the ground. How high is the kite? How far away is the kite horizontally? Answer to the nearest meter.

 

 

 

 

 

Answers:

  • Sin=9/41 , cos=40/41 ,tan=9/40 , csc=41/9 , Sec=41/40 ,cot =40/9
  • 24 km
  • 80°
  • 80 m high, 157 m away

 

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