Signs of Functions — Hard Practice Quiz

A Trigonometry cheat sheet for Signs of Functions — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

All trigonometric functions are positive

$$\text{Quadrant I: All positive}$$

Only sine and cosecant are positive

$$\text{Quadrant II: } \sin \alpha, \csc \alpha \text{ positive}$$

Only tangent and cotangent are positive

$$\text{Quadrant III: } \tan \alpha, \cot \alpha \text{ positive}$$

Only cosine and secant are positive

$$\text{Quadrant IV: } \cos \alpha, \sec \alpha \text{ positive}$$

Practice quiz

  1. If an angle $\theta$ satisfies both $\sin \theta < 0$ and $\cos \theta > 0$, in which quadrant must $\theta$ lie?

    • Quadrant I
    • Quadrant II
    • Quadrant III
    • Quadrant IV

    Answer: Quadrant IV

  2. Given an angle $\alpha$ such that $\tan \alpha > 0$ and $\csc \alpha < 0$, identify the quadrant in which $\alpha$ must terminate.

    • Quadrant I
    • Quadrant II
    • Quadrant III
    • Quadrant IV

    Answer: Quadrant III

  3. An angle $\beta$ is known to have $\sec \beta < 0$ and $\cot \beta > 0$. In which quadrant is $\beta$ located?

    • Quadrant I
    • Quadrant II
    • Quadrant III
    • Quadrant IV

    Answer: Quadrant III

  4. If an angle $\phi$ satisfies the condition $\sin \phi \cdot \cos \phi < 0$, which of the following quadrants are possible locations for $\phi$?

    • Quadrant I and III
    • Quadrant II and IV
    • Quadrant I and II
    • Quadrant III and IV

    Answer: Quadrant II and IV

  5. Given that angle $\alpha$ is in Quadrant II and angle $\beta$ is in Quadrant IV, determine the sign of the expression $\tan \alpha \cdot \cos \beta$.

    • Positive
    • Negative
    • Zero
    • Cannot be determined

    Answer: Negative

  6. An angle $\theta$ satisfies the conditions $\frac{\sin \theta}{\cos \theta} > 0$ and $\cos \theta < 0$. In which quadrant does $\theta$ lie?

    • Quadrant I
    • Quadrant II
    • Quadrant III
    • Quadrant IV

    Answer: Quadrant III

  7. For an angle $\alpha$, if $\sin \alpha \cdot \tan \alpha > 0$, which of the following quadrants are possible for $\alpha$?

    • Quadrant I and II
    • Quadrant II and III
    • Quadrant III and IV
    • Quadrant I and IV

    Answer: Quadrant I and IV

  8. If an angle $\theta$ has both $\csc \theta < 0$ and $\sec \theta < 0$, in which quadrant must $\theta$ be located?

    • Quadrant I
    • Quadrant II
    • Quadrant III
    • Quadrant IV

    Answer: Quadrant III

  9. Given an angle $\alpha$ for which $\sin \alpha < 0$ and $\tan \alpha < 0$, what is the sign of $\cos \alpha$?

    • Positive
    • Negative
    • Zero
    • Cannot be determined

    Answer: Positive

  10. For an angle $\gamma$, if $\cos \gamma \cdot \cot \gamma > 0$, which of the following quadrants are possible for $\gamma$?

    • Quadrant I and II
    • Quadrant II and III
    • Quadrant III and IV
    • Quadrant I and IV

    Answer: Quadrant I and II

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