I don't get how you get these answers:
cos180 degree = -1
sin45 degree = root(2)/2
cos(-135) = -root(2)/2
cot(-60) = -1/root(3) or -root(3)/3
What does cos or sin do to make the answer different?
please help
Draw a triangle. In this case, it extends to the left on the x-axis, and the height is zero. Cos180° = x/h = -1/1 = -1
Draw a triangle. The legs are 1,1, hypotenuse=√2.
sin = y/h = 1/√2 = √2/2
cos -135 = - cos 45 = -√2/2
Draw a triangle
cot -60 = x/y = 1/-√3 = -1/√3
how do you know what is the side lengths for the cot-60 angle?
Great question! To understand how cos and sin functions work and generate different answers, let's start by understanding what they represent.
In trigonometry, cos (short for cosine) and sin (short for sine) are functions that relate angles in a right triangle to the ratios of its side lengths. The cosine function, cos(x), gives you the ratio of the length of the adjacent side to the hypotenuse, while the sine function, sin(x), gives you the ratio of the length of the opposite side to the hypotenuse.
Now, let's break down each example you provided:
1. cos180 degrees = -1:
To find cos180 degrees, we use the cosine function. In a unit circle (a circle with radius 1), the x-coordinate of a point on the circumference corresponds to the cosine value of the angle measured from a reference line. Since 180 degrees is exactly opposite the reference line, the x-coordinate of the corresponding point is -1. Therefore, cos180 degrees = -1.
2. sin45 degrees = √2/2:
To find sin45 degrees, we use the sine function. In a unit circle, the y-coordinate of a point on the circumference corresponds to the sine value of the angle measured from the reference line. For 45 degrees, the coordinates of the corresponding point on the unit circle are (√2/2, √2/2). Therefore, sin45 degrees = √2/2.
3. cos(-135) = -√2/2:
To find cos(-135), we use the cosine function. The cosine of a negative angle is the same as the cosine of its positive counterpart. Since -135 degrees is equivalent to 225 degrees (as the unit circle is periodic), the cosine value remains the same as cos225 degrees. The coordinates for 225 degrees on the unit circle are (-√2/2, -√2/2), giving us cos(-135) = -√2/2.
4. cot(-60) = -1/√3 or -√3/3:
To find cot(-60), we need to evaluate the reciprocal of the tangent function. Tangent (tan(x)) is the ratio of the length of the opposite side to the adjacent side in a right triangle. Since -60 degrees falls in the second quadrant of the unit circle, where both the x and y-coordinates are negative, the coordinates for -60 degrees on the unit circle are (-√3/2, -1/2). The cotangent (cot(x)) is obtained by taking the reciprocal of the tangent: cot(-60) = 1/tan(-60) = 1/(-1/2) = -2/1 = -2. However, we can simplify this by rationalizing the denominator, which results in -1/√3 or -√3/3.
In summary, the cos and sin functions generate different answers based on the ratios of specific side lengths in a right triangle, which correspond to angles on the unit circle. They provide crucial information for understanding the relationships between angles, sides, and ratios in trigonometry.