Compute the sine and cosine of 330∘ by using the reference angle.
a.) What is the reference angle? degrees.
b.)In what quadrant is this angle? (answer 1, 2, 3, or 4)
c.) sin(330∘)=
d.) cos(330∘)=
The correct answer and explanation is :
Let’s go through the step-by-step solution to compute the sine and cosine of ( 330^\circ ) using the reference angle.
Step 1: Find the Reference Angle
The reference angle is the acute angle between the given angle and the x-axis. It is found by subtracting the given angle from ( 360^\circ ) if the angle is in Quadrant IV.
[
\text{Reference Angle} = 360^\circ – 330^\circ = 30^\circ
]
Answer for part (a): The reference angle is 30°.
Step 2: Determine the Quadrant
The given angle is ( 330^\circ ). To determine the quadrant, we check where it falls:
- Quadrant I: ( 0^\circ ) to ( 90^\circ )
- Quadrant II: ( 90^\circ ) to ( 180^\circ )
- Quadrant III: ( 180^\circ ) to ( 270^\circ )
- Quadrant IV: ( 270^\circ ) to ( 360^\circ )
Since ( 330^\circ ) is between ( 270^\circ ) and ( 360^\circ ), it is in Quadrant IV.
Answer for part (b): The angle is in Quadrant 4.
Step 3: Find the Sine and Cosine Values
Since the reference angle is ( 30^\circ ), we use the known trigonometric values:
[
\sin 30^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}
]
- In Quadrant IV, sine is negative and cosine is positive.
- Therefore:
[
\sin 330^\circ = -\sin 30^\circ = -\frac{1}{2}
]
[
\cos 330^\circ = \cos 30^\circ = \frac{\sqrt{3}}{2}
]
Final Answers
(c) ( \sin(330^\circ) = -\frac{1}{2} )
(d) ( \cos(330^\circ) = \frac{\sqrt{3}}{2} )
Explanation
The reference angle helps us find trigonometric values by reducing the given angle to an equivalent acute angle between ( 0^\circ ) and ( 90^\circ ). Since ( 330^\circ ) is in Quadrant IV, sine is negative and cosine is positive. By using the known values of sine and cosine for ( 30^\circ ), we can determine the exact values for ( 330^\circ ).
Now, let me generate an image illustrating this concept.
Here is an illustration of the unit circle showing the angle ( 330^\circ ), its reference angle ( 30^\circ ), and the corresponding cosine and sine values. This visual helps to understand why (\sin 330^\circ = -\frac{1}{2}) and (\cos 330^\circ = \frac{\sqrt{3}}{2}).