Trigonometry – Unit Circle, Identities, Compound Angles, Solving Equations Quiz - ICSE Class 11 Mathematics

Free practice quiz: Trigonometry – Unit Circle, Identities, Compound Angles, Solving Equations (ICSE Class 11 Mathematics). 40 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 11 Mathematics practice quiz has 40 multiple-choice questions on Trigonometry – Unit Circle, Identities, Compound Angles, Solving Equations, and 40 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Sample questions

1. If $\sin(\theta) = \frac{3}{5}$ and $\theta$ is in the second quadrant, find the value of $\cos(\theta)$.

   1. $\frac{4}{5}$
   2. $-\frac{4}{5}$
   3. $-\frac{3}{5}$
   4. $\frac{3}{5}$
   Show answer

   Answer: B. $-\frac{4}{5}$

   Given $\sin(\theta) = \frac{3}{5}$. Since $\theta$ is in the second quadrant, $\cos(\theta)$ will be negative. Using the identity $\sin^2(\theta) + \cos^2(\theta) = 1$, we have $(\frac{3}{5})^2 + \cos^2(\theta) = 1$. This gives $\frac{9}{25} + \cos^2(\theta) = 1$, so $\cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25}$. Therefore, $\cos(\theta) = \pm\sqrt{\frac{16}{25}} = \pm\frac{4}{5}$. Since $\theta$ is in the second quadrant, $\cos(\theta) = -\frac{4}{5}$.

2. If $\sin(\theta) = \frac{1}{2}$ and $\theta$ is in the second quadrant, find the value of $\cos(\theta)$.

   1. $\frac{\sqrt{3}}{2}$
   2. $-\frac{\sqrt{3}}{2}$
   3. $\frac{1}{2}$
   4. $-\frac{1}{2}$
   Show answer

   Answer: B. $-\frac{\sqrt{3}}{2}$

   Given $\sin(\theta) = \frac{1}{2}$. Since $\theta$ is in the second quadrant, $\cos(\theta)$ will be negative. Using the identity $\sin^2(\theta) + \cos^2(\theta) = 1$, we have $(\frac{1}{2})^2 + \cos^2(\theta) = 1$. This gives $\frac{1}{4} + \cos^2(\theta) = 1$, so $\cos^2(\theta) = 1 - \frac{1}{4} = \frac{3}{4}$. Therefore, $\cos(\theta) = \pm\sqrt{\frac{3}{4}} = \pm\frac{\sqrt{3}}{2}$. Since $\theta$ is in the second quadrant, $\cos(\theta) = -\frac{\sqrt{3}}{2}$.

3. If $\tan(\theta) = \frac{3}{4}$ and $\theta$ is in the third quadrant, find the value of $\sin(\theta)$.

   1. $-\frac{3}{5}$
   2. $\frac{3}{5}$
   3. $-\frac{4}{5}$
   4. $\frac{4}{5}$
   Show answer

   Answer: A. $-\frac{3}{5}$

   Given $\tan(\theta) = \frac{3}{4}$. Since $\theta$ is in the third quadrant, both $\sin(\theta)$ and $\cos(\theta)$ are negative. We can construct a right-angled triangle where the opposite side is 3 and the adjacent side is 4. The hypotenuse is $\sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5$. Therefore, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. As $\theta$ is in the third quadrant, $\sin(\theta)$ is negative. So, $\sin(\theta) = -\frac{3}{5}$.

4. If $\cos(\theta) = \frac{4}{5}$ and $\theta$ is in the fourth quadrant, find the value of $\sin(\theta)$.

   1. $-\frac{3}{5}$
   2. $\frac{3}{5}$
   3. $-\frac{4}{5}$
   4. $\frac{4}{5}$
   Show answer

   Answer: A. $-\frac{3}{5}$

   Given $\cos(\theta) = \frac{4}{5}$. Since $\theta$ is in the fourth quadrant, $\sin(\theta)$ will be negative. Using the identity $\sin^2(\theta) + \cos^2(\theta) = 1$, we have $\sin^2(\theta) + (\frac{4}{5})^2 = 1$. This gives $\sin^2(\theta) + \frac{16}{25} = 1$, so $\sin^2(\theta) = 1 - \frac{16}{25} = \frac{9}{25}$. Therefore, $\sin(\theta) = \pm\sqrt{\frac{9}{25}} = \pm\frac{3}{5}$. Since $\theta$ is in the fourth quadrant, $\sin(\theta) = -\frac{3}{5}$.

5. If $\sin(\theta) = -\frac{3}{5}$ and $\theta$ is in the third quadrant, find the value of $\cos(\theta)$.

   1. $\frac{4}{5}$
   2. $-\frac{4}{5}$
   3. $\frac{3}{5}$
   4. $-\frac{3}{5}$
   Show answer

   Answer: B. $-\frac{4}{5}$

   Given $\sin(\theta) = -\frac{3}{5}$. Since $\theta$ is in the third quadrant, $\cos(\theta)$ will be negative. Using the identity $\sin^2(\theta) + \cos^2(\theta) = 1$, we have $(-\frac{3}{5})^2 + \cos^2(\theta) = 1$. This gives $\frac{9}{25} + \cos^2(\theta) = 1$, so $\cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25}$. Therefore, $\cos(\theta) = \pm\sqrt{\frac{16}{25}} = \pm\frac{4}{5}$. Since $\theta$ is in the third quadrant, $\cos(\theta) = -\frac{4}{5}$.

More questions in this quiz

Answers and explanations are in the full quiz.

1. Solve for $x$ in the equation $2\sin(x) = 1$ in the interval $[0, 2\pi]$.
2. If $\tan(\theta) = 2$, and $\theta$ is in the third quadrant, find the value of $\sin(\theta)$.
3. Solve for $x$ in the equation $\cos(x) = -1$ in the interval $[0, 2\pi]$.
4. If $\sin(\theta) = \frac{1}{3}$ and $\theta$ is in the first quadrant, find the value of $\tan(\theta)$.
5. Solve for $x$ in the equation $\sin(x) = -\frac{\sqrt{3}}{2}$ in the interval $[0, 2\pi]$.
6. If $\cos(\theta) = -\frac{1}{4}$ and $\theta$ is in the second quadrant, find the value of $\sin(\theta)$.
7. Solve for $x$ in the equation $\tan(x) = -1$ in the interval $[0, 2\pi]$.
8. Solve for $x$ in the equation $\sin(x) = \frac{\sqrt{3}}{2}$ in the interval $[0, 2\pi]$.
9. If $\tan(\theta) = -\frac{1}{2}$ and $\theta$ is in the fourth quadrant, find the value of $\cos(\theta)$.
10. Solve for $x$ in the equation $\cos(x) = \frac{1}{2}$ in the interval $[0, 2\pi]$.
11. If $\sin(\theta) = -\frac{4}{5}$ and $\theta$ is in the third quadrant, find the value of $\tan(\theta)$.
12. Solve for $x$ in the equation $\tan(x) = \sqrt{3}$ in the interval $[0, 2\pi]$.
13. If $\cos(\theta) = \frac{1}{3}$ and $\theta$ is in the fourth quadrant, find the value of $\sin(\theta)$.
14. Solve for $x$ in the equation $\cos(x) = -\frac{\sqrt{3}}{2}$ in the interval $[0, 2\pi]$.
15. Solve for $x$ in the equation $\sin(x) = -1$ in the interval $[0, 2\pi]$.
16. If $\cos(\theta) = -\frac{1}{2}$ and $\theta$ is in the third quadrant, find the value of $\tan(\theta)$.
17. Solve for $x$ in the equation $\cos(x) = 0$ in the interval $[0, 2\pi]$.
18. If $\sin(\theta) = \frac{3}{5}$ and $\theta$ is in the first quadrant, find the value of $\tan(\theta)$.
19. Solve for $x$ in the equation $\tan(x) = 0$ in the interval $[0, 2\pi]$.
20. If $\sin(\theta) = -\frac{1}{2}$ and $\theta$ is in the fourth quadrant, find the value of $\cos(\theta)$.
21. Solve for $x$ in the equation $\sin(x) = 1$ in the interval $[0, 2\pi]$.
22. Solve for $x$ in the equation $\cos(x) = -1$ in the interval $[-\pi, \pi]$.
23. If $\tan(\theta) = -2$ and $\theta$ is in the second quadrant, find the value of $\sin(\theta)$.
24. Solve for $x$ in the equation $\cos(x) = 1$ in the interval $[0, 2\pi]$.
25. If $\sin(\theta) = \frac{1}{3}$ and $\theta$ is in the second quadrant, find the value of $\cos(\theta)$.
26. Solve for $x$ in the equation $\tan(x) = -\sqrt{3}$ in the interval $[0, 2\pi]$.
27. If $\cos(\theta) = -\frac{3}{5}$ and $\theta$ is in the second quadrant, find the value of $\tan(\theta)$.
28. Solve for $x$ in the equation $\sin(x) = \frac{1}{2}$ in the interval $[-\pi, \pi]$.
29. Solve for $x$ in the equation $\cos(x) = -\frac{1}{2}$ in the interval $[0, 2\pi]$.
30. If $\tan(\theta) = -\frac{3}{4}$ and $\theta$ is in the fourth quadrant, find the value of $\sin(\theta)$.
31. Solve for $x$ in the equation $\sin(x) = -\frac{1}{2}$ in the interval $[0, 2\pi]$.
32. If $\cos(\theta) = -\frac{1}{3}$ and $\theta$ is in the third quadrant, find the value of $\sin(\theta)$.
33. Solve for $x$ in the equation $\tan(x) = 1$ in the interval $[0, 2\pi]$.
34. If $\tan(\theta) = \frac{1}{2}$ and $\theta$ is in the third quadrant, find the value of $\cos(\theta)$.
35. Solve for $x$ in the equation $\cos(x) = 0$ in the interval $[-\pi, \pi]$.

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