use properties of the trigonometric function to find the exact value of each expression. Do no use a calculator.

sec ( - pi/18) * cos (37 pi)/18

I do not see how to do this problem. Every time I do it out I get a reference angle of twenty degrees which isn't on the unit circle. I do it on my calculator and get answer of one. If you could show me how to do this problem that would be great. I think the part I'm getting stuck on is the - pi/8 there must be some other rule that I do not know about when there are negetives

sec(-pi/18)*cos(37PI/18)

Remember that cos(-theta)=cos(theta)
so sec(theta)=sec(-theta)
sec(pi/18)cos(37Pi/18)
sec(2PI +Pi/18)cos(37PI/18)
sec(37pi/18)cos(37PI/18)
1/cos * cos= 1

how did you go from the first line to the next in these lines

sec(2PI +Pi/18)cos(37PI/18)
sec(37pi/18)cos(37PI/18)
1/cos * cos= 1

any trig function is periodic by 2PI.

sin (theta)=sin(theta+2PI)

To find the exact value of the expression sec(-pi/18) * cos(37pi/18), we can use the properties of trigonometric functions.

First, let's focus on the angle -pi/18. The negative sign indicates that we are dealing with a clockwise rotation on the unit circle. To better visualize this, we need to rewrite the angle in a more familiar form.

Recall that the unit circle is divided into 360 degrees or 2pi radians. Since pi radians is equal to 180 degrees, we can rewrite -pi/18 as (-180 degrees)/18 or -10 degrees.

Now, let's find the reference angle for -10 degrees. The reference angle is the positive acute angle formed between the terminal side of the angle and the x-axis. In this case, the reference angle is 10 degrees (or pi/18 radians).

Since the angle is negative, we need to determine the quadrant where the terminal side lies. The reference angle of 10 degrees falls in the first quadrant. When the reference angle is positive, the value of sec is positive in the first and fourth quadrants.

Next, let's focus on the angle 37pi/18. The reference angle for 37pi/18 can be found by subtracting the nearest multiple of 2pi from the angle. In this case, the nearest multiple of 2pi is 36pi/18, which simplifies to 2pi. Subtracting 2pi from 37pi/18 gives us pi/18 as the reference angle.

Now, let's determine the quadrant where the terminal side of pi/18 lies. The reference angle of pi/18 falls in the first quadrant. When the reference angle is positive, the value of cos is positive in the first and fourth quadrants.

Taking everything into account, we have:
sec(-pi/18) * cos(37pi/18) = sec(-10 degrees) * cos(pi/18)

Since we found that sec is positive in the first and fourth quadrants, and cos is positive in the first and fourth quadrants, the product of sec(-10 degrees) and cos(pi/18) will be positive.

Finally, since sec is the reciprocal of cosine, we can simplify the expression as follows:

sec(-10 degrees) * cos(pi/18) = 1/cos(-10 degrees) * cos(pi/18)

Using the identity sec(x) = 1/cos(x), we can further simplify:

1/cos(-10 degrees) * cos(pi/18) = cos(pi/18) / cos(-10 degrees)

Now, if you refer to a trigonometric identity called the angle addition identity, you can rewrite cos(pi/18) / cos(-10 degrees) as cos((pi/18) + (-10 degrees)).

Therefore, the exact value of the expression sec(-pi/18) * cos(37pi/18) is equal to cos((pi/18) + (-10 degrees)).

It is important to note that the exact value calculation involves manipulating the given angles using trigonometric identities and understanding the properties of trigonometric functions. Since the expression provided does not result in a simple numeric answer, we cannot evaluate it without using a calculator.