I do not understand for my life how to solve this problem.
Evaluate the expression without the aid of a calculator.
arccos[(-3^1/2)/2)
I'm not sure how my teacher did it. I know she said to find which quadrant that cos can be in and graph the triangle to find the angle.
first of all
3^(1/2) = √3
arccos[(-3^1/2)/2) really says :
find the angle theta, so that
cos theta = -√3/2
are you familiar with the ratio of sides of the 30-60-90 triangle ?
if so, then you should recognize that cos 30º = √3/2
but our cosine is negative so the angle must be in quadrants II or III by the CAST rule. (and 30º = pi/6 radians)
so theta is pi - pi/6 = 5pi/6
or
theta is pi + pi/6 = 7pi/6
then arccos[(-3^1/2)/2) = 5pi/6 or 7pi/6
No worries, I can help you understand how to solve this problem step by step.
First, let's break down the problem:
You are asked to evaluate the expression arccos[(-3^1/2)/2], which means finding the angle whose cosine equals (-3^1/2)/2.
To solve this, we can use the concept of the unit circle and the symmetry of trigonometric functions.
Here's how you can approach the problem:
1. Identify the denominator of the given expression: 2.
This tells us that the angle we are looking for lies on the unit circle where the x-coordinate is (-3^1/2)/2.
2. Determine the quadrant where the given x-coordinate lies:
The denominator is positive, so the x-coordinate value (-3^1/2)/2 should be positive.
- In the first quadrant (0° to 90°), both x and y coordinates are positive.
- In the second quadrant (90° to 180°), the x-coordinate is negative, but the y-coordinate is positive.
- In the third quadrant (180° to 270°), both x and y coordinates are negative.
- In the fourth quadrant (270° to 360°), the x-coordinate is positive, but the y-coordinate is negative.
Since we identified that the x-coordinate is positive, we can narrow down the options to the first and fourth quadrants.
3. Graph the triangle:
- Draw the unit circle on a coordinate plane.
- Place a point on the x-axis at (-3^1/2)/2. This is your reference point.
- Draw a vertical line from this point to the unit circle, creating a right triangle.
4. Calculate the values of the triangle's sides:
- The hypotenuse of the triangle is always 1 (as it is the radius of the unit circle).
- The adjacent side is the x-coordinate (-3^1/2)/2.
- The opposite side is the y-coordinate. Since the hypotenuse is 1 and the adjacent side is (-3^1/2)/2, we can use the Pythagorean theorem to find the opposite side (y-coordinate).
5. Determine the angle using the arccos function:
Use the equation cos(angle) = adjacent/hypotenuse, which in this case is (-3^1/2)/2. By rearranging the equation, we can solve for the angle.
angle = arccos((-3^1/2)/2)
6. Solve for the angle using a calculator:
The value of arccos((-3^1/2)/2) is approximately 150.52 degrees. You can use a calculator or an online tool to find this value.
Remember, these steps are provided to help you understand how to solve a problem like this step by step. In practice, you can use a calculator to find the value more efficiently.