Question

Cos930°=??????????......

Answer 1

$cos930 = cos(360 \times 2 + 30) = cos30 = - \frac{ \sqrt{3} }{2}$

Answer 2

The cosine of 930 degrees is equal to the cosine of the reference angle, which is 930 degrees minus the largest multiple of 360 degrees that can be subtracted from it to obtain an angle in the range from 0 to 360 degrees.

So, let's find the reference angle:

930° - 360° = 570° (largest multiple of 360 less than 930)

Now, since cosine has a period of 360 degrees, the cosine of any angle is the same as the cosine of its reference angle. Therefore,

cos(930°) = cos(570°)

To calculate the cosine of 570 degrees, we need to convert it to an acute angle. Since 570 degrees is in the third quadrant, and the cosine function is positive in the fourth quadrant (180° to 270°), we can use the identity:

cos(180° + x) = -cos(x)

where x is the acute angle in the fourth quadrant.

So,

cos(570°) = -cos(180° + 570°)

cos(570°) = -cos(750°)

Now, let's find the reference angle for 750 degrees:

750° - 360° = 390° (largest multiple of 360 less than 750)

Now, since cosine has a period of 360 degrees, the cosine of any angle is the same as the cosine of its reference angle. Therefore,

cos(750°) = cos(390°)

To calculate the cosine of 390 degrees, we need to convert it to an acute angle. Since 390 degrees is in the second quadrant, and the cosine function is negative in the second quadrant (90° to 180°), we can use the identity:

cos(180° - x) = -cos(x)

where x is the acute angle in the second quadrant.

So,

cos(390°) = -cos(180° - 390°)

cos(390°) = -cos(210°)

Now, let's find the reference angle for 210 degrees:

210° - 180° = 30° (smallest angle in the second quadrant)

Finally, the cosine of 930 degrees is:

cos(930°) = -cos(750°) = -(-cos(210°)) = cos(30°)

Since the cosine of 30 degrees is a well-known value, we have:

cos(930°) = cos(30°) = √3 / 2 ≈ 0.866