Question

The sine of angle θ is 0.3.

What is cos(θ)? Explain how you know.

Answer 1

To find the value of cos(θ) when the sine of θ is 0.3, we can use the relationship between sine and cosine in a right triangle. The value of cos(θ) is approximately 0.954.

Step-by-step explanation:

To find the value of cos(θ) when the sine of θ is 0.3, we can use the relationship between sine and cosine in a right triangle. The sine function is defined as the ratio of the opposite side to the hypotenuse, which means sin(θ) = Ay/A. From this, we can rewrite the equation as Ay = 0.3A. Now, using the Pythagorean theorem (A^2 = Ax^2 + Ay^2), we can substitute Ay with 0.3A and solve for Ax: Ax^2 = A^2 - (0.3A)^2 = 0.91A^2. Therefore, Ax = √(0.91A^2). Finally, the relationship between cosine and the adjacent side is given by cos(θ) = Ax/A, which can be simplified to cos(θ) = √0.91. So, the value of cos(θ) is approximately 0.954.

Answer 2

cos(θ) = -0.95

Step-by-step explanation:

Remember the relation:

sin(θ)^2 + cos(θ)^2 = 1

So if we have:

sin(θ) = 0.3

we can replace that in the above equation to get:

0.3^2 + cos(θ)^2 = 1

now we can solve this for cos(θ)

cos(θ)^2 = 1 - 0.3^2 = 0.91

cos(θ) = ±√0.91

cos(θ) = ± 0.95

Now, yo can see that there are two solutions, which one is the correct one?

Well, you can see that the endpoint of the segment that defines θ is on the second quadrant.

cos(x) is negative if the endpoint of the segment that defines the angle is on the second or third quadrant.

Then we can conclude that in this case, the correct solution is the negative one.

cos(θ) = -0.95