Sign of Trigonometric Functions

Concept First: The Unit Circle and Coordinates

The sign of a trigonometric function for a given angle $x$ is determined entirely by the quadrant in which the terminal side of the angle lies. To see why, we return to the unit circle definition.

Let $P(a, b)$ be a point on the unit circle (radius 1) centred at the origin $O$, such that the angle $\angle AOP = x$ (measured from the positive $x$-axis). By definition, $\cos x = a$ and $\sin x = b$. Since $P$ lies on the unit circle, $a$ and $b$ are the $x$- and $y$-coordinates of $P$, respectively.

The key observation is that $a$ and $b$ can be positive, negative, or zero, depending on which quadrant $P$ falls in. The sign of $\cos x$ is simply the sign of $a$, and the sign of $\sin x$ is the sign of $b$. From there, the signs of the other four trigonometric functions follow directly from their definitions in terms of $\sin x$ and $\cos x$.

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Sign of $\cos(-x)$ and $\sin(-x)$

Consider a point $P(a, b)$ on the unit circle corresponding to angle $x$. Now consider the angle $-x$, which is the same magnitude as $x$ but measured clockwise from the positive $x$-axis. The point $Q$ corresponding to angle $-x$ will be the reflection of $P$ across the $x$-axis. Therefore, if $P$ has coordinates $(a, b)$, then $Q$ has coordinates $(a, -b)$.

From this geometric fact, we immediately get two fundamental identities:

• $\cos(-x) = a = \cos x$
• $\sin(-x) = -b = -\sin x$

These are the even-odd identities for cosine and sine: cosine is an even function, and sine is an odd function.

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Bounds of Sine and Cosine

Since every point $P(a, b)$ on the unit circle satisfies $-1 \leq a \leq 1$ and $-1 \leq b \leq 1$, we have the fundamental bounds:

These bounds are crucial: sine and cosine never exceed 1 in absolute value.

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Sign of Trigonometric Functions in Each Quadrant

We now determine the sign of $\sin x$ and $\cos x$ in each quadrant by examining the signs of $a$ and $b$.

First Quadrant ($0 < x < \frac{\pi}{2}$): Both $a$ and $b$ are positive. Therefore:

• $\sin x > 0$
• $\cos x > 0$

Second Quadrant ($\frac{\pi}{2} < x < \pi$): $a$ is negative, $b$ is positive. Therefore:

• $\sin x > 0$
• $\cos x < 0$

Third Quadrant ($\pi < x < \frac{3\pi}{2}$): Both $a$ and $b$ are negative. Therefore:

• $\sin x < 0$
• $\cos x < 0$

Fourth Quadrant ($\frac{3\pi}{2} < x < 2\pi$): $a$ is positive, $b$ is negative. Therefore:

• $\sin x < 0$
• $\cos x > 0$

From these, we can summarise:

• $\sin x$ is positive for $0 < x < \pi$ (first and second quadrants) and negative for $\pi < x < 2\pi$ (third and fourth quadrants).
• $\cos x$ is positive for $0 < x < \frac{\pi}{2}$ (first quadrant) and for $\frac{3\pi}{2} < x < 2\pi$ (fourth quadrant), and negative for $\frac{\pi}{2} < x < \frac{3\pi}{2}$ (second and third quadrants).

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Signs of the Other Trigonometric Functions

The signs of $\tan x$, $\cot x$, $\sec x$, and $\csc x$ follow from their definitions:

The sign of $\tan x$ is the sign of the ratio $\frac{\sin x}{\cos x}$. So:

• In Quadrant I: $\frac{+}{+} = +$
• In Quadrant II: $\frac{+}{-} = -$
• In Quadrant III: $\frac{-}{-} = +$
• In Quadrant IV: $\frac{-}{+} = -$

Similarly, $\cot x$ has the same sign as $\tan x$ (since it is its reciprocal). $\sec x$ has the same sign as $\cos x$ (reciprocal), and $\csc x$ has the same sign as $\sin x$ (reciprocal).

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Complete Sign Table

The following table summarises the signs of all six trigonometric functions in the four quadrants. …