Express in functions of theta:

a. sin ( 810- theta)
b. cot ( theta - 360)
c cos (-180- theta)

a.

810 ° = 720 ° + 90 ° = 2 * 360 ° + 90 °

So :

sin ( 810 ° - theta ) = sin ( 90 ° - theta )

sin ( A - B ) = sin A * cos B - cos A * sin B

In this case A = 90 ° , B = theta.

sin 90 ° = 1

cos 90 ° = 0

sin ( 90 ° - theta ) = sin 90 ° * cos theta - cos 90 ° * sin theta = 1 * cos theta - 0 * sin theta = cos theta

So :

sin ( 810 ° - theta ) = cos theta

2.

cot ( theta - 360 ° ) = cos ( theta - 360 ° ) / sin ( theta - 360 ° )

sin ( theta - 360 ° ) = sin theta

cos ( theta - 360 ° ) = cos theta

So :

cot ( theta - 360 ° ) = cos theta / sin theta = cot theta

cot ( theta - 360 ° ) = cot theta

3.

cos ( - 180 ° - theta )

All trigonometric functions of - 180 ° is same like trigonometric functions of 180 °

sin - 180 ° = sin 180 °

cos - 180 ° = cos 180 °

sin 180 ° = 0

cos 180 ° = - 1

cos ( A - B ) = sin A * sin B + cos A * cos B

In this case A = - 180 ° , B = theta

cos ( - 180 ° - theta ) = sin ( - 180 ) ° * sin theta + cos ( - 180 ° ) * cos theta =

sin 180 ° * sin theta + cos 180 ° * cos theta = 0 * sin theta + ( - 1 ) * cos theta =

0 - cos theta = - cos theta

cos ( - 180 ° - theta ) = - cos theta

a. sin (810 - theta)

Well, let's think about it. We know that sine is a periodic function and that sin (theta + 2pi) = sin (theta). So, let's first add 810 to theta:

sin (810 - theta) = sin (theta + 810)

Now, we can subtract 2pi from 810 and find an equivalent angle:

810 - 2pi = 810 - 6.283 = 803.717

So, sin (810 - theta) = sin (theta + 803.717)

b. cot (theta - 360)

Cotangent is the reciprocal of tangent, and we know that the tangent function has a periodicity of pi radians or 180 degrees. Therefore, we can add or subtract 2pi or 360 degrees to find an equivalent angle:

cot (theta - 360) = cot (theta + 0)

So, cot (theta - 360) = cot (theta)

c. cos (-180 - theta)

Cosine is also a periodic function, and we know that cos (theta + 2pi) = cos (theta). Let's add 2pi to the given angle:

cos (-180 - theta) = cos (theta + 2pi)

Now, we subtract 2pi from -180 to find an equivalent angle:

-180 - 2pi = -180 - 6.283 = -186.283

So, cos (-180 - theta) = cos (theta - 186.283)

a. sin(810 - θ)

To express sin(810 - θ) in terms of θ, we can use the following trigonometric identity:

sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

In this case, A = 810 and B = θ. Therefore, the expression becomes:

sin(810 - θ) = sin(810)cos(θ) - cos(810)sin(θ)

b. cot(θ - 360)

To express cot(θ - 360) in terms of θ, we can use the following trigonometric identity:

cot(A - B) = cot(A)cot(B) + 1/tan(A)tan(B)

In this case, A = θ and B = 360. Therefore, the expression becomes:

cot(θ - 360) = cot(θ)cot(360) + 1/tan(θ)tan(360)

It's important to note that cot(360) = cot(0) = 1, and tan(360) = tan(0) = 0. Therefore, the expression simplifies to:

cot(θ - 360) = cot(θ) + 0 = cot(θ)

c. cos(-180 - θ)

To express cos(-180 - θ) in terms of θ, we can use the following trigonometric identity:

cos(-A) = cos(A)

In this case, A = 180 + θ. Therefore, the expression becomes:

cos(-180 - θ) = cos(180 + θ)

To express the given trigonometric functions in terms of theta, let's use the trigonometric identities and the properties of trigonometric functions.

a. sin (810 - theta):

We know the trigonometric identity sin (a - b) = sin(a) * cos(b) - cos(a) * sin(b).

Here, a = 810 and b = theta.

Using the identity, we can rewrite sin (810 - theta) as:

sin (810 - theta) = sin(810) * cos(theta) - cos(810) * sin(theta)

Now, we need to determine the values of sin(810) and cos(810).

To find sin(810), we can use the property that sin(x + 360n) = sin(x), where n is an integer.

So, sin(810) = sin(810 - 2 * 360) = sin(90) = 1

Similarly, cos(810) = cos(810 - 2 * 360) = cos(90) = 0

Substituting these values, we get:

sin (810 - theta) = 1 * cos(theta) - 0 * sin(theta) = cos(theta)

Therefore, sin (810 - theta) = cos(theta).

b. cot (theta - 360):

We know that cot (a - b) = cot(a) * cot(b) + 1

Here, a = theta and b = 360.

Using the identity, we can rewrite cot (theta - 360) as:

cot (theta - 360) = cot(theta) * cot(360) + 1

Now, we need to determine the values of cot(theta) and cot(360).

To find cot(theta), we use the identity cot(x + 360n) = cot(x), where n is an integer.

So, cot(theta) = cot(theta - 360) = cot(0) = ∞ (undefined)

We know that cot(360) = cot(0) = ∞ (undefined) as well.

Substituting these values:

cot (theta - 360) = ∞ * ∞ + 1 = ∞ + 1

Since ∞ + 1 is still ∞, cot (theta - 360) is undefined.

c. cos (-180 - theta):

We know that cos (-x) = cos(x), which means that cosine is an even function.

Using this property, we can rewrite cos (-180 - theta) as:

cos (-180 - theta) = cos(180 + theta)

Now, cos(180 + theta) can be evaluated using the angle addition formula:

cos(180 + theta) = cos(180) * cos(theta) - sin(180) * sin(theta)

Since cos(180) = -1 and sin(180) = 0, we can substitute these values:

cos(180 + theta) = -1 * cos(theta) - 0 * sin(theta) = -cos(theta)

Therefore, cos(-180 - theta) = -cos(theta).

sin ( 810- theta)

I am sure you meant: sin(180 - θ)
which is sinθ

cot(θ - 360) ....

recall that tan(-x) = -tanx
and tan(360-x) = -tanx
and of course the cot behaves just like the tan

cot(θ-360)
= -cot(360-θ)
= -(-cotθ)
= cotθ

cos(-180 - θ)
= cos(-(180 + θ))
= cos(180+θ) , since cos(-x) = cosx
= -cosθ , (180+θ) is in the 3rd quad

The above can be easily seen by the CAST rule.
You should also verify each answer you get for these type of questions by picking some angle and using your calculator