7.3 Cosine

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

Why do we need another ratio? Can't we just use Sine? The short and easy answer is "Yes", for finding missing angles and sides in a right triangle we could get by only using sine. However, there are situations where we will need to use Cosine to solve and find solutions to problems related to triangles.

Cosine 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.

Use Cos to find the measure of angle A:

Solution:

Step 1, Set up the ratio

Step 2, Re-write as inverse cosine

Step 3, Graphing Calculator

Press:

then

and you will see

Now you can enter the fraction

Press:

then

and you should now see

Step 4, 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 36.87^o.

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

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.

Use Cos to find the measure of angle B:

Solution:

Step 1, Set up the ratio

Step 2, Re-write as inverse cosine

Step 3, Graphing Calculator

Enter the information into the calculator and we get

Step 4, The answer

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

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 )

Use Cos to find all missing measures for this triangle.

Solution:

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

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

or

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

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...

To Find Side AC	To Find the Hypotenuse

Here I need b & c	Here I only need c

In order to find the missing sides I will need to find the hypotenuse first and then the other side. Again, if we do not know the length of the hypotenuse, it is the first measurement we need to find

To solve for c we can solve the ratio by using this little trick: "Multiply the numbers on the diagonal, then divide by the other number".

We enter it into the graphing calculator as follows:

and

You should now see:

We get the answer of 4.999988091, but since we are working with a triangle where we know what the side lengths should be we can let c = 5.

Now that I know c = 5 I can re-write the sine ratio to find the missing side.

so:

is now

We enter it into the graphing calculator as follows:

then

You should now see:

We get the answer of 3.999994641, but since we are working with a triangle where we know what the side lengths should be we can let b = 4.

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

To Find Side AB	To Find the Hypotenuse

Here I need a & c	Here I only need c
Solve here =>	Starting with it gives us again we see we have, c = 5
Now that we know c = 5, we can set-up as so we get again we see we have, a = 3	Then... <= Solve here

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 )

Use Cos to find all missing measures for this triangle.

Solution:

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

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

or

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

To find the missing sides, we will need to set-up a cosine ratio for each missing side using the hypotenuse.

To Find Side AC	To Find side BC

Here I only need a	Here I only need b

This time it does not matter which side we find first, since we are only missing one variable in each ratio.

To Find Side AC	To Find side BC
here we get	here we get

Again we see that since we were using rounded numbers we get answers that are close to the exact values that we know are the lengths of the sides of the right triangle.

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 3^rd side!

Practice Problems

Given the Right Triangle:

Draw out and label the right triangle
Use a Cosine ratio to find the missing measures
Round all answers to 2-decimal places

Assignment:

Use Cosine 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

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: