Trigonometric Ratios

Trigonometric Ratios

SOH CAH TOA

Why do we need Trig Ratios?

The Pythagorean Theorem relates all three sides of a Right Triangle, but so far (except for SRT's) we don't know how to find angles given sides. The trigonometric ratios provide us with that connection....we can use them to solve for a missing side or angle using what we have already learned about Ratios & Similarity.

Trig ratios can also be use in many applications such as Physics, Carpentry, Navigation,...even gaming!

SOH CAH TOA

There are three Primary Trig ratios. Each Ratio compares two sides of a Right Triangle to an Angle in the Triangle (but not the right angle) The acronym SOH CAH TOA helps us to remember the definitions.

Multiple Choice

Which letter represents the OPPOSITE side to angle a?

• A

• B

• C

Multiple Choice

Which letter represents the ADJACENT side to angle a?

• A

• B

• C

Multiple Choice

Which letter represents the HYPOTENUSE?

• A

• B

• C

Multiple Choice

From ∠C, what side is AB?

• Adjacent

• Hypotenuse

• Opposite

• Right

The Sine Ratio (SOH)

In a Right Triangle: The Sine of an Angle is the ratio of the opposite side to the hypotenuse. See in the example problem that it can be used to solve for missing sides. More on this later, for now lets look at setting up the ratio...

            Sine (SOH)

The ratio Depends on the Angle and always refers to an Ang.  The Acronym for Sine is SOH
so the sine of an angle is the side opposite divided by the hypotenuse.

    
 $\sin\Theta=\frac{o}{h}$   
 $\sin A=\frac{21}{29}$  
 $\sin\ C\ =\frac{20}{29}$     

Multiple Choice

Which ratio is the Sine of Angle C?

• ²⁴/₃₀

• ¹⁸/₃₀

• ³⁰/₂₄

• ³⁰/₁₈

Multiple Choice

What is the Sine of Angle Z?

Sin Z =?

• 24/32

• 32/24

• 32/40

• 24/40

Multiple Choice

What is Sin A?

• 32/40

• 32/24

• 24/40

• 24/32

Multiple Choice

What is Sin C?

• 21/35

• 28/21

• 35/28

• 28/35

Multiple Select

 Which triangle below shows the ratio as $\sin\left(x\right)=\frac{b}{c}$  ?

The Cosine Ratio

The Cosine of an angle is the ratio of the adjacent side to the hypotenuse.

           Cosine  (CAH)

 $\cos\ \theta=\frac{a}{h}$  

 $\cos\ A\ =\frac{4}{5}$          $\cos\ B=\frac{3}{5}$   

Multiple Choice

What is the Cosine of Angle C?

• 9/41

• 9/40

• 40/41

• 40/9

Multiple Choice

Find cos B

• 3/5

• 3/4

• 4/5

• 5/4

Multiple Choice

What is the reduced ratio for cos X?

• 15/36 = 5/12

• 15/39 = 5/13

• 36/39 = 2/13

• 36/15 = 12/5

Multiple Choice

Find Cos Z

• 18/30 = 3/5

• 18/24 = 3/4

• 24/30 = 4/5

• 24/18 = 4/3

Multiple Select

 Which triangle below shows the ratio as $\cos\ \left(x\right)=\frac{a}{b}$  ?

Multiple Select

 Which triangle below shows the ratio as $\cos\ \left(x\right)=\frac{a}{c}$  ?

The Tangent Ratio (TOA)

The Tangent of an angle is the ratio of the opposite side to the adjacent side. Notice that SIN & COS both compare a side to the hypotenuse, but the tangent compares both legs.

Tangent (TOA)

 $\tan\ \theta=\frac{o}{a}$  

 $\tan\ A=\frac{15}{8}$  

 $\tan\ C=\frac{8}{15}$  

Multiple Choice

What is the Tangent of Angle A?

• 16/20

• 20/16

• 12/16

• 16/12

Multiple Choice

Tan x=?

• 50/40

• 40/30

• 40/50

• 30/40

Multiple Choice

Tan x=?

• 5/4

• 4/3

• 4/5

• 3/4

Multiple Choice

Which triangle below shows the ratio as  $\tan\ \left(x\right)=\frac{t}{u}$  ?

Multiple Choice

 Which triangle below shows the ratio as $\tan\ \left(x\right)=\frac{c}{a}$  ?

Writing Trig Ratio Equations

• All ratios refer to an angle, start there to name side

• All ratios are usually reduced if can

• SIN & COS have Hyp as denominator

• side names refer to an angle

• the Right Angle has no opposite or adjacent, we dont use trig ratios for the Rt Angle (yet)

Multiple Choice

Which equation is a correct use of ratios to solve for x? 

 to do this type of problem, define the sides in relation to the Given Angle-- in reference to 45, x is the hyp and 13 is opposite.

. $\frac{13}{x}=\frac{o}{h}=>SOH$ 

• tan (45) = x/13

• cos (45) = x/13

• sin (45) = 13/x

• sin (45) = x/13

Multiple Choice

What is the correct equation?

• sin 40 = w/28

• cos 40 = w/28

• tan 40 = w/28

• sin 40 = 28/w

Multiple Choice

What is the correct equation to solve for x ?

• tan (58) = x/11

• tan (58) = 11/x

• cos (58) = x/11

• sin (58) = x/11

Multiple Choice

Choose the correct ratio.

• cos(33)=x/14

• sin(33)=x/14

• cos(33)=14/x

• sin(33)=14/x

Putting it all Together

...see what you have learned

Multiple Choice

Which one is the easy way to remember trigonometric ratios?

• Sah Coh Toa

• Soh Cah Toa

• Soh Cah Tao

• Soh Ceh Toa

Multiple Choice

What is the ratio for Sine?

• Hyp/Opp

• Opp/Hyp

• Opp/Adj

• Adj/Opp

Multiple Choice

Sin X

• ⁸/₁₅

• ¹⁷/₁₅

• ¹⁵/₁₇

• ⁸/₁₇

Multiple Choice

Cos X

• ⁸/₁₅

• ¹⁷/₁₅

• ¹⁵/₁₇

• ⁸/₁₇

Multiple Choice

What is the ratio for Tangent

• Opp/Adj

• Adj/Opp

• Opp/Hyp

• Hyp/Opp

Multiple Choice

Tan X

• ⁸/₁₅

• ¹⁷/₁₅

• ¹⁵/₁₇

• ⁸/₁₇

Multiple Choice

Which trigonometric ratio is correct?

• sinA= opp/adj

• cosA= opp/hyp

• tanA= opp/adj

• sinA= adj/opp

Multiple Choice

Choose the correct ratio.

• sinZ=5/13

• sinX=5/13

• cosX=13/12

• tanZ=12/5

Multiple Choice

 Identify the trig ratio that may be used to find the marked unknown side.

•  $Sin\ 38\degree\ =\ \frac{h}{15}$  

•  $Sin\ 38\degree\ =\ \frac{15}{h}$  

•  $Cos\ 38\degree\ =\ \frac{15}{h}$  

•  $Tan\ 38\degree=\ \frac{h}{15}$  

Multiple Select

Which are correct?

•  $\tan\ x=\frac{3}{4}$  

•  $\sin z=\frac{40}{50}$  

•  $\cos\ z=\frac{3}{5}$  
  

•  $30^2+40^2=50^2$  

Congratulations!

You now know the Primary Trig Ratios...next up, using them to find missing sides