Signs of Functions — Practice Quiz

A Trigonometry cheat sheet for Signs of Functions — every key formula with its symbols defined — plus a medium-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. In which quadrant are all trigonometric functions positive?

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

    Answer: Quadrant I

  2. If an angle $\alpha$ is in Quadrant II, which trigonometric functions are positive?

    • All functions
    • Only $\sin \alpha$ and $\csc \alpha$
    • Only $\tan \alpha$ and $\cot \alpha$
    • Only $\cos \alpha$ and $\sec \alpha$

    Answer: Only $\sin \alpha$ and $\csc \alpha$

  3. For an angle $\theta$ in Quadrant III, which of the following statements is true?

    • $\sin \theta > 0$
    • $\cos \theta > 0$
    • $\tan \theta > 0$
    • $\sec \theta > 0$

    Answer: $\tan \theta > 0$

  4. Which trigonometric functions are positive in Quadrant IV?

    • $\sin \alpha$ and $\csc \alpha$
    • $\tan \alpha$ and $\cot \alpha$
    • $\cos \alpha$ and $\sec \alpha$
    • All functions

    Answer: $\cos \alpha$ and $\sec \alpha$

  5. An angle of $210^{\circ}$ lies in which quadrant, and which functions are positive there?

    • Quadrant II; $\sin$ and $\csc$
    • Quadrant III; $\tan$ and $\cot$
    • Quadrant IV; $\cos$ and $\sec$
    • Quadrant I; All functions

    Answer: Quadrant III; $\tan$ and $\cot$

  6. If $\theta = \frac{5\pi}{3}$, which of the following trigonometric functions is negative?

    • $\cos \theta$
    • $\sec \theta$
    • $\sin \theta$
    • None of the above

    Answer: $\sin \theta$

  7. If $\sin \theta > 0$ and $\cos \theta < 0$, in which quadrant does $\theta$ lie?

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

    Answer: Quadrant II

  8. In which quadrant is $\tan \alpha < 0$ and $\sec \alpha > 0$?

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

    Answer: Quadrant IV

  9. Given that $\sin \beta = 0.8$ and $\tan \beta < 0$, in which quadrant is $\beta$?

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

    Answer: Quadrant II

  10. If $\cot \phi > 0$ and $\csc \phi < 0$, in which quadrant does $\phi$ lie?

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

    Answer: Quadrant III

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