Determine the EXACT values of each of the following:

a) sin 240°
b) cos 135°

sin 240

1. where is the angle ? in quadrant III , so it is -
2. How far away from the x-axis ? 60° (the angle in standard position)

= - sin 60°
= = √3/2 , (from the 30-60-90 triangle)

see if you can the second one in the same way.

small typo, answer should have said ....

= - √3/2

To determine the exact values of sine and cosine of certain angles, you can use the unit circle or trigonometric identities.

a) sin 240°:
To find the exact value of sin 240°, we will use the unit circle.
1. Draw a unit circle, which is a circle centered at the origin with a radius of 1 unit.
2. Mark the angle 240° on the unit circle. This angle falls in the third quadrant.
3. Draw a vertical line from the point where the angle intersects the unit circle to the x-axis.
4. The length of this line is the value of sin 240°.

Looking at the unit circle, we can see that the y-coordinate of the point where the angle intersects the unit circle is equal to the sin 240°.
In the third quadrant of the unit circle, the y-coordinate is negative.

Therefore, sin 240° = -0.866.

b) cos 135°:
To find the exact value of cos 135°, we can use the trigonometric identity:
cos x = sin(90° - x).

Using this identity, we can rewrite cos 135° as sin(90° - 135°):
cos 135° = sin(90° - 135°).

Now, we can find the exact value of sin(90° - 135°) by using the result from part (a).

sin(90° - 135°) = sin(-45°).

From the unit circle, we know that sin(-45°) is equal to sin 45°, which is the same as sin π/4.
Using the unit circle again, we find that sin π/4 = 0.707.

Therefore, cos 135° = sin(90° - 135°) = sin(-45°) = sin 45° = 0.707.