Understanding Sin θ and Cos θ in the First Quadrant

Let's explore the values of $\sin \theta$ and $\cos \theta$ for angles ranging from $0^\circ$ to $90^\circ$. This range of angles, $0^\circ \le \theta \le 90^\circ$, corresponds to the first quadrant in the coordinate plane when we consider trigonometric functions in terms of the unit circle.

The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system. For any angle $\theta$, we can find a point on the unit circle where the terminal side of the angle intersects the circle. If this point has coordinates $(x, y)$, then:

• $\cos \theta = x$ (the x-coordinate of the point)
• $\sin \theta = y$ (the y-coordinate of the point)

Signs of Coordinates in the First Quadrant

The first quadrant is defined by the region where both the x-coordinate and the y-coordinate are positive. This region is for angles $\theta$ such that $0^\circ \le \theta \le 90^\circ$.

• For any point $(x, y)$ in the first quadrant, $x > 0$ and $y > 0$.

Values of Sin θ and Cos θ for 0° ≤ θ ≤ 90°

Since $\cos \theta = x$ and $\sin \theta = y$, and both $x$ and $y$ are positive for angles in the first quadrant ($0^\circ \le \theta \le 90^\circ$), it follows that:

• $\cos \theta$ is positive.
• $\sin \theta$ is positive.

Let's look at the boundary values:

• When $\theta = 0^\circ$, the point on the unit circle is (1, 0). So, $\cos 0^\circ = 1$ (positive) and $\sin 0^\circ = 0$ (non-negative, considered positive in this context of signs).
• When $\theta = 90^\circ$, the point on the unit circle is (0, 1). So, $\cos 90^\circ = 0$ (non-negative) and $\sin 90^\circ = 1$ (positive).

For all angles between $0^\circ$ and $90^\circ$ (exclusive), both $\sin \theta$ and $\cos \theta$ are strictly positive. Including the boundary values, they are always positive or zero, but typically when discussing the sign in a quadrant, we refer to the strict sign within the open interval $(0^\circ, 90^\circ)$. The options provided consider zero as positive in this context.

Therefore, for $0^\circ \le \theta \le 90^\circ$, the value of $\sin \theta$ and $\cos \theta$ is always positive.