Trigonometric Ratios

Trig Ratios 2- Calculator & Solving Triangles

Scientific Calculators

• To do calculations, you will need a scientific calculator ( sin, cos, tan)

• you can also use your phone calc, turn it sideways

• you can also use Desmos, Geogebra or Google Calc

• have your calc ready, we're gonna need it

Solving for a Side

suppose you have this problem, and youre asked to solve for x. The Pythagorean Theorem can always be used when you have two sides, but we only have a side and an angle...

Setup the Ratio

Notice that in Reference to the angle of 18, x is adjacent and 17 is hypotenuse. Since Adj/Hyp or a/h is the cosine, we can write  

 $\cos\left(18\right)=\frac{x}{17}$  

and if we multiply both sides by 17 

 $17\cos\left(18\right)\ =x$  

Time to Calculate

Try entering that into your calculator

 $17\cos\left(18\right)\ =x$  

if you need help using your calculator, try google 

Multiple Choice

 Calculate

$$17\cos\left(18\right)\ =x$$

• 0.956

• 0.951

• 17.21

• 16.16

Let's try another One

practice = Mo Betta

How do we Solve for a

hint: start with the given angle, then write a ratio

Multiple Choice

Which will correctly Solve for a?

• $$\sin\left(20\right)=\frac{a}{35}$$  

• $$\cos\left(35\right)=\frac{a}{20}$$  

• $$\sin\left(35\right)=\frac{a}{20}$$  

• $$\cos\left(20\right)=\frac{a}{35}$$  

Multiple Choice

Which statement is a correct rearrangement to solve for a?

$$\sin\left(35\right)=\frac{a}{20}$$  

• $$20\left(\sin35\right)=a$$  

• $$\ \frac{20\left(\sin35\right)}{20}=a$$  

• $$35\sin\left(20\right)=a$$  

• $$\ \frac{\left(\sin35\right)}{20}=a$$  

Multiple Choice

Solve for the length of side a (we just set it up)

• 11.97

• 11.47

• 0.57

• 0.34

What if X is in the denominator?

      setup the ratio
1) ____ $\left(57\right)=\frac{o}{h}$  

      substitute, rearrange to solve

2)  $\sin\left(57\right)=\frac{10.8}{x}$  

When x is the denominator... use algebra

 $\sin\left(57\right)=\frac{10.8}{x}$             multiply both sides by x

 $x\ \sin\left(57\right)=10.8$          then divide by sin(57)

 $x=\frac{10.8}{\sin\left(57\right)}$         ....or just cross multiply

Multiple Choice

Which equation could be used to solve for the variable....

Multiple Choice

now Solve for x...

• 5.88

• 19.8

• 12.88

• 13.2

• 9.06

Multiple Choice

Set up an equation that could be used to solve for the variable.

Multiple Choice

Solve for x

• 5.5

• 19.05

• 9.52

• 7.8

Multiple Choice

What is the length of side x?

• 6.57 cm

• 7 cm

• 7.44 cm

• 26.33 cm

Multiple Choice

Solve for x. Round to the nearest tenth.

• 15.3

• 16.6

• 24.1

• 24.3

Multiple Choice

Solve for x. Round to the nearest tenth.

• 19.0
• 20.0
• 21.0
• 20.5

Multiple Choice

Solve for x. Round to the nearest tenth.

• 74.9
• 3.6
• 63.5
• 54.1

Solving for an Angle using Trig Ratios

In all of the previous problems, we have been given an angle and a side to solve for a missing side. But what if we have two sides but dont know the angles? Remember that Trig ratios relate angles to sides, so we can use what we already know, but we need one critical piece of information to understand how it works....

Old School Trig Tables

Suppose we had a problem which gave us this statement..

 $\cos A=\frac{1}{2}$  

if we used the trig table, we could see that the cosine of 60 is 1/2

So A=  $60^o$  
..our calculator can do this for any angle...This process is called the Inverse of the Trig Function

Inverse Trig Ratios

...are used to find angles

the inverse sine

 $\sin^{-1}\left(\frac{o}{h}\right)=\theta$  

..is also called the Arcsine

the inverse of cosine is arccosine and inverse tanget is arctangent

Inverse Trig to find Angles: Find angles A and B

1) start with the ratio for what is given..in this case any of the three ratios could be used.
 $\sin\left(A\right)=\frac{3}{5}\ \ \ \ \ \ \ \ A\ =\ \sin^{-1}\left(0.6\right)$ 
so    $A=36.9^o$  

but how did we find this?...
   

Use the Inverse to find Angles

We could use tables to work backwards, but its easier to use a calculator. The Inverse function will do that...

Using Calculator to Find Angle

The shift (2nd) button on TI's allows you to use the functions above each button. Above sin is inverse sine (arcsine) To calculate 

 $\sin\left(A\right)=\frac{3}{5}\ \ \ \ \ \ \ \ A\ =\ \sin^{-1}\left(0.6\right)$  

you would enter  
2nd         sin     (   $3\div5$  )   =

Multiple Choice

Set up an equation that could be used to solve for the variable.

Multiple Choice

Rearrange that equation to solve for   $$\theta$$  

• $$\theta=\ \sin^{-1}\left(\frac{9}{6}\right)$$  

• $$\theta=\ \sin^{-1}\left(\frac{6}{9}\right)$$  

• $$\theta=\ \cos^{-1}\left(\frac{9}{6}\right)$$  

• $$\theta=\ \cos^{-1}\left(\frac{6}{9}\right)$$  

Multiple Choice

 solve for   $$\theta$$  

• $$26^o$$  
  

• $$41.8^o$$

• $$48.2^o$$  

• $$15.6^o$$  

• error

Multiple Choice

Set up the problem so that you can solve for X.

• sin X = 41/28

• tan X = 41/28

• cos X = 28/41

• sin X = 28/41

Multiple Choice

Which equation solves for X?

• $$X=\cos^{-1}\left(\frac{28}{41}\right)$$

• $$X=\sin^{-1}\left(\frac{28}{41}\right)$$
  

• $$X=\cos^{-1}\left(\frac{41}{28}\right)$$
  

• $$X=\sin^{-1}\left(\frac{41}{28}\right)$$
  

Multiple Choice

Solve for X

• $$43.1^o$$   

• $$46.9^o$$   

• $$55.7^o$$  
  

• $$34.3^o$$  
  

Time to see what you've learned

...get your calculator ready

Multiple Choice

Solve for x.           hint:      $$1-1-\sqrt{2}$$  

• x = 90º

• x = 60º

• x = 45º

• x = 30º

Multiple Choice

Solve for x.                  hint      $$1-2-\sqrt{3}$$  

• x = 90º

• x = 60º

• x = 45º

• x = 30º

Multiple Choice

Which equation would you use to solve for the unknown angle?

• x = tan^–1(24/7)

• x = tan^–1(7/24)

• x = tan^–1(25/7)

• x = tan^–1(7/25)

Multiple Choice

Which equation would you use to solve for the unknown angle?

• x = sin^–1(4/5)

• x = cos^–1(4/5)

• x = cos^–1(5/4)

• x = tan^–1(4/5)

Multiple Choice

What equation can be used to solve for x?

• Sin(40°)= x/40

• Sin(40º)=x/19

• Cos(40º)=x/19

• Tan(40°)=19/x

Multiple Choice

Use inverse trig ratios (SOH CAH TOA) to solve for the missing angle

• 32°

• 44°

• 61°

• 50°

Multiple Choice

Find the measure of the indicated angle (?) round to the nearest tenth.

• 9.4
• 55.2
• 20.15
• 19.4

Multiple Choice

Solve for m∠N

• 14

• 65

• 76

• 53

Multiple Choice

Solve for x. Round to the nearest tenth.

• 103.5
• 4.7
• 55.0
• 23.4

Multiple Choice

Solve for x. Round to the nearest tenth.

• 9.8

• 0.1

• 0.038

• 9.9

Multiple Choice

Solve for the missing angle

• 50

• 0.053

• 36.9

• 95

Multiple Choice

Find the measure of the missing angle.

• 64^o

• 26^o

• 61^o

• .008^o

Multiple Choice

Find the measure of the indicated angle.

• 9.39

• 55.2

• 20.2

• 19.4

Solve the Triangle: means to find all the sides and angles.

• You can use angle Sum Theorem (=180) if you have two known angles

• You can use the Pythagorean Theorem if you have 2 sides.

• You can use SRT for 30-60-90 and 45-45-90

• Trig Ratios can be used to find missing sides

• Inverse Trig Ratios are used to find angles

Multiple Select

We have been using the 3-4-5 Triangle frequently, which are True statements?

• $$3^2+4^2=5^2$$

• $$\sin\ A=\frac{3}{5}$$

• $$5\sin A=3$$
  

• $$A=\sin^{-1}\left(\frac{3}{5}\right)$$
  

• $$A=36.87^o$$

Multiple Select

For the 3-4-5 Triangle, which are True?

• $$5^2-4^2=3^2$$

• $$\cos\ B=\frac{3}{5}$$

• $$B=\sin^{-1}\left(\frac{4}{5}\right)$$

• $$\cos\ A=\ \sin\ B$$

• $$B=53.1^o$$

Pauhana

..can go beach now