What angle in the first quadrant could you reference to help you find sine and cosine of each of the following angles ?
330
120
113
203
To find the sine and cosine of angles in the first quadrant, you can refer to the reference angle of each angle. The reference angle is the acute angle formed between the terminal side of an angle and the x-axis.
For the given angles:
330 degrees:
The reference angle is 360 degrees - 330 degrees = 30 degrees.
Since the angle is in the fourth quadrant (330 degrees - 360 degrees), the sine and cosine values will be negative in the first quadrant.
The sine of the reference angle (30 degrees) is the same as the sine of the original angle: sin(330 degrees) = sin(30 degrees) = -0.5
The cosine of the reference angle (30 degrees) is the same as the cosine of the original angle: cos(330 degrees) = cos(30 degrees) = 0.866
120 degrees:
The reference angle is the same as the original angle since it is in the first quadrant: 120 degrees.
The sine of the reference angle is the same as the sine of the original angle: sin(120 degrees) = sin(120 degrees) = 0.866
The cosine of the reference angle is the same as the cosine of the original angle: cos(120 degrees) = cos(120 degrees) = 0.5
113 degrees:
The reference angle is the same as the original angle since it is in the first quadrant: 113 degrees.
The sine of the reference angle is the same as the sine of the original angle: sin(113 degrees) = sin(113 degrees) = 0.913
The cosine of the reference angle is the same as the cosine of the original angle: cos(113 degrees) = cos(113 degrees) = 0.407
203 degrees:
The reference angle is the total angle in the first quadrant minus the given angle: 180 degrees - 203 degrees = -23 degrees.
Since the angle is in the third quadrant, both the sine and cosine values will be negative in the first quadrant.
The sine of the reference angle (23 degrees) is the same as the sine of the original angle: sin(203 degrees) = sin(23 degrees) = -0.407
The cosine of the reference angle (23 degrees) is the same as the cosine of the original angle: cos(203 degrees) = cos(23 degrees) = -0.913
To find the sine and cosine of an angle, we can use reference angles in the first quadrant. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis.
1. Angle 330:
Since 330 degrees is outside of the first quadrant, we need to find its reference angle within the first quadrant. To obtain the reference angle, we subtract 360 degrees (a full circle) from the given angle until we reach an angle in the first quadrant. In this case, subtracting 360 from 330 gives us 330 - 360 = -30 degrees. However, negative angles are equivalent to their positive counterparts, so we add 360 degrees to -30, resulting in 330 degrees as the reference angle in the first quadrant.
2. Angle 120:
The angle 120 degrees is already within the first quadrant, so we can directly use it to find the sine and cosine.
3. Angle 113:
Similarly, angle 113 degrees falls within the first quadrant, so we can use it directly to determine the sine and cosine.
4. Angle 203:
To find the reference angle for angle 203 degrees, we subtract 180 degrees (half a circle) from the given angle. Therefore, 203 - 180 = 23 degrees becomes the reference angle in the first quadrant.
In summary, the reference angles for each given angle in the first quadrant are:
330 degrees --> 330 degrees
120 degrees --> 120 degrees
113 degrees --> 113 degrees
203 degrees --> 23 degrees
Now that we have the angles in the first quadrant, we can use trigonometric functions (sine and cosine) to find their values.
I ask myself, "how far from the x-axis is my given angle"
After I decide which quadrants the given angle is in, I go
in II: 180 - angle
in III: angle - 180
in IV: 360 - angle
e.g. for 330° , which is in IV
reference angle = 360-330 = 30°
...
for 203, which is in III
reference angle = 203-180 = 23°
do the others the same way