7.4 Tangent

Now that we can identify and solve for missing measures of a right triangle with Sine and Cosine, we can look at using Tangent to solve for missing measures in a right triangle.

Yet another ratio? Can't we just use Sine & Cosine?  The short and easy answer is "Nope!"  Sine and Cosine work well if you know the Hypotenuse & a Side, or an Acute Angle with either a Side or the Hypotenuse.  What if we only knew the 2-sides ( Legs ) of the Right Triangle and no acute angle measures or the length of the Hypotenuse, well then we would need to use Tangent.

Tangent Ratio

We need to re-write the ratio to help us out.  Since we know that:
             
We will now use θ rather than the variable label of the angle/vertex.
For angle A: For angle B:

or, better still...
or, better still...

Example A ( Missing Angle Measure )

Given Right Triangle ABC with side lengths of 3,4,5.

Solution:

Use Tan to find the measure of angle A:

Step 1, Set up the ratio   

Step 2, Re-write as inverse tangent    

Step 2,  Graphing Calculator

     Press: then and you will see

    

     Now you can enter the fraction

     Press:  then then then

     and you should now see

    
Step 3, The answer

     We were given a long decimal number as an answer,
     so we will round it to 2-decimal places.

     So the degree measure of angle A is about 36.87o.


Note:  When we need to find the degree measure of an angle, we use the inverse tangent on the calculator.  The tangent function will give us the ratio of the side opposite to the adjacent.


Watch the video:

 

Quick Question...

Instructions:

  • Step 1 Identify the Angle to be Given
               Skip if no angles are to be given.
  • Step 2 Identify the Side(s) to be Given
               If you skipped the angles, select two sides
  • Step 3 Identify the Missing Measure you want to find.
  • Select the number of problem to do and Start

To use the app:

  • Click for full screen mode
  • Clickto check your answer.
  • Clickto goto the next problem
  • Clickto stop the app
  

 

Example B ( Missing Angle Measure )

Given Right Triangle ABC with side lengths of 3,4,5.

Solution:

Use Tan to find the measure of angle B:

Step 1, Set up the ratio       

Step 2, Re-write as inverse tangent    

Step 2,  Graphing Calculator

     Enter the information into the calculator and we get

    

Step 3, The answer

     So the degree measure of angle B is about 53.13o.




Watch the video:

Do a Quick Question

Now let us see what happens if I only know the degree measure of one acute angle and the length of any angle and the length of any one-side

Example C ( Missing Side & Hypotenuse Measures )

Solve this triangle for all missing measures.

Solution:

To begin with, if we know only 1 of the acute angles we can  subtract it from 90o to find the other acute angle.

If we know A = 36.87o,
then B = 90o - 36.87o = 53.13o

or

If we know B = 53.13o,
then A = 90o - 53.13o = 36.87o

To find the missing sides, we will need to set-up sine ratio for each missing side.

Remember:         

Here we only know the side opposite angle A. 

Which gives us two options to find side AC
1.  To Find Side AC 2.  To Find Side AC
 
   

In either case you see that we will get the exact same answer.


Now if we only know the side opposite angle B...

Which gives us two options to find side AB
1.  To Find Side AB 2.  To Find Side AB
   
   

But what about the Hypotenuse?

Since Tangent is the ratio between the side opposite and the side adjacent to the angle of interest, it can not be used to find the hypotenuse.  In order to find the hypotenuse you will need to use Sine, Cosine, or the Pythagorean Theorem.































Watch the video:

Do a Quick Question

Now let us see what happens if I only know the degree measure of one acute angle and the length of the hypotenuse.

Example D ( Missing Side Measures )

Solve this triangle for all missing measures.

Solution:

To begin with, if we know only 1 of the acute angles we can  subtract it from 90o to find the other acute angle.

If we know A = 36.87o,
then B = 90o - 36.87o = 53.13o

or

If we know B = 53.13o,
then A = 90o - 53.13o = 36.87o


Remember:         

Notice that the Hypotenuse is no where to be seen in this ratio.  So if we only know the length of the the hypotenuse, we do not have enough information to use Tangent to solve for any of the missing measure.  We would need to use use Sine, Cosine, or the Pythagorean Theorem in order to solve for the missing measures.


Watch the video:

Do a Quick Question

And by the way, one last thing to remember...

If we know the lengths of any 2-sides of a right triangle and neither of the acute angle measures, use the Pythagorean Theorem to find the 3rd side!

 

Practice Problems

Given the Right Triangle:


Assignment:

Use Tangent to solve for the Missing Measures.

Instructions:

  • Screenshot each question and paste it into the
    Jamboard and show all work for each  problem.
  • Screenshot the score for each group of question
    onto the group page.
  • Round all answers to 2-decimal places
  • If you can not use a Tangent ratio to solve, explain what you can use, to solve it and solve it using other means.

 

1-3     Do:
4-6     Do:
7-9     Do:

10-12 Do:
13-15 Do:
16-18 Do:

19-21 Do:
22-24 Do:
25-27 Do: