### Video Transcript

In this video, we will learn how to
use the Pythagorean identities to find the values of trigonometric functions. We will begin by recalling some key
formulae and definitions.

We know that the trig functions are
periodic and have an infinite number of solutions. However, in this video, we will
focus on solutions between zero and 360 degrees using the CAST diagram. The 𝑥- and 𝑦-axes create four
quadrants as shown. Quadrant one has positive 𝑥- and
positive 𝑦-values; quadrant two has negative 𝑥- and positive 𝑦-values; quadrant
three, negative 𝑥 and negative 𝑦; and quadrant four, positive 𝑥 and negative
𝑦.

When dealing with our trig
functions, we measure in a counterclockwise direction from the positive 𝑥-axis. This means that quadrant one
contains angles between zero and 90 degrees; quadrant two, between 90 and 180
degrees; and so on. We can label the quadrants with the
acronym CAST as shown. In quadrant one, labeled A, the sin
of angle 𝜃, cos of angle 𝜃, and tan of angle 𝜃 are all positive. In quadrant two, the sin of angle
𝜃 is positive. However, the cos of angle 𝜃 and
tan of angle 𝜃 are both negative. In quadrant three, when 𝜃 lies
between 180 and 270 degrees, the tan of 𝜃 is positive, whereas the sin of 𝜃 and
cos of 𝜃 are both negative. Finally, in quadrant four, the cos
of 𝜃 is positive and the sin and tan of 𝜃 are negative.

In this video, we will also need to
recall the three reciprocal trig identities. The csc of angle 𝜃 is equal to one
over the sin of angle 𝜃. The sec of angle 𝜃 is equal to one
over the cos of angle 𝜃. And the cot of angle 𝜃 is equal to
one over the tan of angle 𝜃. If the functions sin 𝜃, cos 𝜃,
and tan 𝜃 are positive or negative, then their reciprocal will also be positive or
negative. This means that all six functions
are positive in the first quadrant. In the second quadrant, between 90
and 180 degrees, the sin of 𝜃 and csc of 𝜃 will be positive, whereas all four
other functions will be negative. This pattern continues in quadrant
three and four.

In the questions that follow in
this video, we will need to use this information to decide whether our answers are
positive or negative.

We will now consider the three
Pythagorean identities. Firstly, we have sin squared 𝜃
plus cos squared 𝜃 is equal to one. Our second identity is tan squared
𝜃 plus one is equal to sec squared 𝜃. Whilst we don’t need to consider
the proofs of the identities in this video, we can get from the first identity to
the second identity by dividing each term by cos squared 𝜃. sin squared 𝜃 divided by cos
squared 𝜃 is tan squared 𝜃, as sin 𝜃 over cos 𝜃 equals tan 𝜃. Dividing cos squared 𝜃 by cos
squared 𝜃 gives us one. From the reciprocal identities, we
know that one divided by cos squared 𝜃 is sec squared 𝜃. The third Pythagorean identity is
one plus cot squared 𝜃 is equal to csc squared 𝜃. This can be obtained by dividing
each term of the first identity by sin squared 𝜃.

sin squared 𝜃 divided by sin
squared 𝜃 is equal to one. cos squared 𝜃 divided by sin
squared 𝜃 is equal to one over tan squared 𝜃. And this is equal to cot squared
𝜃. Finally, one divided by sin squared
𝜃 is equal to csc squared 𝜃. We will now use these three
identities together with the CAST diagram to solve some trig equations.

Find cos 𝜃 given sin 𝜃 is equal
to negative three-fifths, where 𝜃 is greater than or equal to 270 degrees and less
than 360 degrees.

There are lots of ways of solving
this problem. In this video, we will use the
Pythagorean identity sin squared 𝜃 plus cos squared 𝜃 is equal to one. Before substituting in our value of
sin 𝜃, it is worth noting that 𝜃 must be greater than or equal to 270 degrees and
less than 360 degrees. Using our knowledge of the CAST
diagram, this means that 𝜃 must lie in the fourth quadrant. In this quadrant, the cos of angle
𝜃 is positive, whereas the sin of 𝜃 and tan of 𝜃 are negative. This ties in with the fact that we
are told that sin 𝜃 is negative three-fifths. We know that our answer for cos of
𝜃 must be positive.

We can now substitute the value of
sin 𝜃 into our Pythagorean identity. This gives us negative three-fifths
squared plus cos squared 𝜃 is equal to one. Squaring a negative number gives a
positive answer. Therefore, negative three-fifths
squared is equal to nine twenty-fifths. We can then subtract this from both
sides of our equation. cos squared 𝜃 is equal to 16 over
25 or sixteen twenty-fifths. Square rooting both sides of this
equation, we see that cos of 𝜃 is equal to positive or negative the square root of
16 over 25. When square rooting a fraction, we
simply square root the numerator and denominator separately. The square root of 16 is four, and
the square root of 25 is five. As the cos of 𝜃 must be positive,
we can conclude that the cos of 𝜃 is four-fifths.

In our next question, we need to
evaluate the sine function given the cosine function and the quadrant of an
angle.

Find the value of sin 𝜃 given cos
of 𝜃 is equal to negative 21 over 29, where 𝜃 is greater than 90 degrees and less
than 180 degrees.

In order to answer this question,
we’ll use the Pythagorean identity sin squared 𝜃 plus cos squared 𝜃 is equal to
one. We are told that 𝜃 lies between 90
and 180 degrees, so it is worth considering our CAST diagram. The angle lies in the second
quadrant, which means that the sin of angle 𝜃 must be positive. The cos of angle 𝜃 and tan of
angle 𝜃 must be negative. This ties in with the fact that we
are told that the cos of 𝜃 is equal to negative 21 over 29. It also helps us in that we know
the answer for sin 𝜃 must be positive.

We can substitute the value of cos
𝜃 into the Pythagorean identity. Squaring negative 21 over 29 gives
us 441 over 841. We can then subtract this from both
sides of our equation such that sin squared 𝜃 is equal to one minus 441 over
841. The right-hand side simplifies to
400 over 841. We can then square root both sides
of our equation so that sin of 𝜃 is equal to positive or negative the square root
of 400 over 841.

Square rooting the numerator gives
us 20 and the denominator 29. As sin of 𝜃 must be positive, if
the cos of 𝜃 is negative 21 over 29 and 𝜃 lies between 90 and 180 degrees, then
the sin of 𝜃 is equal to 20 over 29.

In our next question, we will use
the Pythagorean identities to evaluate an expression.

Find the value of sin 𝜃 cos 𝜃
given sin 𝜃 plus cos 𝜃 is equal to five-quarters.

In order to solve this problem, we
will need to recall the Pythagorean identities. We will begin with the equation sin
𝜃 plus cos 𝜃 is equal to five over four. We can square both sides of this
equation. The left-hand side becomes sin 𝜃
plus cos 𝜃 multiplied by sin 𝜃 plus cos 𝜃. The right-hand side is equal to 25
over 16 as we simply square the numerator and denominator separately.

Distributing the parentheses or
expanding the brackets using the FOIL methods, we get sin squared 𝜃 plus sin 𝜃 cos
𝜃 plus sin 𝜃 cos 𝜃 plus cos squared 𝜃. We can group or collect the middle
terms. sin squared 𝜃 plus two sin 𝜃 cos
𝜃 plus cos squared 𝜃 is equal to 25 over 16. One of the Pythagorean identities
states that sin squared 𝜃 plus cos squared 𝜃 is equal to one. This means we can rewrite the
left-hand side of our equation as two sin 𝜃 cos 𝜃 plus one. We can then subtract one from both
sides of this equation. 25 over 16 minus one is equal to
nine over 16.

We can then divide both sides of
this new equation by two, giving us sin 𝜃 cos 𝜃 is equal to nine over 32. If sin 𝜃 plus cos 𝜃 is equal to
five over four, then sin 𝜃 multiplied by cos 𝜃 is equal to nine over 32.

In our final question, we will need
to use a different Pythagorean identity.

Find the sec of 𝜃 minus the tan of
𝜃 given the sec of 𝜃 plus the tan of 𝜃 is equal to negative 14 over 27.

We recall that the difference of
two squares states that 𝑥 squared minus 𝑦 squared is equal to 𝑥 plus 𝑦
multiplied by 𝑥 minus 𝑦. This means that sec 𝜃 plus tan 𝜃
multiplied by sec 𝜃 minus tan 𝜃 is equal to sec squared 𝜃 minus tan squared
𝜃. We are told in the question the
value of sec 𝜃 plus tan 𝜃. It is equal to negative 14 over
27. And we need to calculate the value
of sec 𝜃 minus tan 𝜃.

One of the Pythagorean identities
states that tan squared 𝜃 plus one is equal to sec squared 𝜃. If we subtract tan squared 𝜃 from
both sides of this equation, we have sec squared 𝜃 minus tan squared 𝜃 is equal to
one. This is equivalent to the
right-hand side of our equation. Negative 14 over 27 multiplied by
sec 𝜃 minus tan 𝜃 is equal to one. We can then divide both sides of
this equation by negative 14 over 27. We know that dividing by a fraction
is the same as multiplying by the reciprocal of this fraction. Therefore, the right-hand side is
equal to one multiplied by negative 27 over 14.

If the sec of 𝜃 plus the tan of 𝜃
is equal to negative 14 over 27, then the sec of 𝜃 minus the tan of 𝜃 is equal to
negative 27 over 14. These values are the reciprocal of
one another.

We will now summarize the key
points from this video. The three Pythagorean identities
are as follows. sin squared 𝜃 plus cos squared 𝜃
is equal to one. tan squared 𝜃 plus one is equal to
sec squared 𝜃. One plus cot squared 𝜃 is equal to
csc squared 𝜃. We have seen in this video that we
can use these identities to find the values of trig functions. When solving any trig equations, it
is also important to recall which of the functions are positive and negative in each
quadrant. One way of doing this is using the
CAST diagram as shown.