Ratios & Identities: Sign of Trigonometric Ratios

Trigonometric ratios are fundamental in understanding the relationships between the angles and sides of a triangle, particularly right-angled triangles. These ratios also extend to the unit circle, which allows us to define them for any angle, including those greater than 90° (or π/2 radians). The sign of these trigonometric ratios is crucial as it tells us about the angle's position relative to the coordinate axes.

Understanding the Trigonometric Ratios

The primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). There are also three reciprocal ratios: cosecant (csc), secant (sec), and cotangent (cot). These ratios are defined as follows for a right-angled triangle:

• $\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$
• $\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$
• $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$
• $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite side}}$
• $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent side}}$
• $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent side}}{\text{opposite side}}$

Sign of Trigonometric Ratios in Different Quadrants

The sign of trigonometric ratios depends on the quadrant in which the terminal side of the angle lies. The four quadrants of the coordinate plane are as follows:

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

The signs of the trigonometric functions in each quadrant can be summarized in the following table:

Quadrant	sin(θ)	cos(θ)	tan(θ)	csc(θ)	sec(θ)	cot(θ)
I	+	+	+	+	+	+
II	+	-	-	+	-	-
III	-	-	+	-	-	+
IV	-	+	-	-	+	-

A mnemonic to remember this is "All Students Take Calculus," where:

• All (All functions are positive in Quadrant I)
• Students (Sine is positive in Quadrant II)
• Take (Tangent is positive in Quadrant III)
• Calculus (Cosine is positive in Quadrant IV)

Examples

Example 1: Determining the Sign of Trigonometric Ratios

Determine the sign of $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$ if $\theta$ is an angle in the third quadrant.

Solution:

Using the table above, we can determine that:

• $\sin(\theta)$ is negative in the third quadrant.
• $\cos(\theta)$ is also negative in the third quadrant.
• $\tan(\theta)$ is positive in the third quadrant.

Example 2: Using the Unit Circle

Find the sign of $\cos(\theta)$ when $\theta = 150^\circ$.

Solution:

150° corresponds to the second quadrant on the unit circle. Referring to the table:

• $\cos(\theta)$ is negative in the second quadrant.

Therefore, $\cos(150^\circ)$ is negative.

Example 3: Calculating Exact Values

Calculate $\sin(240^\circ)$ and determine its sign.

Solution:

240° is in the third quadrant, where the sine function is negative. To find the exact value, we can use the reference angle, which is $240^\circ - 180^\circ = 60^\circ$.

Since $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, and sine is negative in the third quadrant, we have:

• $\sin(240^\circ) = -\frac{\sqrt{3}}{2}$

Conclusion

Understanding the sign of trigonometric ratios is essential for solving problems in trigonometry. It helps us determine the correct value of a trigonometric function based on the angle's quadrant. Remembering the signs associated with each quadrant and using reference angles can simplify the process of finding exact values of trigonometric functions for any given angle.