Trigonometric Ratios

Trigonometric Ratios

Review

Trigonometric Ratios

(aka Trig)

Triangle parts

sides: the line segments that make the triangle shape

vertices: the point where two sides meet

angle: a figure made up of 2 line segments

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Poll

How confident do you feel about naming sides and angles in a triangle?

• I feel good:)

• I think I've got it.

• I'm getting better.

• I don't understand:(

When talking about trigonometry in Geometry class, we need to focus on RIGHT TRIANGLES. Let's explore....

RIGHT TRIANGLE BASICS

legs: the two sides of the triangle that make the right angle

hypotenuse: the side that is OPPOSITE the right angle; always will be the LONGEST side of the triangle

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Poll

How confident do you feel about naming sides in a right triangle?

• I got it!

• I think I've got it!

• It's getting better.

• I'm still confused.

When we talk about trigonometry, we have to change up our labeling of the sides slightly.

Let's discuss these labels......

HYPOTENUSE

The hypotenuse in a right triangle is ALWAYS named that:) ALWAYS label that side FIRST! It is ALWAYS across from the right angle (the box). You can abbreviate this to "hypo", "hyp", or simply "h" for short.

ACUTE ANGLE

You will always be looking at ONE of the acute angles that are formed in the right triangle. Whichever one is given is the one you base your labeling on....CIRCLE IT so it will stand out:)

OPPOSITE SIDE

The side directly across from the angle you circled will be labelled as the opposite side. You can abbreviate this to "opp" or just "o" if you prefer:)

ADJACENT SIDE

The adjacent (aka "next to", "neighboring", or "side-by-side") side is the side that remains. It will TOUCH the circled acute angle in the picture. You can abbreviate this as "adj" or simply "a" for short.

Multiple Choice

In the right triangle GHI here, name the hypotenuse.

• HI

• GI

• HG

• angle H

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Multiple Choice

In this right triangle, what is the hypotenuse?

• BC

• AB

• AC

Multiple Choice

In this right triangle, what side is opposite the 35⁰ acute angle marked?

• BC

• AB

• AC

Multiple Choice

In this right triangle, what side is adjacent to the 35⁰ acute angle?

• BC

• AB

• AC

Poll

How comfortable do you feel about labeling the sides of a triangle based on hypotenuse, opposite, and adjacent sides?

• I've got it!

• It's getting better.

• I'm still lost.

Once we label the sides of the right triangle based on the acute angle that is given, we then need to use our three trig functions to solve.

Sine of the angle 

definition:    $\frac{opposite\ side}{hypotenuse}$  

 $\sin\ S\ =\ \frac{BK}{SK}$  

Cosine of the angle

definition:    $\frac{adjacent\ side}{hypotenuse}$  

 $\cos\ S\ =\ \frac{SB}{SK}$  

Tangent of the angle

definition:   $\frac{opposite\ side}{adjacent\ side}$ 


 $\tan\ S\ \ =\ \frac{BK}{SB}$   

Fill in the Blanks

Fill in the Blanks

Fill in the Blanks

Multiple Choice

In the right triangle here, find sin D.

• 12/13

• 13/5

• 5/13

• 5/12

Multiple Choice

In the right triangle here, find tan D.

• 12/13

• 13/5

• 5/13

• 5/12

Multiple Choice

In the right triangle here, find cos K.

• 12/13

• 13/5

• 5/13

• 5/12

SOMETIMES you may need to REDUCE your trig ratio.....

Let's look at an example.....

Find the sine of   $\angle\ K$  .

 $\sin\theta\ =\ \frac{oposite\ side}{hypotenuse}$  

 $\sin\ K\ =\ \frac{8}{10}$  

 $\sin\ K\ =\ \frac{4}{5}$  

Multiple Choice

In the right triangle here, find sin M.

• 4/5

• 3/5

• 4/3

• 5/3

Multiple Choice

In the right triangle, find tan K.

• $\ \frac{4}{3}$

• $\frac{3}{4}$

• $\frac{3}{5}$

• $\frac{4}{5}$

Poll

How do you feel about your ability to find the three trig ratios of a right triangle?

• I've got it!

• It's getting better.

• It makes more sense now.

• I still don't understand.

CHALLENGE

Let's try to find the sin K.

Focus on angle K.

•  $\sin\ \theta\ =\ \frac{opposite\ side}{hypotenuse}$  

•  $\sin\ K\ =\ \frac{MT}{MK}$  

• MT = 8

• MK =  ???

We need to find the length of KM. HOW?

We need to use the PYTHAGOREAN THEOREM!!!!

 $a^2\ +\ b^2\ =\ c^2$  

Here, we are LOOKING for the hypotenuse, or "c".

 $8^2\ +\ 15^2\ =\ x^2$  
 $64\ +\ 225\ =\ x^2$  
 $289\ =\ x^2$  
 $\sqrt{289\ }\ =\ \sqrt{x^2}$ 

 $17\ =\ x$   

Find sin K.

No we can finish the problem.

 $\sin\ K\ =\ \frac{opposite\ side}{hypotenuse}$ 

 $\sin\ K\ =\ \frac{8}{17}$   

Multiple Choice

What is the cos M?

• $\frac{9}{15}\ =\ \frac{3}{5}$

• $\frac{12}{15}\ =\ \frac{4}{5}$

• $\frac{9}{12\ }=\frac{3}{4}$

Poll

How do you feel about right triangle trig after this lesson?

• SO much better!

• It;s getting better.

• Um...I guess ok?

• Trig? What's that???

ACTIVITY

Go back and look over the activity from last week on writing trig rations. (the worksheet AND the IXL assignment) Let me know if you have any questions you don't understand.