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Using the Inverse of Sine, Cosine, and Tangent

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Objective

  • Review over the Sine, Cosine, and Tangent function
  • Review over using the Sine, Cosine, and Tangent function on a calculator
  • Go over how to use the inverse functions to find a missing angle
  • Do some examples
  • Homework

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Now, for the last time, let’s go back to �SOHCAHTOA

SOHCAHTOA

A really weird acronym that will help you remember the basic trig functions

Let’s break them down:

S – Sine

O- Opposite

H- Hypotenuse

C- Cosine

A- Adjacent

H – Hypotenuse

T- Tangent

O- Opposite

A- Adjacent

So, what does this all mean?

It means this:

 

 

 

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Example 1:

 

 

20

35

13

Well, we remember from SOHCAHTOA that:

 

We can see that the side opposite the angle is 20

And, we can see that the hypotenuse is the biggest side, which we know is 35

So:

 

Or:

 

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Example 2:

 

 

15

22

12

Well, we remember from SOHCAHTOA that:

 

We can see that the side that is adjacent of the angle is 15

And, we can see that the hypotenuse is the biggest side, which we know is 22

So:

 

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Example 3:

 

 

 

9

10

Well, we remember from SOHCAHTOA that:

 

 

And, we can see that the side that is adjacent to the angle is 9

So:

 

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FINDING THE MISSING SIDE USING SINE

So to find the missing side using sine, we need to have two things.

We need an angle (and it’s actual measurement),

And we need either the hypotenuse of the triangle, or the opposite side of the angle.

Again, instead of explaining, here’s an example:

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EXAMPLE 1:

Find the missing side of the triangle if:

​

5.7

 

 

The hypotenuse = 5.7

 

So, we type into our calculators:

 

Then hit equal to get:

~0.707�Now we can set up the equation:

We know from SOHCAHTOA that:

 

Now we plug in what we know letting the unknown side be x:

 

 

 

5.7 * * 5.7

x = 4.03

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EXAMPLE 2:

Find the missing side of the triangle if:

​

3

 

 

The opposite side = 3

 

So, we type into our calculators:

 

Then hit equal to get:

~0.866�Now we can set up the equation:

We know from SOHCAHTOA that:

 

Now we plug in what we know letting the unknown side be x:

 

 

 

x * * x

0.866x = 3

______ _____

0.866 0.866

x = 3.5

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EXAMPLE 3:

Find the missing side of the triangle if:

​

5.7

 

 

The hypotenuse = 5.7

 

So, we type into our calculators:

 

Then hit equal to get:

~0.707�Now we can set up the equation:

We know from SOHCAHTOA that:

 

Now we plug in what we know letting the unknown side be x:

 

 

 

5.7 * * 5.7

x = 4.03

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EXAMPLE 4:

Find the missing side of the triangle if:

​

3

 

 

The adjacent side = 3

 

So, we type into our calculators:

 

Then hit equal to get:

~0.866�Now we can set up the equation:

We know from SOHCAHTOA that:

 

Now we plug in what we know letting the unknown side be x:

 

 

 

x * * x

0.866x = 3

______ _____

0.866 0.866

x = 3.5

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Example 5:

Find the missing side of the triangle if:

​

5.7

 

 

The adjacent side = 5.7

 

So, we type into our calculators:

 

Then hit equal to get:

~1.73�Now we can set up the equation:

We know from SOHCAHTOA that:

 

Now we plug in what we know letting the unknown side be x:

 

 

 

5.7 * * 5.7

x = 9.86

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Example 6:

Find the missing side of the triangle if:

​

3

 

 

The opposite side = 3

 

So, we type into our calculators:

 

Then hit equal to get:

~0.577�Now we can set up the equation:

We know from SOHCAHTOA that:

 

Now we plug in what we know letting the unknown side be x:

 

 

 

x * * x

0.577x = 3

______ _____

0.577 0.577

x = 5.2

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So, when do we use the inverse of a trig function?

First, we need to understand what the inverse of a function is.

So, according to google:

“An inverse function is a function that undoes the action of another function.”

So, in other words, if we used one of the trig functions (sine, cosine, or tangent) to find a missing side

Then we’re going to use the inverse of the trig functions to find the missing angle.

Sounds harder than it actually is, so let’s look at an example:

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Example 1:

  •  

6.2

 

The hypotenuse = 6.2

The adjacent side = 3.9

3.9

So, we know that we have the adjacent side, and the hypotenuse.

And what trig function deals with that?

Cosine!

So, we figure out what the cosine of this angle is first:

 

Now, to find the missing side, we normally use the cosine of the angle.

In this case, to find the angle, we’re going to take the cosine of the angle, and plug it into the inverse cosine.

So:

​

 

 

And when we plug this into our calculator, we get:

 

So, the angle we’re looking for is 51 degrees.

 

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Example 2:

  •  

6.1

 

The hypotenuse = 6.1

The adjacent side = 3.5

3.5

So, we know that we have the adjacent side, and the hypotenuse.

And what trig function deals with that?

Cosine!

So, we figure out what the cosine of this angle is first:

 

Now, to find the missing side, we normally use the cosine of the angle.

In this case, to find the angle, we’re going to take the cosine of the angle, and plug it into the inverse cosine.

So:

​

 

 

And when we plug this into our calculator, we get:

 

So, the angle we’re looking for is 55 degrees.

 

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Example 3:

  •  

10.6

 

The hypotenuse = 10.6

The opposite side = 7.1

7.1

So, we know that we have the opposite side, and the hypotenuse.

And what trig function deals with that?

Sine!

So, we figure out what the Sine of this angle is first:

 

Now, to find the missing side, we normally use the sine of the angle.

In this case, to find the angle, we’re going to take the sine of the angle, and plug it into the inverse sine.

So:

​

 

 

And when we plug this into our calculator, we get:

 

So, the angle we’re looking for is 42 degrees.

 

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Example 4:

  •  

343

 

The hypotenuse = 343

The opposite side = 334

334

So, we know that we have the opposite side, and the hypotenuse.

And what trig function deals with that?

Sine!

So, we figure out what the sine of this angle is first:

 

Now, to find the missing side, we normally use the sine of the angle.

In this case, to find the angle, we’re going to take the sine of the angle, and plug it into the inverse sine.

So:

​

 

 

And when we plug this into our calculator, we get:

 

So, the angle we’re looking for is 77 degrees.

 

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EXAMPLE 5:

  •  

6.2

 

The opposite side = 6.2

The adjacent side = 3.9

3.9

So, we know that we have the opposite side, and the adjacent side.

And what trig function deals with that?

Tangent!

So, we figure out what the Tangent of this angle is first:

 

Now, to find the missing side, we normally use the tangent of the angle.

In this case, to find the angle, we’re going to take the tangent of the angle, and plug it into the inverse tangent.

So:

​

 

 

And when we plug this into our calculator, we get:

 

So, the angle we’re looking for is 58 degrees.

 

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EXAMPLE 6:

  •  

5

 

The adjacent side = 5

The opposite side = 3.6

3.6

So, we know that we have the adjacent side, and the opposite side.

And what trig function deals with that?

Tangent!

So, we figure out what the tangent of this angle is first:

 

Now, to find the missing side, we normally use the tangent of the angle.

In this case, to find the angle, we’re going to take the tangent of the angle, and plug it into the inverse tangent.

So:

​

 

 

And when we plug this into our calculator, we get:

 

So, the angle we’re looking for is 36 degrees.