What is the handy name that helps us remember the trigonometric functions for right triangles?

**SOH - CAH - TOA**

Consider the triangle already labelled: HOW do we know which side is the hypotenuse?

The **hypotenuse** is the __longest__ side, and it is __across the right angle__.

Consider the same amazing triangle. How do we know which side is the **opposite**?

Aside from the right angle, we need another angle inside the triangle (it is usually given to us) - here it's "alpha". The **opposite** side is opposite (right across from) angle alpha. This side will always depend on the angle you are focusing on. ALWAYS LABEL YOUR SIDES!

Here's our favourite triangle again! Have another look! How do we know which side is the adjacent?

Adjacent means "next to". So the **adjacent** side** next to the angle** - here it's angle alpha again! This side will always depend on the angle you are focusing on ALWAYS LABEL YOUR SIDES.

On the brand new triangle below, find the hypotenuse, the opposite and the adjacent sides:

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- Hypotenuse = side CA

(longest side, opposite the right angle)

- Opposite = side CB (opposite angle "theta")
- Adjacent = side AB ("next to" angle theta)

What does SOH - CAH - TOA stand for?

SOH - CAH - TOA

What's the general process for finding an unknown side of a right triangle using SOHCAHTOA (trigonometry)?

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- Label ALL your sides! H = hypotenuse, O = opposite, A = adjacent.
- See which of the SOHCAHTOA functions fits the information you have (you can only have ONE unknown)
- Write down the equation that you need and plug-in the information that you have.
- Rearrange it so that your unknown is the subject of the equation (meaning, it's left alone on one side of the equation).
- Type in the other side of the equation into your calculator and solve.
- Unless told otherwise, always express your answer to 3 significant figures (don't forget to check if you need to round up!)

CAREFUL! Make sure your calculator is in **degrees**!

Consider the following triangle. Which function would you use to find side h?

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- What sides do you have?

h (opposite), 3 (adjacent), and the angle is 46º

- What function fits
**opposite**and**adjacent**?

TOA. tan(angle) = opposite/adjacent

tan(46º) = h/3

tan(46º)x3 = h

(now put the left hand side of this equation in you calculator - make sure it's in **degrees**!)

How do you find side x?

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- What sides do you have?

x (hypotenuse), 6 (adjacent), angle 20.5º

- What function fits hypotenuse and adjacent?

CAH - cos(angle) = adj/hyp

cos(20.5) = 6/x

- Since x is in the bottom part of the fraction, multiply both side by x so you can cancel it on the left side of the equation:

cos(20.5) (x) = 6

Now divide both sides by cos (20.5)

x = 6/cos(20.5) = 6.41 (3 s. f.)

What do we ALWAYS need to do to trig functions in order to find an unknown angle?

To find an unknown angle, we ALWAYS need to take the **inverses** of the trig functions.

For example: **sin**(x) = 0.5

x = **sin ^{-1 }**(0.5) = 30º

Find angle theta:

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- What sides do we have? Opposite & adjacent
- What function fits opp & ajd?

TOA: tan("theta") = opp/adj = 8/5

"theta" = tan^{-1} (8/5) = 57.99 = 58.0º (3 s.f.)

Find angle theta:

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- What sides do we have? 3 (opposite) & 6.5 (hypotenuse)
- What function fits opp & hyp?

SOH: **sin**(theta) = opp/hyp = 3/6.5

"theta" = **sin ^{-1}**(3/6.5) = 27.48 = 27.5º (3 s.f.)

What is an "angle of elevation"?

It's the angle between the floor and an upward line.

What is an "angle of depression"?

It's the angle between the ceiling and a downward line.

What are angles of elevation and depression?

What is the sine formula?

What is the cosine formula?

What is the tangent formula?

What type of triangle do you need to have in order to use SOH CAH TOA?

A right angled triangle.

How do you know which to use sine, cosine, or tangent?

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- Write down SOH-CAH-TOA.
- Circle all of the letters for the side you know. (For example, circle all of the Os.)
- Circle all of the letters for the side you're looking for.
- Use whichever one has both letters circled.

What is the handy name that helps us remember the trigonometric functions for right triangles?

**SOH - CAH - TOA**

How do you liberate an angle from a trig function?

Use the inverse!