Without using a calculator, compute the sine and cosine of 210° by using the reference angle. What is the reference angle? degrees. In what quadrant is this angle? (answer 1, 2, 3, or 4) sin(210°) = cos(210) (Type sqrt(2) for 2 and sqrt(3) for V3.)
The Correct Answer and Explanation is :
To compute the sine and cosine of 210° using the reference angle, we first need to determine the reference angle and the quadrant in which 210° lies.
Step 1: Find the reference angle.
- A reference angle is the acute angle formed by the terminal side of the given angle and the x-axis.
- 210° is in the third quadrant, so we subtract 180° from it to find the reference angle:
[
\text{Reference angle} = 210° – 180° = 30°
]
Thus, the reference angle is 30°.
Step 2: Determine the quadrant.
- Since 210° is between 180° and 270°, it lies in the third quadrant.
Step 3: Find sine and cosine values.
In the third quadrant:
- Sine and cosine are both negative.
- The sine and cosine values for 30° (the reference angle) are:
[
\sin(30°) = \frac{1}{2}, \quad \cos(30°) = \frac{\sqrt{3}}{2}
]
Since 210° is in the third quadrant, we make both values negative: - (\sin(210°) = -\frac{1}{2})
- (\cos(210°) = -\frac{\sqrt{3}}{2})
Final Answer:
- Reference angle: 30°
- Quadrant: 3
- (\sin(210°) = -\frac{1}{2})
- (\cos(210°) = -\frac{\sqrt{3}}{2})
Explanation:
In trigonometry, the reference angle helps us find the sine and cosine of any angle. For angles in the third quadrant, both sine and cosine are negative. By using the reference angle (30°) and knowing the standard values of sine and cosine for this angle, we can adjust the signs based on the quadrant. The third quadrant has negative sine and cosine values, so we apply this to the reference angle values to get the correct values for 210°.