cos theta = 5/13 with -pi/2 < theta <0 find each of the following

cos theta (- 11pi/6)

help plz?

To find the value of cos(theta) at theta = -11pi/6, we need to understand the concept of periodicity and the unit circle in trigonometry.

1. Periodicity: The cosine function is periodic with a period of 2pi, which means it repeats every 2pi radians. In other words, the cosine function has the same value at two angles if the angles differ by an integer multiple of 2pi.

2. Unit Circle: The unit circle is a circle with a radius of 1 unit centered at the origin (0, 0) in a coordinate plane. The angle theta is measured from the positive x-axis counterclockwise to the terminal side of an angle in standard position.

To find the value of cos(theta) at theta = -11pi/6, we will perform the following steps:

Step 1: Determine the reference angle:
- To find the reference angle, subtract -11pi/6 from a full circle (2pi) since we are given that -pi/2 < theta < 0.
- Reference angle = (2pi - (-11pi/6)) = (12pi/6 + 11pi/6) = 23pi/6

Step 2: Identify the quadrant:
- Since -pi/2 < theta < 0, the angle lies in the fourth quadrant.

Step 3: Determine the sign of the cosine:
- In the fourth quadrant, the cosine value is positive.

Step 4: Calculate the cosine value:
- Based on the given information that cos(theta) = 5/13, we know that the ratio of the cosine to the hypotenuse in a right triangle is 5/13.
- Let's call the adjacent side 'x' and the hypotenuse '13' to form the right triangle.
- Since cosine(theta) = adjacent/hypotenuse, we have cos(theta) = x/13.
- We are now solving for 'x'. Using the Pythagorean theorem, we know that x^2 + (5/13)^2 = 1^2.
- Solving the equation, we find x^2 = 1 - (5/13)^2.
- x^2 = (13^2 - 5^2)/13^2
- x^2 = (169 - 25)/169
- x^2 = 144/169
- x = +sqrt(144/169) or x = -sqrt(144/169) (but we choose the positive value since cosine is positive in the fourth quadrant).
- x = 12/13

Step 5: Calculate the value of cos(theta) at theta = -11pi/6:
- We have cos(theta) = adjacent/hypotenuse = x/13 = (12/13)/1 = 12/13

Thus, the value of cos(theta) at theta = -11pi/6 is 12/13.

Remember, understanding the unit circle and trigonometric identities is crucial to accurately solve trigonometry problems.