What does the ASTC rule stand for?

ASTC stands for All Silver Tea Cups (or Add Sugar To Coffee). It is a mnemonic to remember which trigonometric functions are positive in each quadrant: All (Quadrant I), Sine (Quadrant II), Tangent (Quadrant III), and Cosine (Quadrant IV).

Why is tangent positive in Quadrant III?

In Quadrant III, both the x-coordinate (cosine) and y-coordinate (sine) are negative. Since tangent is the ratio of sine to cosine (y/x), dividing a negative number by a negative number results in a positive value. Thus, tangent (and cotangent) is positive in Quadrant III.

What are the signs of reciprocal functions like secant or cosecant?

Reciprocal trigonometric functions share the exact same sign as their primary counterparts. Secant (1/cos) has the same sign as cosine, cosecant (1/sin) has the same sign as sine, and cotangent (1/tan) has the same sign as tangent.

Trigonometric Signs & Quadrants (ASTC Rule) Visualizer

Active: Quadrant I (0° to 90°)

Trig Sign Table

Ratio	Q I	Q II	Q III	Q IV
sin (y)	+	+	-	-
cos (x)	+	-	-	+
tan (y/x)	+	-	+	-
csc (1/y)	+	+	-	-
sec (1/x)	+	-	-	+
cot (x/y)	+	-	+	-

Current Values (θ = 45°)

sin+ 0.707

cos+ 0.707

tan+ 1.0

csc+ 1.414

sec+ 1.414

cot+ 1.0

Variable Adjuster

Angle in Degrees (θ)45°

0360

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Sweep Speed: 2x

Trigonometric Signs (ASTC)

ASTC

The algebraic sign (+ or -) of a trigonometric function is determined by the quadrant of the angle. The ASTC rule is a handy mnemonic: All positive in Q I, Sine in Q II, Tangent in Q III, Cosine in Q IV.

\text{All} \rightarrow \text{Silver} \rightarrow \text{Tea} \rightarrow \text{Cups}

Whiteboard Solver Steps

Step 1

Determine Quadrant & Reference Angle

An angle starting from the positive x-axis sweeps counterclockwise. - Quadrant I: $0^\circ < \theta < 90^\circ$ - Quadrant II: $90^\circ < \theta < 180^\circ$ - Quadrant III: $180^\circ < \theta < 270^\circ$ - Quadrant IV: $270^\circ < \theta < 360^\circ$ . Here, $45^\circ$ lands in Quadrant I (0° to 90°).

\theta = 45^\circ \implies \text{Quadrant } I

Step 2

Apply the ASTC (All Silver Tea Cups) Mnemonic

To quickly identify which functions are positive in each quadrant, remember: - All (Q I): All functions are positive (+). - Silver (Q II): Only Sine and its reciprocal Cosecant are positive (+). - Tea (Q III): Only Tangent and its reciprocal Cotangent are positive (+). - Cups (Q IV): Only Cosine and its reciprocal Secant are positive (+).

\text{Active Mnemonic: } \mathbf{A} \implies \mathbf{All (+)}

Step 3

Analyze coordinate signs (x, y)

Recall that coordinates on the Unit Circle are $(x, y) = (\cos\theta, \sin\theta)$ . - In Q I: $x > 0, y > 0$ (Both Positive) - In Q II: $x < 0, y > 0$ (Cosine negative, Sine positive) - In Q III: $x < 0, y < 0$ (Both negative) - In Q IV: $x > 0, y < 0$ (Cosine positive, Sine negative)

x = \cos\theta \approx 0.7071, \quad y = \sin\theta \approx 0.7071

Step 4

Compute Signs for Reciprocal & Quotient Functions

The signs of the remaining functions follow simple rules of division: - Tangent ( $y/x$ ) and Cotangent ( $x/y$ ) are positive where $x$ and $y$ share the same sign (Q I and Q III). - Secant ( $1/x$ ) shares the exact sign of Cosine ( $x$ ). - Cosecant ( $1/y$ ) shares the exact sign of Sine ( $y$ ).

\begin{aligned}\tan\theta = \frac{y}{x} = 1.0000 \\ \csc\theta = \frac{1}{y} = 1.4142 \\ \sec\theta = \frac{1}{x} = 1.4142 \\ \cot\theta = \frac{x}{y} = 1.0000\end{aligned}

RV Anveshana