The questions says it has 2 solutions. It wants me to find what x would be if cos(x)=-.49 and it is given that x is on the scale between 0 and 2pi.
I know that cosine is the x value, and since cosime is negative in this case the two solutions need to be in the 2nd and 3rd quadrants.
Would the correct way to do this be by making a right triangle and solving the sides?
what angle has a cosine = .49 ??
cos^-1 .49 = 60.66 degrees
so our answers are 60.66 degrees above and below the negative y axis (180 deg)
180 - 60.66 = 119.34 degrees
180 + 60.66 = 240.66 degrees
0.00695
Yes, you are on the right track! To find the two solutions for the equation cos(x) = -0.49 in the interval between 0 and 2π, you can use a right triangle to visualize the problem and solve for the values of x.
In this case, since the cosine function is negative (-0.49), you correctly mentioned that the solutions will be in the second and third quadrants. By using a right triangle, you can determine the angles whose cosine value is -0.49. Here's how you can proceed:
1. Draw a right triangle and label one of the acute angles as α. The hypotenuse will have a length of 1 (as it is the radius of the unit circle), and the side adjacent to angle α will have a length of -0.49 (as cosine is the ratio of the adjacent side to the hypotenuse).
- The hypotenuse represents the radius of the unit circle (which is always 1).
- The adjacent side represents the x-coordinate of the chosen point on the unit circle.
2. Using the Pythagorean theorem, find the length of the opposite side of the triangle. The Pythagorean theorem states that the sum of the squares of the length of the two sides of a right triangle is equal to the square of the length of the hypotenuse.
- Let's denote the length of the opposite side as y.
- Using the Pythagorean theorem: y^2 + (-0.49)^2 = 1^2
- y^2 + 0.2401 = 1
- y^2 = 0.7599
- y ≈ √0.7599
- y ≈ ±0.8723
3. Now, you have the ratio of the opposite side (y) to the hypotenuse (1). Remember that sine is the ratio of the opposite side to the hypotenuse. Therefore, sin(α) ≈ ±0.8723.
4. Next, determine the angles in the second and third quadrants whose sine is approximately ±0.8723. You can use the inverse sine function (commonly denoted as sin^-1) or a calculator to find the exact values of these angles.
- Using the inverse sine function, sin^-1(0.8723) ≈ 60.77 degrees (approximately 1.062 in radians)
- Since sine is a periodic function, there is another angle in the third quadrant with the same sine value as in the second quadrant.
- So, the second solution in the third quadrant would be 180 degrees minus the angle we found: 180 - 60.77 ≈ 119.23 degrees (approximately 2.080 in radians).
5. Finally, to find the values of x within the given interval of 0 to 2π, convert the angles to radians by multiplying by π/180.
- The two solutions for x are approximately 1.062 radians (in the second quadrant) and 2.080 radians (in the third quadrant).
Therefore, the two solutions for the equation cos(x) = -0.49 in the interval between 0 and 2π are x ≈ 1.062 and x ≈ 2.080 radians.