### Video Transcript

In this video, we will learn how to
write trigonometric functions like sine, cosine, and tangent and their reciprocals
in terms of cofunctions and use their properties to recognize the equivalents of two
trigonometric functions. We recall that if the angle 𝜃 is
in standard position on a set of axes with a unit circle centered at the origin,
then we can use the coordinates of the intersection between the terminal side of the
angle and the unit circle to define sin 𝜃 and cos 𝜃. By looking at this in more detail,
we will be able to define the sine and cosine of any angle.

Since tan 𝜃 is equal to sin 𝜃
over cos 𝜃, we can use these values to evaluate the tangent of any angle or any
reciprocal trigonometric function. This geometric interpretation
allows us to discover identities of the trig functions.

Let’s begin by considering the
periodicity of these functions. A full rotation of 360 degrees or
two 𝜋 radians counterclockwise of the angle 𝜃 will not change the position of its
initial or terminal side. This will be true for any number of
full rotations. Therefore, sin of 𝜃 is equal to
sin of 𝜃 plus 360 multiplied by 𝑛. And the cos of 𝜃 is equal to the
cos of 𝜃 plus 360 multiplied by 𝑛. As this also holds for any
clockwise rotation, these identities are valid for any integer 𝑛.

When dealing in radians, we would
replace 360 with two 𝜋. Since the tangent function is
periodic every 180 degrees or 𝜋 radians, the tan of 𝜃 is equal to tan of 𝜃 plus
180 multiplied by 𝑛. This is once again true for any
integer 𝑛, and we can replace 180 with 𝜋 when dealing in radians.

We can also take the reciprocal of
both sides of any of these identities to find similar identities for the reciprocal
trigonometric functions. These are the cosecant, secant, and
cotangent, respectively. We can determine other
trigonometric identities by using geometric transformations on the unit circle. For example, we can reflect angle
𝜃 in the 𝑦-axis. Since this is a reflection in the
vertical axis, the two triangles are congruent. We can use this to determine the
coordinates of the intersection between the hypotenuse and the unit circle. These are negative cos 𝜃, sin
𝜃.

We also notice that the hypotenuse
is the terminal side of the angle 180 degrees minus 𝜃 in standard position. This means that the sin of 180
degrees minus 𝜃 is equal to the sin of 𝜃 and the cos of 180 degrees minus 𝜃 is
equal to negative cos 𝜃. Using the fact that tan of 𝜃 is
equal to sin of 𝜃 over cos of 𝜃, the tan of 180 degrees minus 𝜃 is equal to
negative tan of 𝜃.

This leads us to a specific example
of related angles, in this case when two angles sum to give 180 degrees or 𝜋
radians. Since these angles are known as
supplementary angles, the three identities are known as the supplementary angle
trigonometric identities. Once again, by taking the
reciprocal of both sides, we can find similar identities for the reciprocal
trigonometric functions.

We can follow the same process to
find trigonometric identities of other related angles. Firstly, we have the point in the
third quadrant with coordinates negative cos 𝜃, negative sin 𝜃 as shown. This corresponds to the angle 180
degrees plus 𝜃 in standard position. The sin of 180 degrees plus 𝜃 is
therefore equal to negative sin 𝜃. And the cos of 180 degrees plus 𝜃
is equal to the negative of cos 𝜃. As dividing a negative by a
negative gives a positive answer, the tan of 180 degrees plus 𝜃 is equal to the tan
of 𝜃.

We can repeat this for the related
angles 𝜃 and 360 degrees minus 𝜃. In this case, we have the sin of
360 degrees minus 𝜃 is equal to negative sin 𝜃. The cos of 360 degrees minus 𝜃 is
equal to the cos of 𝜃. And the tan of 360 degrees minus 𝜃
is equal to the negative tan of 𝜃.

Identifying whether the sine,
cosine, and tangent of any angle is positive or negative in each quadrant can be
summarized using a CAST diagram. In the first quadrant, when 𝜃 lies
between zero and 90 degrees, all three of our functions — the sine, cosine, and
tangent — are positive. In the second quadrant, the sin of
𝜃 is positive, whereas the cos and tan of 𝜃 are negative. In the third quadrant, between 180
and 270 degrees, the sine and cosine functions are negative, whereas the tangent is
positive. And finally, in the fourth
quadrant, the sin of 𝜃 is negative, the cos of 𝜃 is positive, and the tan of 𝜃 is
negative.

We will now look at an example
where we’ll use the related angle identities to simplify a trigonometric
expression.

Simplify the cos of 𝜃 plus the cos
of 180 degrees minus 𝜃.

There are a few ways of simplifying
this expression. For example, we could recall the
supplementary angle identity. That tells us that the cos of 180
degrees minus 𝜃 is equal to negative cos of 𝜃. Substituting this into our
expression, we would have the cos of 𝜃 plus negative cos of 𝜃. As this is the same as subtracting
cos of 𝜃 from cos of 𝜃, our answer would be zero.

Whilst this method appears
straightforward, it relies on us remembering the supplementary angle identity
shown. There are many such identities, and
trying to remember all these is difficult. It is therefore helpful to
understand where these identities come from using the unit circle. If we sketch an acute angle 𝜃 in
standard position, then the angle 180 degrees minus 𝜃 is the supplementary angle to
𝜃. Together, these form a straight
line on the 𝑥-axis as shown.

Reflecting our triangle in the
vertical axis, we have the angle 180 degrees minus 𝜃 in standard position. Since the triangles are congruent,
their bases will be equal in length. This leads us to the identity the
cos of 180 degrees minus 𝜃 is equal to the negative of cos of 𝜃. As this is true for any angle 𝜃
measured in degrees, we know that our answer is correct. cos of 𝜃 plus cos of 180 degrees
minus 𝜃 is equal to zero.

Before looking at another example,
we will consider complementary angles. We recall that complementary angles
are angles which sum to 90 degrees. These are useful in trigonometry
because if 𝜃 is one angle of a right triangle, then the other angle is 90 degrees
minus 𝜃. We can use this information to find
a set of identities called the cofunction identities. By firstly drawing 𝜃 in standard
position, we can construct the following congruent triangle. This is the angle 90 degrees minus
𝜃 in standard position. Since the triangles are congruent,
we can equate the corresponding sides such that the sin of 90 degrees minus 𝜃
equals cos 𝜃 and the cos of 90 degrees minus 𝜃 is sin 𝜃.

Using the fact that the tan of 𝜃
is equal to sin 𝜃 over cos 𝜃 and the cot of 𝜃 is the reciprocal of this, we have
the tan of 90 degrees minus 𝜃 is equal to the cot of 𝜃. Once again, we can take the
reciprocal of both sides of any of these identities to find identities for the
reciprocal trigonometric functions.

The cofunction identities are one
example of what are known as the corelated angle identities. The other corelated identities we
will consider in this video are 𝜃 and 90 degrees plus 𝜃.

Returning to our unit circle, we
will begin by rotating the triangle 90 degrees counterclockwise about the
origin. This gives us a congruent triangle
where the angle 90 degrees plus 𝜃 is in standard position. The sin of 90 degrees plus 𝜃 is
therefore equal to cos 𝜃. The cos of 90 degrees plus 𝜃 is
equal to negative sin of 𝜃. And the tan of 90 degrees plus 𝜃
is equal to negative cot of 𝜃. We could also develop this further
for rotations of 180 and 270 degrees counterclockwise.

We will now look at one final
example where we need to apply a cofunction identity.

Using the fact that the cos of 𝜃
is equal to the sin of 90 degrees minus 𝜃, which of the following is equivalent to
the cos of 35 degrees? Is it (A) one over the sin of 35
degrees? (B) sin of 35 degrees. (C) sin of 55 degrees. (D) sin of 145 degrees. Or (E) negative sin of 35
degrees.

We can begin this question by
substituting 𝜃 equals 35 degrees into both sides of our equation. This gives us the cos of 35 degrees
is equal to the sin of 90 degrees minus 35 degrees. The right-hand side simplifies to
the sin of 55 degrees, which from the five options given is option (C). This suggests that this is the
correct answer. However, we can check by sketching
the angle 35 degrees in standard position on the unit circle.

The coordinates of the point marked
on the unit circle are the cos of 35 degrees, sin of 35 degrees. As this is a right triangle, the
third angle is equal to 55 degrees. If we drew this angle of 55 degrees
in standard position, it would be congruent to the triangle drawn for 35 degrees in
standard position. The point on the unit circle has
coordinates cos 55 degrees, sin 55 degrees. Since the height of this second
triangle has the same length as the base of the first triangle, the sin of 55
degrees must be equal to the cos of 35 degrees. We can therefore conclude that
option (C) is correct. If cos of 𝜃 is equal to the sin of
90 degrees minus 𝜃, then the cos of 35 degrees is equal to the sin of 55
degrees.

We will now summarize the key
points from this video. We saw in this video that we can
show properties of the trigonometric functions by considering the congruency of
triangles formed by angles in standard position on the unit circle. In particular, we saw that the sin,
cos, and tan of 180 degrees plus 𝜃 are equal to negative sin 𝜃, negative cos 𝜃,
and tan 𝜃, respectively. We also had the corelated angle
identities. These were the sin, cos, and tan of
90 degrees minus 𝜃 and 90 degrees plus 𝜃. All of these identities and more
can be directly shown from the symmetries of the unit circle.