Find two exact values of theta in degrees (0 less than or equal to theta < 360°) if cos theta = -(square root of 2)/2.
cosØ = -√2/2
so Ø is in quadrants II or III
you should know the trig ratios of the 45-45-90 as well as the 30-60-90 degree triangles.
so you know the angle in standard position is 45°, because cos45 = √2/2
so Ø = 180-45 or 135° (in II)
or Ø = 180+45 = 225° (in III)
To find the two exact values of theta in degrees if cos theta = -(square root of 2)/2, we can use the unit circle to determine the angles that correspond to this cosine value.
1. Start by drawing a unit circle, which is a circle with a radius of 1 unit.
2. On the unit circle, locate the point on the x-axis that corresponds to the cosine value of -(square root of 2)/2. This point is (-√2/2, 0).
3. The angle theta can be measured by the reference angle formed between the positive x-axis and the line connecting the origin to the point (-√2/2, 0). Let's call this angle alpha.
4. Since the cosine value is negative, the reference angle alpha can be found in the second and third quadrants of the unit circle.
5. In the second quadrant, the reference angle alpha is the angle formed between the positive x-axis and the line connecting the origin to the point (-√2/2, 0). Since the x-coordinate is negative, the y-coordinate is positive. The reference angle alpha in this case is 135°.
6. In the third quadrant, the reference angle alpha is the angle formed between the positive x-axis and the line connecting the origin to the point (-√2/2, 0). Since both the x-coordinate and y-coordinate are negative, the reference angle alpha in this case is 225°.
Therefore, the two exact values of theta in degrees are 135° and 225°.
To find the angles in degrees where cosine is equal to -(√2)/2, we can use the unit circle or inverse trigonometric functions.
1. Using the unit circle:
Cosine represents the x-coordinate of a point on the unit circle, and -(√2)/2 is associated with the special angles 45° and 225° in the unit circle.
- Angle 1: The first angle is 45° in the 1st quadrant. When you draw a line from the origin to the point on the unit circle with an x-coordinate of -(√2)/2, it intersects the unit circle at 45°.
- Angle 2: The second angle is 225° in the 3rd quadrant. When you draw a line from the origin to the point on the unit circle with an x-coordinate of -(√2)/2, it intersects the unit circle at 225°.
Therefore, the two angles in degrees where cos(θ) = -(√2)/2 are 45° and 225°.
2. Using the inverse cosine function (arccos):
The inverse cosine function can directly give us the angle in radians where the cosine is equal to a specific value. To convert that to degrees:
- Angle 1: arccos(-(√2)/2) = π/4 radians. Multiply by 180/π to convert to degrees: (π/4) * (180/π) ≈ 45°.
- Angle 2: arccos(-(√2)/2) = 5π/4 radians. Multiply by 180/π to convert to degrees: (5π/4) * (180/π) ≈ 225°.
Hence, the two angles in degrees where cos(θ) = -(√2)/2 are 45° and 225°.