Give three different angles, in degrees, where the cos of the angle is 0
The Correct Answer and Explanation is:
The three different angles (in degrees) where the cosine of the angle is 0 are:
90°, 270°, and 450°.
Explanation:
The cosine function, written as cos(θ), is one of the primary trigonometric functions. It represents the x-coordinate of a point on the unit circle that corresponds to the angle θ, measured from the positive x-axis. On the unit circle, cosine is 0 whenever the terminal side of the angle lies on the y-axis, because at those points the x-coordinate is zero.
Let’s examine this more closely using the unit circle, which has a radius of 1 and is centered at the origin (0, 0):
- At 90°, the point on the unit circle is (0, 1). The x-value is 0, so cos(90°) = 0.
- At 270°, the point is (0, -1). Again, the x-value is 0, so cos(270°) = 0.
- At 450°, which is one full revolution (360°) plus 90°, the point is again (0, 1), just like at 90°, so cos(450°) = 0.
This pattern continues every 180° after 90°, i.e., at 90°, 270°, 450°, 630°, etc. These are called co-terminal angles because they end at the same position on the unit circle.
Mathematically, the cosine of an angle θ is zero when:cos(θ)=0 whenever θ=90°+180°×nwhere n is an integer\cos(θ) = 0 \text{ whenever } θ = 90° + 180° \times n \quad \text{where } n \text{ is an integer}cos(θ)=0 whenever θ=90°+180°×nwhere n is an integer
Thus, the cosine function is zero at regular intervals, and you can find an infinite number of angles where this happens. However, in this problem, we only need three, so 90°, 270°, and 450° are sufficient.
