Question

What is the relation between the sine and cosine values of angles in each quadrant? How would you use the 60° angle to find sine and cosine of 120°, 240°, and 300°? What angles could we find sine and cosine for using information for π/4 and π/6?

Answer 1

Explanation:

The values of cosine and sine share a relationship with the identities that form as a result of the nature of their symmetry. Their values are constant for specific angles, and their sign is constant for specific quadrants. The values of sine are always negative in quadrants III and IV because the negative sine of theta is equal to the sine of negative theta. This is defined by the sine function being odd. The values of cosine are negative in quadrants II and III because the cosine of negative theta is equal to the cosine of theta. This is defined by the cosine function being even. Because of this, the values of sine and cosine are consistent for a specific angle, with the sign being consistent for each particular quadrant. The symmetry of the cosine and sine functions proves them to correspond to the x-axis and y-axis respectively, especially in the context of the unit circle. The use of the 60 degree angle can be used to find the value of sine and cosine values for 120 degrees, 240 degrees, and 300 degrees because of the consistency of values for sine and cosine due to their symmetry. All of these angles act as reflections across the plane that correspond with the symmetry of sine and cosine. The primary value of 60 degrees can be found through the fact that a 30-60-90 triangle is a special triangle, which allows for the values for sine and cosine to be accurately described. The value for sine will be and the value of cosine will be for 60 degrees. Then, cosine will be - if the angle is 120 degrees, since it acts as a reflection across the y axis. Sine and cosine will become negative for a 240 degree rotation because it acts as a reflection about the origin. Sine becomes -

with a rotation of 300 degrees because the sine of -60, which is equivalent to the sine of 300, is equal to the negative of sine of 60. With the information from π/4, a 45-45-90 triangle, the sine and cosine values for 3π/4, 5π/4. and 7π/4 can be found because of their relation to reflections of π/4. With the information from π/6, a 30-60-90 triangle, the sine and cosine values for 5π/6, 7π/6, and 11π/6 can be found due to their relation to reflections of π/6 across the y axis, x axis, and across the origin.

Answer 2

The sine, cosine, and tangent of 60 deg. is the same as 120, 240, and 300. 30 deg, 45 deg, and 60 deg. can each be matched to other counter parts across the quadrants. The only ones that have their own sin, cos, and tan are 90, 180, 270, and 360. The pi over 4 is a radian and can be found using an equation= π/4*180/π The two πs will cancel out and you are left with 180/4 which will give you the degrees. You would do the same equation for the π/6. This can also be used in reverse and you can find the radian using the degrees. Just multiply the degrees by pi/180 and then find a common multiple to reduce the fraction/radian down. pi will stay in your answe