1. Find the amplitude, if it exists, and the period of y=tan 1/4È

Answer: No limits in amplitude, y=tanÈ/4 period is 4pi

2.Solve x = Arctan(-sqrt3)
Work: Pythagorean theorem: sqrt(1^2+3)= sqrt4=2 in length. Triangle is 1, sqrt 3,2

Answer: -60°

3. In triangleABC, A = 35°, a = 43, and c = 20. Determine whether triangle ABC has no solution, one solution, or two solutions. Then solve the triangle. Round to the nearest tenth

Answer: One solution; 57.8

I agree.

Thanks! I always second guess myself when doing the first few problems out of new lessons

To find the amplitude, if it exists, and the period of the function y = tan(1/4 * Θ), we need to understand some properties of the tangent function.

The amplitude is a property specific to trigonometric functions like sine and cosine. However, the tangent function does not have an amplitude. It is an unbounded function that takes on all real values.

On the other hand, the period of the tangent function is π, which means the pattern of the function repeats every π radians or 180 degrees. To find the period, you divide 2π (360 degrees) by the coefficient of Θ, which is 1/4 in this case.

Therefore, the period of y = tan(1/4 * Θ) is 4π radians or 720 degrees.

For the second question, to solve x = arctan(-sqrt(3)), we need to find the angle that has a tangent equal to -sqrt(3).

We know that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side in a right triangle. In this case, we have -sqrt(3) for the tangent.

Using the Pythagorean theorem, we can create a right triangle with the opposite side length of -sqrt(3), the adjacent side length of 1, and the hypotenuse length of 2.

Since the tangent function gives us the ratio of the opposite side to the adjacent side, we can determine that the angle is -60 degrees.

Finally, for the third question, we are given information about a triangle ABC, specifically angle A, side a, and side c.

To determine whether triangle ABC has no solution, one solution, or two solutions, we need to use the sine rule. The sine rule states that in a triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

Using the given information, we know the side opposite angle A (side a) is 43, and the side opposite angle C (side c) is 20.

To check if there is a solution, we can set up the equation: sin(A)/a = sin(C)/c

Plugging in the known values, we have: sin(35°)/43 = sin(C)/20

Rearranging the equation, we can cross multiply: sin(C) = (sin(35°) * 20) / 43

Using a calculator, evaluate the right side of the equation to get sin(C) ≈ 0.388

Now, we need to find the angle C using the inverse sine function: C = arcsin(0.388)

Using a calculator again, you will find C ≈ 22.1°

Since we have found a unique value for angle C that satisfies the equation, the triangle has one solution.

To solve the triangle, we have angles A = 35°, B = 180° - A - C ≈ 122.9°, and C ≈ 22.1°.

Side b can be found using the sine rule: b = (a * sin(B)) / sin(A)

Plugging in the known values, we have: b = (43 * sin(122.9°)) / sin(35°)

Evaluate using a calculator, and you will find that b ≈ 70.7 units.

Thus, the solution for triangle ABC, rounded to the nearest tenth, is A = 35°, B ≈ 122.9°, C ≈ 22.1°, a = 43 units, b ≈ 70.7 units, and c = 20 units.