Given that sin 𝑦 = −2√3

Question

Given that sin 𝑦 = −2√3

𝑎𝑛𝑑 tan 𝑦 = √3 , 𝑓𝑖𝑛𝑑 𝑦
A. 60°
B. 120°
C. 210°
D. 240°

Answer 1

We know that sin 𝑦 = −2√3 and tan 𝑦 = √3

Using the fact that sin^2 𝑦 + cos^2 𝑦 = 1, we can find cos 𝑦:

sin^2 𝑦 + cos^2 𝑦 = 1
(-2√3)^2 + cos^2 𝑦 = 1
12 + cos^2 𝑦 = 1
cos^2 𝑦 = -11 (this is not possible, since the square of a real number cannot be negative)

Therefore, there is no real value of 𝑦 that satisfies sin 𝑦 = −2√3 and tan 𝑦 = √3. This may be a mistake in the problem statement.

Answer 2

Yes, there is a typo.

siny = -√3/2
tany = √3
gives y in QIII, so y = 240°

Answer 3

Ah, that makes more sense! Thank you for the correction.

Given that sin 𝑦 = -√3/2 and tan 𝑦 = √3, we know that y is in the third quadrant, where sine is negative and tangent is positive.

Using the unit circle, we can see that the only angle in the third quadrant with sine equal to -√3/2 is y = 240°. Therefore, the answer is D. 240°.

Answer 4

To find the value of 𝑦, we can use the information given about sin 𝑦 and tan 𝑦.

First, let's find the cosine of 𝑦 using the Pythagorean identity:
cos^2 𝑦 = 1 - sin^2 𝑦

Since sin 𝑦 = -2√3, we have:
cos^2 𝑦 = 1 - (-2√3)^2
= 1 - 4(3)
= 1 - 12
= -11

Since cosine is always between -1 and 1, this result is not possible. Therefore, there is no real value of 𝑦 that satisfies the given conditions.

Answer 5

To find the value of 𝑦, we can use the given information about sin 𝑦 and tan 𝑦. Let's break down the process step by step:

Step 1: Use the given value of sin 𝑦 = −2√3. Since sin 𝑦 is negative, 𝑦 must be in either the third or fourth quadrant of the unit circle.

Step 2: Since tan 𝑦 = √3, we can use the fact that tan 𝑦 = sin 𝑦 / cos 𝑦. Substituting the given value of sin 𝑦 = −2√3, we get -2√3 = (√3)/cos 𝑦. Simplifying further, we get -2 = 1/cos 𝑦.

Step 3: To solve for cos 𝑦, we can take the reciprocal of both sides of the equation. Doing so, we get cos 𝑦 = -1/2.

Step 4: Now that we have the values of sin 𝑦 and cos 𝑦, we can determine the quadrant in which 𝑦 lies. Since sin 𝑦 is negative (-2√3) and cos 𝑦 is negative (-1/2), 𝑦 must lie in the third quadrant.

Step 5: To find the angle 𝑦 in degrees, we can use the inverse trigonometric function. Since 𝑦 is in the third quadrant, we can use the inverse sine function (sin⁻¹) to find 𝑦. In this case, sin⁻¹ (-2√3) will give us the angle in radians.

Step 6: Convert the angle from radians to degrees. To convert from radians to degrees, multiply the angle in radians by (180/π) or approximately 57.3.

Step 7: Calculate the value of 𝑦 in degrees. We have already determined in step 5 that sin⁻¹ (-2√3) gives us the angle in radians. Now, multiply this angle by (180/π) to convert it to degrees.

By following these steps, we can find the value of 𝑦 in degrees and determine the correct option from the given choices.