• A Level Maths

Demo

• A Level
• Trigonometry (Introduction)

1. Express the following in terms of (positive) acute angles:

1. $\sin240^{\circ}$
2. $\cos(-50^{\circ})$
3. $\tan600^{\circ}$

1. $-\sin120^{\circ}$
2. $\cos50^{\circ}$
3. $\tan60^{\circ}$

2. Write the following in terms of $\sin\theta$, $\cos\theta$, or $\tan\theta$ only:

1. $\sin(180^{\circ}-\theta)$
2. $\cos(\theta-540^{\circ})$
3. $\tan(540^{\circ}-\theta)$

1. $\sin\theta$
2. $-\cos\theta$
3. $-\tan\theta$

3. Find the exact values of:

1. $\cos495^{\circ}$
2. $\tan(-225^{\circ})$
3. $\sin(-60^{\circ})$

1. $-\dfrac{\sqrt{2}}{2}$
2. $-1$
3. $-\dfrac{\sqrt{3}}{2}$

4. Simplify:

1. $1-\cos^2\frac{1}{2}\theta$
2. $5\sin^23\theta+5\cos^23\theta$

1. $\sin^2\frac{1}{2}\theta$
2. $5$

5. Simplify:

1. $(1+\sin x)^2+(1-\sin x)^2+2\cos^2x$
2. $\sin^4\theta+\sin^2\theta\cos^2\theta$

1. $4$
2. $\sin^2\theta$

6. Prove the following identities:

1. $(\sin\theta+\cos\theta)^2\equiv2\sin\theta\cos\theta + 1$
2. $\dfrac{1}{\cos\theta}-\cos\theta\equiv\sin\theta\tan\theta$

See video

7. Prove the following identities:

1. $(2\sin\theta-\cos\theta)^2+(\sin\theta+2\cos\theta)^2\equiv5$
2. $\sin^2x\cos^2y-\cos^2x\sin^2y\equiv\sin^2x-\sin^2y$

See video

8. Given $\tan\theta=\dfrac{5}{12}$ and $\theta$ is acute, find the exact value of:

1. $\sin\theta$
2. $\cos\theta$

1. $\dfrac{5}{13}$
2. $\dfrac{12}{13}$

9. Given $\cos\theta=-\dfrac{3}{5}$ and $\theta$ is obtuse, find the exact value of:

1. $\sin\theta$
2. $\tan\theta$

1. $\dfrac{4}{5}$
2. $-\dfrac{4}{3}$

10. Given $\sin\theta=-\dfrac{7}{25}$ and $270^{\circ}<\theta<360^{\circ}$, find the exact value of:

1. $\cos\theta$
2. $\tan\theta$

1. $\dfrac{24}{25}$
2. $-\dfrac{7}{24}$

11. For the following pair of equations, find a single equation in terms of $x$ and $y$, but not $\theta$: $$x=\sin\theta \quad y=\cos^2\theta$$

$x^2+y=1$

12. For the following pair of equations, find a single equation in terms of $x$ and $y$, but not $\theta$: $$x=\sin\theta+\cos\theta \quad y=\cos\theta-\sin\theta$$

$x^2+y^2=2$

13. Find the angle marked $x$ in the following triangle.

$79.9^{\circ}$

14. By first finding $x$, find the area of the following triangle.

$63.2$

15. By first finding $x$, find the area of the following triangle.

$258$

16. Find the reflex angle marked $x$ in the following triangle.

$302^{\circ}$

17. Calculate the area of a triangle with side lengths $10$, $13$, and $18$.

$63.5$

18. Find the area of the quadrilateral $ABCD$.

$25.6$