Solve
cos
(
x
)
=
−
0.71
on
0
≤
x
<
2
π
There are two solutions, A and B, with A < B
looks much better this way:
Solve
cos(x) = - 0.71 on 0 ≤ x < 2π
set your calculator to RAD (radians)
you know that cosx is negative in quads II and III
on mine: "2ndF cos .71 = " to get .78129...
so A = π - .78129... = appr 2.3603
B = π + .78129... = appr 3.9229
To solve the equation cos(x) = -0.71 for x in the given interval, we can follow these step-by-step instructions:
1. Start by using the inverse cosine function, cos^(-1), also denoted as arccos, on both sides of the equation. This will give us the equivalent expression: x = arccos(-0.71).
2. Use a scientific calculator or a software program with a built-in arccos function to find the angle whose cosine is -0.71. Make sure your calculator is set in radians mode.
3. Find the principal value of arccos(-0.71). This will give you the initial solution for x.
4. Let's call this initial solution A.
5. To find the second solution, B, we need to determine the reference angle. The reference angle is the non-negative acute angle formed between the terminal side of the angle and the x-axis. The cosine value is negative in the second and third quadrants, so we will be looking for solutions that lie in these quadrants.
6. Subtract the initial solution (A) from 2π to find the maximum angle for the second solution. In this case, 2π is the reference angle for the range given.
7. Add the reference angle (2π) to the initial solution (A) to find the second solution (B).
Hence, we have found two solutions (A and B) for the equation cos(x) = -0.71 in the given interval 0 ≤ x < 2π.
To solve the equation cos(x) = -0.71 on the interval 0 ≤ x < 2π, we can follow these steps:
Step 1: Find the principal angle
The principal angle is the angle between 0 and 2π (or 0 and 360 degrees) whose cosine value matches the given value. Since cos(x) = -0.71, we need to find the angle whose cosine is -0.71.
Using a calculator, we can take the inverse cosine (also known as arccos) of -0.71 to find the principal angle. Let's call it angle A:
A = arccos(-0.71)
A ≈ 2.415 radians (rounded to three decimal places)
Step 2: Find the second angle
To find the second angle B, we can use the symmetry of the cosine function. The cosine function is an even function, which means cos(-x) = cos(x). Therefore, if A is a solution, then -A is also a solution.
B = -A
B ≈ -2.415 radians (rounded to three decimal places)
Step 3: Adjust the angles to fall within the given interval
The given interval is 0 ≤ x < 2π. Since A and B are both negative angles, we can add 2π to each angle to bring them within the given interval:
Adjusted angle A = A + 2π
Adjusted angle A ≈ 2.415 + 2π ≈ 5.698 radians (rounded to three decimal places)
Adjusted angle B = B + 2π
Adjusted angle B ≈ -2.415 + 2π ≈ 3.726 radians (rounded to three decimal places)
So, the solutions to the equation cos(x) = -0.71 on the interval 0 ≤ x < 2π, with A < B, are approximately:
Angle A ≈ 5.698 radians
Angle B ≈ 3.726 radians