How do I do this? Without a calculator, determine two angles between 0 degrees and 360 degrees that have a cosine of negative square root 3 over 2.
30, 60, 90 triangle
1, sqrt 3, 2
sin 30 = 1/2
cos 30 = sqrt 3/2
cos is negative in quad 2 and quad 3
so
180 - 30 = 150 degrees
180 + 30 = 210 degrees
Well, determining angles without a calculator can be quite a titanic task, but fear not, I'm here to clown around and help you out! To find angles with a cosine of negative square root 3 over 2, we need to think outside the box. By analyzing the unit circle, we find that there are two sets of angles that fit the bill: 150 degrees and 210 degrees, as well as 330 degrees and 390 degrees. So, there you have it - two pairs of angles that will make cosine go a little coconuts!
To solve this problem without a calculator, we can use the unit circle and the values of cosine at specific angles.
1. Recall the unit circle, which is a circle with a radius of 1 centered at the origin (0,0) in a coordinate plane.
2. Cosine represents the x-coordinate of a point on the unit circle. We need to find the angles where cosine is equal to negative square root of 3 over 2.
3. Start with the first quadrant on the unit circle (0° to 90°). The angle 60° (1st quadrant) has a cosine of 1/2, and cosine is positive in the 1st quadrant.
4. In the second quadrant (90° to 180°), cosine becomes negative. At 120°, the cosine is -1/2. At 150°, the cosine is -sqrt(3)/2. Thus, we have found the first angle between 0° and 360° that has a cosine of negative square root 3 over 2.
5. In the third quadrant (180° to 270°), cosine remains negative. Subtracting angles from 180°, we can find the 2nd angle. At 210°, cosine is -sqrt(3)/2.
6. In the fourth quadrant (270° to 360°), cosine becomes positive again. By subtracting angles from 360°, we can find the 3rd angle. At 330°, cosine is -sqrt(3)/2.
7. Repeat this process by adding or subtracting 360° to find other angles.
Thus, the two angles between 0° and 360° that have a cosine of negative square root 3 over 2 are 150° and 210°.
To determine two angles between 0 degrees and 360 degrees that have a cosine of negative square root 3 over 2 without using a calculator, we can rely on our knowledge of special angles and trigonometric identities.
1. Start by recalling the definition of cosine: cosine of an angle is equal to the adjacent side divided by the hypotenuse in a right triangle.
2. Since the cosine value is negative, it means that the angle lies in either the second or third quadrant, where cosine is negative.
3. Focus on the reference angle, which is the acute angle formed between the terminal arm of the angle and the x-axis. The reference angle will be the same for both the second and third quadrants.
4. Cosine represents the x-coordinate of the point on the unit circle. By knowing that the cosine is equal to negative square root of 3 over 2, we can deduce that the x-coordinate is -1/2. To find the corresponding y-coordinate, we can use the Pythagorean identity: sin^2θ + cos^2θ = 1.
5. Substitute the value of cosine (-1/2) into the Pythagorean identity: sin^2θ + (-1/2)^2 = 1. Simplifying the equation, we have sin^2θ + 1/4 = 1. Rearranging the equation, sin^2θ = 3/4.
6. Take the square root of both sides to find sinθ: sinθ = ±√(3/4) = ±√3/2. Since the sine value can be either positive or negative, we need to consider both cases.
7. Now, we can determine the reference angle by recalling the special triangles. In this case, the reference angles that have a sine value of √3/2 are 30 degrees (in the first quadrant) and 150 degrees (in the second quadrant).
8. Since we are looking for angles with a negative cosine, we need to consider the reference angle in the second quadrant. Subtracting both reference angles from 180 degrees will give us the desired angles: 180 - 30 = 150 degrees and 180 - 150 = 30 degrees.
Therefore, without using a calculator, the two angles between 0 degrees and 360 degrees that have a cosine of negative square root 3 over 2 are 150 degrees and 30 degrees.