audio read-through Signs of All the Trigonometric Functions

Sometimes in problem-solving you only need the sign (plus or minus) of a trigonometric value—not its size.

With three already-studied concepts, you have access to the signs of all the trigonometric functions, in all the quadrants:

$$\frac{\text{POS}}{\text{NEG}} = \text{NEG}$$

(positive divided by negative is negative)

$$\frac{\text{NEG}}{\text{POS}} = \text{NEG}$$

(negative divided by positive is negative)

$$\frac{\text{NEG}}{\text{NEG}} = \text{POS}$$

(negative divided by negative is positive)

These concepts are reviewed in-a-nutshell here, for your convenience. Need more info? Follow the links given above.

Signs (Plus/Minus) of Sine and Cosine in All Quadrants

By definition, $\,\cos \theta\,$ and $\,\sin\theta\,$ give the $\,x\,$ and $\,y\,$ values (respectively) of points on the unit circle, as shown below.

unit circle definitions of sine and cosine

It follows that:

Recall also:

Signs (Plus/Minus) of Tangent in All Quadrants

signs of tangent in all quadrants

By definition, $\displaystyle\,\tan\theta :=\frac{\sin\theta}{\cos\theta}\,.$ Therefore:

In quadrant I, tangent is positive: $$\cssId{s33}{\frac{POS}{POS} = POS}$$

In quadrant II, tangent is negative: $$\cssId{s35}{\frac{POS}{NEG} = NEG}$$

In quadrant III, tangent is positive: $$\cssId{s37}{\frac{NEG}{NEG} = POS}$$

In quadrant IV, tangent is negative: $$\cssId{s39}{\frac{NEG}{POS} = NEG}$$

Memory Device for the Signs (Plus/Minus) of Sine, Cosine, and Tangent in All Quadrants

memory device for signs of sine, cosine, tangent in all quadrants

The (optimistic!) memory device

All Students Take Calculus

answers the question:

Where are the (first three) trigonometric functions positive?

Of course, you need to remember to start with the word ‘ALL’ in quadrant I, and then proceed counterclockwise.

This memory device answers the question:

Where are the sine, cosine, and tangent POSITIVE?

Signs (Plus/Minus) of Cotangent, Secant, and Cosecant in All Quadrants

signs of cotangent, secant, and cosecant in all quadrants

The reciprocal of a number retains the sign of the original number. Therefore, in all the quadrants:

Concept Practice