1. Find the value of Sin^-1(-1/2)

a. -30 degrees
b. 30 degrees
c. 150 degrees
d. 330 degrees

2. Find the exact value of cos(-420 degrees)
a. -1/2
b. 1/2
c. sqrt of 3/2
d. -sqrt of 3/2

3. p(-9/41, 40/41) is located on the unit circle. Find sin theta.
a. 40/41
b. -9/41
c. -9/40
d. -40/9

4. In triangle ABC, C = 60 degrees, a=12, and b=5. Find c.

5. Which triangle should be solved by beginning with the law of Cosines?
a. A=115 degrees, a=19, b=13
b. A=62 degrees, B=15 degrees, b=10
c. B=48 degrees, a=22, b=5
d. A=50 degrees, b=20, c=18

1. Both a and d are correct. -30 and 330 degrees are the same angle

2. d

3. b theta = sin^-1 (-9/41)/1, siunce it is on a unit circle.
Note that sqrt[(-9/41)^2 + (40/41)^2] = 1

4. Use the law of cosines when you know two adjacent sides and the included angle (which would have a capital letter designation different from the sides). Only one of the choices fits this criterion.

1. To find the value of Sin^-1(-1/2), you can use the inverse sine function. The inverse sine function (Sin^-1) returns the angle whose sine is a given value. To solve this, follow these steps:

- Enter the value -1/2 into your calculator.
- Press the inverse sine function button (usually labeled as "sin^-1" or "arcsin").
- The calculator will return the angle whose sine is -1/2, which is -30 degrees.
- Therefore, the correct answer is Option a. -30 degrees.

2. To find the exact value of cos(-420 degrees), you can use the periodicity of the cosine function. The cosine function repeats its values every 360 degrees. So, you can find an equivalent angle within one full revolution.

- Subtract 360 degrees from -420 degrees to find the equivalent angle within one full revolution: -420 degrees - (-360 degrees) = -420 degrees + 360 degrees = -60 degrees.
- Now, find the exact value of cos(-60 degrees) using either a calculator or trigonometric identities.
- The exact value of cos(-60 degrees) is 1/2.
- Therefore, the correct answer is Option b. 1/2.

3. To find sin theta for point p(-9/41, 40/41) on the unit circle:

- The unit circle is a circle with a radius of 1 centered at the origin (0, 0).
- The x-coordinate of point p is -9/41 and the y-coordinate is 40/41. These coordinates represent a right triangle formed by the radii from the origin to the point.
- Use the Pythagorean theorem to find the length of the opposite side (y-coordinate). Square both the x-coordinate and y-coordinate, then sum them and take the square root. √[(-9/41)^2 + (40/41)^2] = √(81/1681 + 1600/1681) = √(1681/1681) = 1.
- Since the opposite side (y-coordinate) is equal to 1 and the hypotenuse of a unit circle is also 1, sin theta = opposite/hypotenuse = 1/1 = 1.
- Therefore, the correct answer is Option a. 40/41.

4. To find c in triangle ABC, you can use the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles.

- The Law of Cosines formula is c² = a² + b² - 2ab * cos(C), where c is the side opposite angle C.
- Plug in the values: c² = 12² + 5² - 2 * 12 * 5 * cos(60 degrees).
- Simplify: c² = 144 + 25 - 120 * cos(60 degrees).
- Evaluate: c² = 169 - 120 * (1/2) = 169 - 60 = 109.
- Take the square root of both sides to find c: c = √109.
- Therefore, the correct answer is c = √109.

5. The Law of Cosines is used to solve triangles when you have information about the lengths of the sides and/or angles.

- In this case, the triangle that should be solved by beginning with the Law of Cosines is Option a. A = 115 degrees, a = 19, b = 13.
- The Law of Cosines formula c² = a² + b² - 2ab * cos(C) can be used to find side c using the given values.
- The other options either provide enough information to use other trigonometric relationships like the Law of Sines or are missing data to apply any trigonometric concept.
- Therefore, the correct answer is Option a. A = 115 degrees, a = 19, b = 13.