Example: The six trig ratios

Trigonometry and precalculus

Basic Trigonometry

Basic trigonometric ratios

Basic Trigonometry
Trigonometry 0.5
Basic Trigonometry II
Trigonometry 1
Example: Using soh cah toa
Example: The six trig ratios
Reciprocal trig functions
Example: Using trig to solve for missing information
Trigonometry 1.5
Example: Calculator to evaluate a trig function
Example: Trig to solve the sides and angles of a right triangle
Trigonometry 2
Example: Solving a 30-60-90 triangle
Special right triangles

Additional content

Using Trig Functions
Using Trig Functions Part II

Example: The six trig ratios Worked example where we walk through finding the major trig ratios

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Example: The six trig ratios

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Determine the six trigonometric ratios
for angle A in the right triangle below
so this right over here is angle A, it's at vertex A.
And to help me remember the definitions of the trig ratios,
and these are human constructed definitions that have ended up being
very, very useful for analyzing a whole series
of things in the world.
To help me remember them, I use the word
SOH CAH TOA
Let me write that down.
SOH CAH TOA
sometimes you can think of it as one word but
it's really the 3 parts that define at least
3 of the trig functions for you and then we
can get the other 3 by looking at the first 3.
So SOH tells us that sine of an angle,
in this case it's sine of A,
So sine of A is equal to
the opposite, that's the O, over the hypotenuse.
Well, in this context, what is the opposite to angle A?
Well, we go across the triangle, it opens up on to
side BC, it has length 12, so that is the opposite side.
So, this is going to be equal to twelve and
what's the hypotenuse?
Well, the hypotenuse is the longest side of the triangle.
It's opposite the 90 degree angle, and so we go
opposite the 90 degree angle,
longest side is side AB and it has length 13,
so this right over here is the hypotenuse.
So, the sine of A is 12/13.
So now let's go to CAH.
CAH defines cosine for us.
It tells us that cosine of an angle
in this case, cosine of A, is equal to
the adjacent side to the angle over the hypotenuse.
So, what's the adjacent side to angle A?
Well, if we look at angle A, there are two sides that are
next to it. One of them is the hypotenuse,
the other one has length 5.
The adjacent one is side CA, so it's 5 and
what is the hypotenuse? Well, we've already figure that out.
The hypotenuse is right over here
it's opposite the 90 degree angle.
It's the longest side of the right triangle.
It has length 13. So the cosine of A is 5/13.
Let me label this right over here is the
adjacent side. And this is all specific to angle A.
The hypotenuse would be the same regardless
of the angle you pick but the opposite and
the adjacent is dependent on the angle that we
choose in the right angle.
Now, let's go to TOA. TOA defines tangent for us.
It tells us that the tangent of an angle
is equal to the opposite side over the adjacent side.
So, given this definition, what is the tangent of A?
The opposite side, we have already figured out,
has length 12 and the adjacent side,
we've already figure out has length 5.
So the tangent of A, which is opposite over adjacent is 12/5.
Now, we'll go the to the other 3 trig ratios.
Which you can think of as the reciprocals of

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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?

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When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...

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At 2:33, Sal says "double bonds" but should say "single bonds."

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