7.2 Sine

Now that we can identify each part of the trigonometry ratios in a right triangle: How is knowing this information useful? and/or How can we use this information?

When we were identifying the trigonometry ratios in a right triangle we were only working with the sides not the acute angles in the right triangle. Once we have identified the trigonometry ratios in a right triangle we can use this information to find the measure of each acute angle. We can actually extend this idea so that we can find missing measures in any right triangle, given some information!

If we know...

... the measures of any two sides, we can find the unknown angles and side measures.

... the measure of only 1 acute angle and the length of any side, we can find the unknown angles and side measures.

Before we begin, we will need to identify a couple of things...

Theta θ

We need to identify a symbol that will be used throughout trigonometry and in many other areas of mathematics.

The Greek letter Theta, θ, is used to identify an unknown angle measure. Which is why you see the symbol on the variable button, , on the graphing calculator.

Graphing Calculator

We will be relying the graphing calculator, but we need to set it up before we can use it.

1. Reset the graphing calculator:

Press:

then

The calculator should be reset and you should see something similar to this:

2. Change the mode into degrees.

Press:

You should now see:

Next, we need to select "Degree"

Press:

( twice ) then

then

You should now see:

Note: you will need to do this every time you reset the calculator. It is also a VERY GOOD idea to check this before you begin.

Note: All scientific calculators have this option, make sure that you know how to use your calculator!

Sine Ratio

We need to re-write the ratio to help us out. Since we know that:
and from Right Triangle ABC we get: and
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 ( Given the Legs & Hypotenuse, Find A )

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

Use Sin to find the measure of angle A:

Solution:

Step 1, Set up the ratio

Step 2, Re-write as inverse sine

Step 3, Enter in to the 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 sine on the calculator. The sine function will give us the ratio of the side opposite 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 ( Given the Legs & Hypotenuse, Find B )

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

Use Sin to find the measure of angle B:

Solution:

Step 1, Set up the ratio

Step 2, Re-write as inverse sine

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.

Now you are probably ( hopefully ) wondering why I didn't just subtract the measure of angle A from 90^o to find the missing angle measure.

Well, when I do that, I get the same answer!

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 ( Given a Leg & Angle )

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.

Solution:

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

So that seems fairly simple to do, now what about the sides?

To find a missing leg & hypotenuse, 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. So, if we do not know the length of the hypotenuse, then 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:

then

and

You should now see:

Since 36.87 was a rounded number we will not get the EXACT value for the missing side, just round to 2-decimal places.

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

and

You should now see:

Since 53.13 was a rounded number we will not get the EXACT value for the missing side, just round to 2-decimal places.

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.

So what would happen if we only knew the measure of the side opposite of angle B?

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
	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

OR..

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 ( Given the Hypotenuse & Angle )

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.

Solution:

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

So nothing really changes to solve for the missing angles.

To find the missing legs, we will need to set-up sine ratio for each missing leg using the hypotenuse.

To Find Side AC	To Find side BC

Here I only need b	Here I only need a

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 when using rounded numbers we will get answers that are close to the exact values that we know are the lengths of the sides of the right triangle.

OR..

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 Sine ratio to find the missing measures
Round all answers to 2-decimal places

Assignment:

Use Sine 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: